John Dee and Pedro Nunes


Pedro Nunes (1502 - 1578)

Mathematics, Cosmography and Nautical Science in the 16th century.


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John Dee

(1527 – 1608/9)


John Dee and Pedro Nunes established a intriguing scientific connection. This is a point rarely mentioned in international historiography. Although it is almost certain that Nunes and Dee never met in person, there are strong evidences that confirm a strong intellectual link between the two.  


         John Dee (1527 – 1608/9) 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 of contacts, extending from the Court to the university and to other institutions that promoted the sharing of knowledge about the sciences.

It is not clear when Dee’s interest for Nunes’ work first came about. It can be argued that 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 astronomy — 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.

Later, by 1550/1, John Dee was in Paris where he met, among others, Oronce Finé. It is very likely that, while in Paris, Dee became aware of Nunes' book on the Frenchman's mathematical errors. The reading of this book may well have influenced Dee's interest on the unsolved problems of squaring of the circle and doubling the cube and further work on Euclid's Elements.

Dee owned copies of all of Nunes’ books, with one exception: the Tratado da sphera. The reason for his failure to acquire this book, aside from its linguistic relevance, may relate to a recent clue found at Cambridge University Library. By 1552, after coming back to England, Dee studied in detail a work that had been published in Paris in 1549, attacking some of Nunes’ ideas: Diogo de Sá's De nauigatione [4]. The book is more interesting in what it attacks than in what it proposes. Among other things, the author analysed Nunes’ first treaty step by step according to his own agenda, and launched 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.

However, it may be argued that de Sá’s attack also backfired, since the second and third books included fine translations of 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 demonstrated that he knew 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 reveals his interest in the hierarchy of the sciences and in the mathematical principles of nautical science. When Sá begins his translation of Nunes’ work, Dee 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. Even if embedded within a philosophical discussion, the correct translation and adaptation of Nunes’ treatises seems to have enabled Dee to use these works as a textbook on the topic.

In a 1558 letter to Mercator, Dee implied that he had been corresponding with the Portuguese mathematician for some time. The letter was published in Dee’s first work, Propaedeumata Aphoristica[6]. In this dedicatory letter he recalled the good times both him and 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, industry, and habit, to cultivate diligently the arts most necessary to a Christian state”[8].

In general, researchers have passed over more profound implications of Dee’s declaration.  But this connection has not been missed by some important historians of science: Eva G.R. Taylor and David Waters have addressed the link between the two men in their important work on the history of navigation[9]. Even so, they limit their studies to the influence of Pedro Nunes on Dee’s work on nautical issues.

Unfortunately, the nature of any correspondence going on between Nunes and Dee is unknown, but it is very unlikely that a discussed mathematical topic at the time – the construction of rhumb tables – was left out of these discussions. On the contrary, John Dee’s Canon Gubernauticus or an Arithmeticall Resolution of the Paradoxall Compas [10], was finished sometime during the period that Dee often referred Nunes as the “most learned and grave man”. According to Eva Taylor: “It was, in fact, a practical development on the teaching of Pedro Nuñez on this subject, and its invention belongs to a period when Dee in known to have been in personal touch with the great Portuguese”[11].

 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. These tables were in fact aids to what he called "paradoxall navigation"[12]. Prof. E.G.R. Taylor defined the “paradoxall Compas” as a zenithal equidistant projection chart and the “paradoxal” lines were in fact rhumb lines, for the first time imagined by Pedro Nunes in 1537. John Dee's Canon Gubernauticus calculates the latitude and longitude for the 7 classical rhumbs spiralling across the globe from a point at the equator to a point at 80º latitude[13].

Although pursuing this matter in manuscripts and private discussions in the 40's and 50's, Nunes would only present publicly his full mathematical theory of the rhumb line in his book printed in 1566 (Petri Nonii Salaciencis Opera), which Dee also had in his library. This may also suggest that a sharing of scientific knowledge was going on between both men at the time that Nunes was almost finishing the development of his theory.

Certainly gaining from these contacts, John Dee would in time express his vision of the art of navigation in the 1570’s Euclide’s Preface

“The Arte of Navigation, demonstrated how, by the shortest good way, by the aptest Direction, and in the shortest time, a sufficient Ship, between any two places (in passage Navigable) assigned: may be conducted: and in all storms, and natural disturbances chancing, how, to use the best possible means, whereby to recover the place first assigned. What need, the Master Pilot, hath of other Arts, here before recited, it is easy to know: as, of Hydrography, Astronomy, Astrology, and Horometry. Presupposing continually, the common Base, and foundation of all: namely Arithmetic and Geometry. So that, he be able to understand, and Judge his own necessary Instruments, and furniture Necessary: Whether they be perfectly made or no: and also can, (if need be) make them, himself. (…) And also, be able to Calculate the Planets places for all times.(…)

Sufficiently, for my present purpose, it doth appear, by the premises, how Mathematical, the Art of Navigation, is: and how it needed and also used other Mathematical Arts.”

In Dee’s opinion, the “modern” sea pilot should base his everyday art on scientific knowledge provided by mathematical disciplines, which is in consensus with Nunes’ own vision of the ratio nauigandi. As he had stated in 1537 about its mathematical principles: “(…) And because no rule that is based on speculative or theoretical knowledge can be well practiced and understood if one doesn’t know this same principles" since “nothing is most evident than mathematical demonstration, which, by no means, is possible to be contested”. This would be 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”[14].

         In fact, the ideas that both men sheared would echo throughout Europe and were included in many sixteenth and seventeenth centuries’ navigation textbooks[15]. In England (and why not to say in Europe), John Dee was one of the first to consider navigation as a scientific discipline, which reinforces his important role in what Waters and Taylor called "the English awakening" in maritime affairs. 


Not much has been said about Nunes’ influence on Dee’s mathematical views, in general, and the use of mathematics to study nature. It is hard to believe that Dee considered Nunes “that most learned and grave man who is the sole relic and ornament and prop of the mathematical arts among us” solely because he had learned nautical science with him.

In fact, there is evidence suggesting that this influence was wider. In a letter from the 1590's, Dee claims to have finished a work in 1560, on the areas of plane triangles, in three books (De Triangulorum rectilineorum Areis -- libri -- 3) [16] dedicated to Pedro Nunes and, as late as 1584, Dee and Jacob Kurtz were examining instruments containing improvements (the nonius scale) proposed by Pedro Nunes : “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.” [17].

Again in the Euclide’s Preface, Dee tried to justify and promote the translation of this classic book to vernacular. He then gave several examples of similar translation projects going on in Europe, which included an example for the Iberian Peninsula. He does a very curious remark by suggesting that Nunes taught themes of his Portuguese treatises at the University: "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."[18] 

We know (as Dee also knew) that this was not true since classes at the university were delivered in Latin, but this echoes the fact that Dee was aware of Nunes 1537 works in Portuguese, and very likely, that he was also conscious of Nunes translation project and agreed with the ideas behind it[19]. At the time, this defence of vernaculars as a scientific language was common all across Europe and Nunes was also involved on it. In his 1537 book he states: "Science has no language so, by using any language, it is possible to explain it (...) And therefore, if from one language to other, one can translate any non scientific text, I do not know where, so much fear to put a science text in common language, comes from."[20]


         Pedro Nunes developed his “program for the mathematization of the real world”[21], a precise, sustained and quite explicit program of promotion of the mathematical sciences, starting in the mid 30's until 1566. This was first achieved by applying mathematics to the art of navigation but throughout all his work, that covers subjects spanning from Algebra to Geometry, Nautical Science, Astronomy, Cartography, instruments and even Physics, Nunes extended the “program” with one objective in mind, that is, to show that Mathematics was the main tool to describe the real world. This would be one of the distinctive characteristics of the "scientific revolution" that would have its peak in the seventeenth century.

Naturally, John Dee developed his own scientific program which, in a first stage, was a program based on the use of mathematical principles as tools to describe the natural world, something that Dee would start to promote within his role as a consultant on nautical subjects and go on doing along his life. It is Shumaker and 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"[22]. Even agreeing with these words in what concerns Dee's contribution to pure mathematics, one has to stress that he was perfectly aware of many scientific developments going on in his time and that he influenced and worked side by side with some of the most influential mathematical practitioners in Elizabethan England. His mathematical program would ultimately be published in the Euclide's Preface, which Yates considered, in a broad sense, “the manifesto of Dee’s movement”[23].

full article here: Science Direct.

                      [1] See Chapter XV. Cosmographia Petri Apiani, per Gemmam Frisium apud Louanienses medicum & mathematicu[m] insignem, iam demum ab omnibus vindicata mendis, ac nonnullis quoq[ue] locis aucta, additis eiusdem argumenti libellis ipsius Gemmae Frisii (Antuerpiae: sub scuto Basiliensi, Gregorio Bontio, 1545), Chapter XV, fl. 23v-25v.

[2] An unknown detractor wrote a treatise (nowadays lost) attacking Nunes’ 1537 treatises. Nunes answered back with a manuscript defence. This document is presently kept at the Biblioteca Nazionale di Firenze (Codice palatino nº825). There is modern transcription and study about this manuscript: Joaquim de Carvalho, “Uma obra desconhecida e inédita de Pedro Nunes: [defensão do tratado de rumação do globo para a arte de navegar]”, in: Inedita ac rediuiua, subsídios para a história da filosofia e da ciência em Portugal, (Coimbra, 1952), Separata de Revista da Universidade de Coimbra, n.º 17, (Coimbra: Universidade de Coimbra, 1953), pp. 521-631.

[3] The first known polemics date to the late 1530’s. By 1538, a Portuguese named Manuel Lindo wrote a treatise claiming to have discovered new processes to obtain latitude by measuring extra meridian heights of the sun, something that Nunes had developed at least since 1533. Again he would be attacked by Fernando Oliveira on his work Ars nautica (circa 1570).

[4] Diogo de Sá, De navigatione libri tres quibus mathematicae disciplinae explicantur ab Iacobo a Saa Equite Lusitano nuper in lucem editi (Parisiis: officina Reginaldi Calderii et Claudii eius filii, 1549). Dee’s copy of this book is today at Cambridge University Library, shelfmark: CUL, R*.5.27 (F). See note [B 154] of Julian Roberts and Andrew G. Watson, John Dee's Library Catalogue (London: The Bibliographical Society, 1990).

[5] This discussion was much alive in sixteenth century Europe and became known as quaestio de certitudine mathematicarum. About the quaestio in Portugal and the role of Diogo de Sá's book in this discussion, see: Bernardo Machado Mota, O estatuto da matemática em Portugal nos séculos XVI e XVII, PhD thesis (Universidade de Lisboa, 2008).

[6] John Dee, Propaedeumata aphoristica Ioannis Dee, Londinensis, de praestantioribus quibusdam naturae virtutibus. This book has two known editions: (London: Henry Sutton, 1558) and (London: Reginald Wolfe, 1568).

[7] John Dee on Astronomy. Propaedeumata Aphoristica (I558 and 1568), edited and translated with general notes by Shumaker, Wayne, with introductory essay by Heilbron, John L., (Berkeley: University of California Press, 1978), «Preface», p. ix.

[8] «(...) 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". John Dee on Astronomy. Propaedeumata Aphoristica (I558 and 1568), pp. 114-115.

         It is also possible to find this letter in: M. Van Durme, Correspondance Mercatorienne (Anvers: De Nederlandsche Boekhandel, 1959), pp. 36-39.

         Excerpts from this letter, in Portuguese, can be accessed in: A. Fontoura  da Costa, A Marinharia dos Descobrimentos, p. 233, and: Ana Maria S. Tarrio, «Do humanista Pedro Nunes», Oceanos, 49 (2002) 96-108. For a full portuguese translation see: Fernando B. S. Rua, «Relações entre John Dee e Pedro Nunes: a carta de Dee a Mercator de 20 de Julho de 1558", Clio. Revista do Centro de Historia da Universidade de Lisboa, 10 (2004) 81-109.

[9] See: E. G. R. Taylor, Tudor Geography, 1485-1583 (New York: Octagon Books, 1968); «Canon gubernauticus. An Arithmeticall Resolution of the Paradoxall Compas», in: William Bourne, A Regiment for the Sea and other writings on navigation, E. G. R. Taylor (ed.) (Cambridge: At the University Press, 1963), pp. 419-433; The Haven–Finding Art: a History of Navigation From Odysseus to Captain Cook,  (London: Hollis & Carter, 1971).

         David Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times, (London: Hollis and Carter, 1958).

[10] The manuscript of Dee’s Canon Gubernauticus can be found at the Bodleian Library, Oxford (Ashmolean 242, No. 43). This manuscript has no date and has lots of corrected values. Dee made a reference to the Canon again in The British Complement, of the perfect Art of Navagation; A great volume: in which, are contained our Queen Elizabeth her Arithmeticall Tables Gubernautick for Navagation by the Paradoxall compasse (of me, invented anno 1557) and Navagation by great Circles: and for longitudes, and latitudes; and the variation of the compasse finding most easily, and speedily; yea (if need be) in one minute of time, and sometime, without sight of Sun, Moon or Star; with many other new and needful inventions Gubernautick. Anno 1576. This volume was not printed and his lost.

[11] E. G. R. Taylor, Tudor Geography, p. 95.

[12] About "paradoxall navigation", see: John Davis, Seaman's Secret. Devided into 2. partes, wherein is taught the three kindes of Sayling, Horizontall, Paradoxall, and sayling upon a great Circle: also an Horizontall Tyde Table for the easie finding of the ebbing and flowing of the Tydes, with a Regiment newly calculated for the finding of the Declination of the Sunne, and many other most necessary rules and Instruments, not heeretofore set foorth by any (Londres, Thomas Dawson, 1595). The definition provided suggests only a superficial study from Davis. He certainly owed much of this publication to the documents taken from Dee’s library.  

[13] According to David Waters (The Art of Navigation in England in Elizabethan and Early Stuart Times, p. 372-373), Dee had probably discussed these tables and the solution of the nautical triangle with Thomas Hariot, who developed a solution of Mercator's projection, independently from Edward Wright.

         As far as I know, this connection has not been much studied by the historian community or the fact that, Canon Gubernauticus and Wright's rhumb tables are mostly the same, has been stressed.

[14] Pedro Nunes, Obras, Vol. IV, (Lisboa: Fundação Calouste Gulbenkian, 2008), p. 30.

[15] It is possible to trace Nunes' ideas and influence in works by some of the most important figures of European nautical science of the sixteenth and seventeenth centuries. As examples of printed vernacular books: in Spain, Andrés García de Céspedes, Regimiento de Navegacion (Madrid: Juan de la Cuesta, 1606); in the low countries, Michel Coignet, Instruction nouvelle des poincts plus excellens & necessaires touchant l’art de naviguer (Anvers, 1581); in France, Georges Fournier, Hydrographie contenant la théorie et la practique de toutes les parties de la navigation / composé par le Pere Georges Fournier (Paris: Michel Soly, 1643); in England, Edward Wright, Certaine Errors in Navigation, Arising either of the ordinarie erroneous making or using of the sea Chart, Compasse, Crosse staffe, and Tables of the Sunne, and fixed Starres detected and corrected (Londres: Valentine Sims, 1599).

[16] The last known reference comes in a letter from 1592 (printed in 1599). Dee does here a short reference to a work De Triangulorum rectilineorum Areis -- libri -- 3 -- demonstrati: ad excellentissimum Mathematicum Petrum Nonium conscripti -- Anno -- 1560. In a recent paper presented Stephen Johnson claimed a correspondence between this work and a lost work by Dee, Tyrocinium Mathematicum, also dedicated to Pedro Nunes. In: Stephen Johnson, «John Dee’s Tyrocinium Mathematicum: new evidence for a lost text», John Dee Quatercentenary Conference, 21-22 September 2009.

[17] Jakob Kurtz also discussed improvements to the nonius scale with Cristopher Clavius. About Clavius’ contacts with Kurtz see: Clavius, Corrispondenza, vol. II, part 2, (Pisa: Universite di Pisa; Dipartimento di Matematica, 1992), p. 64, no. 1.

[18] Edward Fenton, The Diaries of John Dee (Oxfordshire: Day Books, 1998), p. 165.

[19] 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 also announced that he was working on a translation of Vitruvius’ De architectura

[20] Pedro Nunes, Obras, vol. I, (Lisboa: Fundação Calouste Gulbenkian, 2002), p. 5.

[21] “Nunes program” is much discussed in: Henrique Leitão, «Ars e ratio: a náutica e a constituição da ciência moderna», in: Maria Isabel Vicente Maroto, Mariano Esteban Piñeiro (coords.), La ciencia y el mar, (Valladolid: Los autores, 2006) 183-207.

[22] John Dee on Astronomy. Propaedeumata Aphoristica (I558 and 1568), p.17.

[23] Frances A. Yates, The Occult Philosophy in the Elizabethan Age (London: Routledge and Kegan Paul, 1979), p. 94.



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