navarro mathematics xvi spain

The Teaching of the Mathematical Disciplines

in Sixteenth-Century Spain*

VICTOR NAVARRO-BROTONSInstituto de Historia de la Ciencia y Documentacion Lopez Pinero, Universidad de Valencia-

CSIC, Valencia, Spain (E-mail: [email protected])

Abstract. This essay examines some aspects of the teaching of mathematics and its applica-

tions in three of the principal sixteenth century Spanish universities (Salamanca, Valencia andAlcala) and in other institutions sponsored by the monarchy, such as the Casa de la Con-tratacion (House of Trade) of Seville and the so-called Academy of Mathematics of Madrid.

All three of the above universities had chairs of mathematics. In the Casa de la Contratacionand other nautical schools the teaching of mathematics was oriented toward providing thefoundations of navigation (nautical astronomy, instruments and maps, etc.). The Academy of

Mathematics was oriented mainly towards subjects related to cosmography and navigation.Although the dierent areas of the teaching of mathematics imposed conditions on the dis-courses and practices of the mathematical disciplines, they did not impede the circulation ofpersons, knowledge, and practices among these areas.

1. Introduction

The teaching of mathematical disciplines and their applications in sixteenthcentury Spain presents common characteristics with other countries orEuropean regions as those of the Italian Peninsula or the Germanic area,and some peculiarities derived from the character of the Spanish monarchyand from its ambitions and imperial projects.1 This teaching was carriedout in dierent places or institutions, from the imparted in schools or byprivate tutors, oriented to the training of the merchants, or to the prepara-tion for dierent trades or occupations that were requiring mathematicalknowledge, until that one developed in the universities. In this work we donot intend to oer an exhaustive panorama of these activities, but only ofthe ones developed in the universities where this teaching took place in ameaningful form, and in the most outstanding centers encouraged by theSpanish monarchy, particularly the Casa de la Contratacion of Seville andthe Academy of Mathematics of Madrid.

* This work has partially nanced by grants from Spanish Ministry of Science andTechnology (BHA 20001456) and fromMinistry of Education and Science (BHA2003-08394-(02-01).

Science & Education (2006) 15:209233 Springer 2006DOI 10.1007/s11191-005-8860-6

Concerning the universities, there is a considerable contrast between themedieval panorama and that of the sixteenth century. One peculiar aspect ofmedieval Spain, compared to the rest of Europe, was the weakness of theuniversities. The monasteries, the cathedrals, the courts and nobility circlesand the Jewish aljamas, along with a few universities, were the placeswhere scientic, medical and philosophical ideas were developed.2 Poor inrents, low in reputation, many Spanish youths avoided the universities, pre-ferring instead to study abroad. But after the middle of the fteenth centurythe fortunes of higher education in Spain improved decisively. The newstructure and dynamic of the Spanish monarchy and its secular and religiousinstitutions provoked a rapid expansion in the number of positions open tograduates.3 During the sixteenth century, a number of universities were cre-ated in the Spanish peninsular and insular kingdoms, including 18 in thekingdom of Castilla and 12 in Aragon. In addtion, the Spanish monarchyorganized universities in their new American territories very quickly.Some new universities became quite prominent, such as those of Alcala andValencia. Others, such as Salamanca and Valladolid, consolidated andgained greater importance. The Law, the subject with the widest opportuni-ties for advancement and prestige, was also the subject most in demand inSpanish universities. In this sense it has been noted that the most signicantcontribution of the universities in sixteenth century Spain, particularly inCastilla, was the formation of a new social class: the letrados.4 The theol-ogy also attracted many students, above all with a view to the ecclesiasticaloces and careers. Nevertheless, in the rst half of the sixteenth century, thenew humanists trends found wide echo in some Spanish universities, thathad professors as noteworthy as Elio Antonio de Nebrija, Hernan NunezEl Pinciano o Francisco Sanchez El brocense. The University of Alcala,responding to the humanists call for studies of sacred texts in their originallanguages, gave Europe its rst institution the Colegio Trilingue wholly dedicated to the study of classical languages. Likewise, the universi-ties met head-on the intellectual problems posed by the emergence ofimperial Spain. On the other hand, the nominalist trends, both in logic (ter-minist logic) and natural philosophy and also in theology, which had scar-cely caused a ripple in Spain at the end of the Middle Ages, were widespreadin the rst half of the sixteenth century in the universities of Salamanca,Alcala, Valencia, Valladolid and Zaragoza. This, in addition to impelling thestudies of logic and natural philosophy, also contributed to encourage theteaching of mathematics. To all that, one must add the creation of the medi-cal schools in the new universities (Alcala and Valencia, especially) and thenew development of the ones already existing, that also contributed to thecultivation and teaching of mathematics, especially, but not solely, by therelationships of medicine with astrology (and cosmology). And if jurists and

VICTOR NAVARRO-BROTONS210

theologians participated actively in the problems outlined by the Habsburgultramarine expansion, the abilities and knowledge of the teachers of mathe-matics was also required to solve the cosmographical problems as well as toform experts in cosmography in the universities.The Casa de la Contratacion and the Academy of Mathematics were

created especially to answer to the scientic and technical demands im-posed by the geographical explorations, the control and rule of the landsconquered, the maintenance of empires and the construction of the state.The Casa de la Contratacion, in particular, was created to administer allthe problems related to navigation to the New World (la carrera de In-dias), included technical aspects such as the design of instruments, thecreation of navigational charts, the centralization and organization of theinformation and the training and license of pilots. The existing literatureon this institution and the teachings imparted in it is very abundant.5 Inthe case of the Academy of Mathematics, the process of its creation, aswell as the type of teachings imparted and its evolution until the middle ofthe seventeenth century has been reconstructed with documents withdetail.6 But what we know about the particular content of the teachings, aswell as about the results of its activities is still scarce. The documentationat our disposal indicates that the bulk of those teachings was located inthe matters related to cosmography, though the initial project was muchmore ambitious. We intend to oer here a panorama of the teaching of themathematics disciplines in these two institutions and to provide some newdata and considerations of the most relevant aspects.

1.1. THE CASA DE LA CONTRATACION

The Casa de la Contratacion was established in 1503 and in 1508 was cre-ated the post of Pilot Major or Chief Pilot, which was responsible for over-seeing the drafting of the Padron real, or master chart, and for trainingand examining the pilots sailing the Indies Route towards the West Indies.In 1523 the post of cosmographer and master of making charts, astrolabesand other navigation devices was created.Two cosmographerchartmaker posts were created in 1537. Mean-

while, in 1524 Charles I established the Consejo de Indias (Council of theIndies), run at the outset almost exclusively by lawyers until Juan deOvandos reforms in 1571. His reforms led to the creation of the positionof Chroniclercosmographer Major of the Indies. Apart from these posts,other mathematicians were named honorary cosmographers and ociallylicensed to make charts and instruments, without being appointed to aparticular post or paid any salary. One of them was Pedro de Medina.Howewer, the were not the only cosmographers and pilots at the service

of the Spanish crown in relation to the Casa de la Contratacion. There

TEACHING OF MATHEMATICAL DISCIPLINES IN SIXTEENTH CENTURY SPAIN 211

were also a considerable number from Portugal and, to a lesser extent,from other countries.7

Among the functions of the Pilot Major was that of training pilots.The Pilot Major instructed and examined the pilots in his house. In 1528Alonso de Chaves (cir.14931587), appointed Cosmographer and Masterchart-maker obtained a license to teach future pilots. The aspiring pilothad to bring proof (testigos o escrituras) that he had sailed for six ormore years to the New World (Indias) and that he had been in Hispani-ola, Cuba, Tierra Firme and New Spain. He had to bring his own chart,astrolabe and quadrant, to demonstrate that he could use them to calculatelatitude and time; and to answer the questions posed by the Pilot Majorand from the pilots present at the examination8. In 1539 Pedro Medina(14931567), who was authorized to examine pilots, also began to teachcosmography and navigation privately, as did other cosmographers linkedto the House.9

The need to regulate and control the content of the teachings in order toavoid the abuses and mutual accusations of corruption among the cosmog-raphers eventually led to the creation in 1552, of the chair of cosmography,with the appointment of Alonso de Chavess son Jeronimo de Chaves(15231574). According to the royal decree of 1552, the subjects to betaught by the chair were: Introduction to the sphere; procedures andregiments (rules) for calculating latitudes by the altitude of the sun andthe pole; the use of charts; the manufacture and use of instruments, particu-larly the compass or seafarers needle, astrolabe, quadrant and cross-staff; knowledge of how to calculate magnetic declination; the use ofday- and night-time clocks; lunations and rules for forecasting tides.10 Thiswas the state of the organization of pilot training at the Casa which soaroused the admiration of Stephen Borough, the English navigator whovisited it in 1558, that he pressed for similar provision in England,recommending the creation of a pilot-major post.11

The content of the teachings and the exams of the pilots can be followedthrough the texts written by the cosmographers, particularly those of Alonsoof Chaves and Pedro of Medina. Alonso of Chavess unpublished QuatriPartitu en cosmographia practica, i por otro nombre llamado espejo de naveg-antes (Mirror for Seamen) dealt with the following subjects: Book 1 treatsall matters pertaining to days of religious celebrations, and to instrumentsfor navigation. Book 2 deals with all aspects of cosmography (astronomicalphenomena, tables of the movement of the sun and the declination of thesun; rules for nding the latitude by measuring the altitude of the NorthStar; declination of some xed stars, etc.) and the practical art of navigation;Book 3 with the age of the moon and the natural movements of oceanwaters; and book 4 with navigators itineraries to all parts of the Indies.12


Pedro Medina prepared several texts on the instruction of pilots andcosmographers, including the Libro de Cosmographia, the Arte de navegar(1545), the Regimiento de navegacion (1552) and the Suma de cosmographia.13

Medinas most important work is the Arte de Navegar (1545), which wastranslated into French, Dutch, and English and republished 15 times in theselanguages. Another widely diused text was Martn Cortes Breve compendiode la sphera y de la arte de navegar (1551), which was translated into Englishat the recommendation of Borough and printed six times in this language.14

Apparently, Cortes was teaching cosmography and the art of navigation tothe pilots of Cadiz, although his book was published in Seville. Cortess textis superior to Medinas in certain aspects concerning the systematizing ofknowledge and techniques, and is more updated in various aspects, such asin the tables of declination of the sun. We can also see the inuence of PedroNunes (15021578) on Martin Cortess discussion of the la carta demarear (chart of navigation) and in his introduction to a procedure fordetermining the altitude of the pole by means of extrameridian (out of themeridian) heights of the sun, also based on Nunes.15

In these works, the general cosmographic framework was usually an exposi-tion based on the Sphere of Sacrobosco, with the pertinent corrections in topicssuch as the doctrine of the zones, the shape of the earth and the arrangement ofseas and lands on its surface. The geographical discoveries not only disprovedbeliefs about the uninhabitability of the torrid zone, but they were decisive forthe construction of the modern concept of the terrestrial globe.In the second half of the century, two important texts relative to the

education of pilots were elaborated: the Itinerario de navegacion de los maresy tierras occidentales by Juan Escalante de Mendoza (cir. 15451595), aships captain, and the Compendio de la Arte de Navegar (1581) by RodrigoZamorano (d. 1620), Pilot Major and professor of the Casa from 1575 until1613. Zamoranos text was re-edited six times in Spanish until 1591 andonce in Dutch (Amsterdam, 1598). It was also translated into English andpublished as an appendix to an edition of Edward Wrights Certain Errorsin Navigation (1599). In his Compendio Zamorano introduced new tables ofthe Suns declination based on the Ephemerides of Johannes Stadius (whichin turn were based on the Prutenic Tables) and revised the values of theregimiento del Norte.16 Zamorano also published an edition in Spanish ofthe rst six books of Euclids Elements (1576) and a treatise on Chronologie(1585).

1.2. THE ACADEMY OF MATHEMATICS

In 1571 the Council of the Indies, the monarchys supreme advisory boardfor the governance and administration of the New World, underwent aprofound reform initiated by its chairman, Juan de Ovando. An important


result was the creation of the post of Chief (or Major) Cosmographerchronicler of the Indies. Ovando appointed to this position his assistant,Juan Lopez de Velasco (ca. 15301598), who attempted to develop Ovandosprogramme of producing an accurate and thorough geographic descrip-tion of both Spain and the New World, including geographic co-ordinatescalculated by astronomic methods.17 At the same time, the urgent necessityof improving navigational charts and instruments was taken up. That taskinvolved several initiatives, one proposed by Juan de Herrera, architect,engineer and Aposentador (Steward) de Palacio.18 After the annexation ofPortugal, Philip II, from Lisbon, approved Herreras initiative along withanother, also from Herrera: the creation of an Academy of mathematics inMadrid. The text that Herrera drafted concerning the objectives of theproposed Academy noted on the dearth of good mathematicians in the king-doms of the Spanish monarchy, and also stated that the aim of the Academywas to train theoretical and practical arithmeticians, geometricians, astrono-mers, musicians with a theoretical background, cosmographers, pilots,architects, designers of fortications, engineers, machinists, gunners, instru-ment-makers, plumbers, levellers, clock experts, perspective experts andsculptors and painters with practice in perspective. It would also take on thetask of training the sons of noble courtiers in mathematics. For eachmatter or activity Herrera recommended a relevant group of texts, whichdisplay his intimate familiarity with the mathematical disciplines and theirapplications. In addition, Herrera recommended that the classes be taught inSpanish.19

Juan Bautista Lavanha (15551624), a Portuguese cosmographer ofnoble descent who had studied in Rome, was chosen to run the academy,and in December 1582 Lavanha was appointed by a royal patent letter, totake charge of matters relating to cosmography, geography and topogra-phy in our court and elsewhere as ordered, and to read mathematics....20

At the same time, Pedro Ambrosio of Onderiz (?-cir.1596) was appointedLavanhas assistant and made responsible for translating scientic texts.Onderiz, who had studied classical languages and had also a good trainingas a mathematician and cosmographer, having studied two years inPortugal (sent there by Herrera), translated into Spanish Euclids Opticsand the pseudo-Euclidean Catoptrics; books XI and XII of the Elements,Teodosios Sphaerica and Archimedess On the equilibrium of planes.21

Two manuscripts of the classes taught by Lavanha and Onderiz in theacademy have survived. One of them is an incomplete Tratado del arte denavegar (Treatise on the art of navigation) copied in 1588 by CamiloMadea, evidently a pupil from Italy.22 An interesting aspect of Lavanhastext is the attention paid to the contributions and proposals of PedroNunes regarding the Art and Science of Navigation (as was entitled


Nuness work).23 Thus, relating to instruments, Lavanha describes Nunessnautical ring as an alternative to the astrolabe, and for the quadrant andastrolabe, suggests using the technique of tracing scales put forward byNunes. He also describes, following Nunes, the shadow instrument usedto calculate the altitude of the sun on earth. Lavanha also discussescartographic projections and how to plot navigational charts.One manuscript by Onderiz entitled Uso de los Globos leydo en

Madrid el ano 1592 (Use of Globes, read in Madrid in 1592), hassurvived.24 As the title of the manuscript suggests, it covers the use ofcelestial and terrestrial globes to determine the position of the Sun, thealtitude of the pole, the declination of stars, the hour of day or night, thezenithal distances of planets and stars, the distances between places (bymeans of a terrestrial globe), and other uses.In 1591, the Academia was subordinated to the Council of the Indies,

although Herrera continued to be in charge of its activities. The post ofCosmographerchronicler was divided in two: Onderiz was namedCosmographer Major of the Indies and Juan Arias de Loyola wasappointed Chronicler. Lavanha moved to Portugal where he took up theposition of Cosmographer Major in that kingdom. Arias and Onderizlooked after the teaching in the Academia until 1595. In that year, follow-ing the death of Onderiz and the cessation of Arias, Julian Firruno fromMilan was appointed to teach all the lessons. Firruno had taught artilleryin Burgos and Seville. After the death of Herrera, in 1597, and on the ini-tiative of the Count of Punoenrostro, an artillery general and member ofthe Council of War, the lectures and the number of professors of the Acad-emy were expanded, although the new professors did not receive ofcialappointments. Thus, Juan Cedillo Daz taught trigonometry, Juan Angellectured on On Floating Bodies of Archimedes, and the soldiers Rodriguezde Muniz and Cristobal de Rojas taught military tactics and fortications.25

In addition to the Elements of Euclid and cosmography, Firruno alsolectured on artillery, for which he prepared a text titled Descripcion y trata-do muy breve y lo mas provechoso de Artillera...(Description and brief Trea-tise and the greatest advantages of Artillery) which is preserved in amanuscript dated 1599.26 Among those interested in these matters, particu-larly nobles, courtiers and knights, was a Swedish diplomat, BotwildNericius, who had studied at the Collegio Romano and maintained corre-spondence with Clavius. In this correspondence, based principally onmathematical matters, Nericius informed Clavius that members of theAcademia de Madrid were interested in his mathematical works. Moreover,Nericius was engaged in an argument with Guidobaldo dal Monte throughClavius on matters of statics. This argument was initiated, it appears, in theAcademia de Madrid between supporters and critics of Guidobaldo.27


When the court moved to Valladolid, the activities in the chair ofMadrid was discontinued, and it is not known whether they were carriedon in Valladolid. Firruno died in 1604. In 1607, the chair recommencedits activities, again in Madrid. The new holder was the CosmographerMajor of the Indias, Andres Garca de Cespedes (d. 1611), a distinguishedcosmographer, astronomer and engineer who succeeded Juan Cedillo Dazin 1611. Garca de Cespedes wrote extensively on these themes and two ofhis works were published: el Libro de instrumentos de geometria (1606) andthe Regimiento de navegacion (1609). The former includes a magnicenttreatise on hydraulics, the most comprehensive on the subject published inSpain during the Renaissance.28 The latter is a collection or summa of allthe knowledge and techniques developed in Portugal and Spain in relationto the art of navigation, besides the incorporation of the contributions byauthors from other countries. The Regimiento also presented the results ofthe amendment, or programme for revising navigational tables, charts andinstruments initiated by Onderiz and completed by Garca de Cespedes.29

Garcia de Cespedes and his collaborators were the rst Cosmographers ofIberian territories to elaborate tables of the Suns declination from newobservations required to calculated the longitude of the Sun and the obliq-uity of the ecliptic. Until then, all authors of nautical tables had beenbased either on the Alphonsine Tables or the Prutenic Tables, by means ofEphemerides calculated by various authors (Zacut, Stoeer, Stadius,Magini,...). The results of Garca de Cespedes established a notableimprovement in the tables precision with a margin of error of less thantwo minutes for the declination of the sun.30

1.3. THE UNIVERSITIES

The three Spanish universities known to have a separate chair for teachingmathematical disciplines in the sixteenth century were Salamanca, Valenciaand Alcala.The University of Valencia was founded in 14991500. Its doors opened

in 1501 and by the middle of the sixteenth century it became the foremostuniversity in the Crown of Aragon with chairs of theology, canon law,civil law, medicine, mathematics, logic, philosophy natural and moral,grammar and humanities.31 Following the ocial foundation of the EstudiGeneral, a chair of mathematics was established in 1503, although we haveno documentary evidence of the subjects taught there in the early decadesof that century. The rst person to occupy the chair was Tomas Duran(ca.14751545), a Dominican from Salamanca who published (Valencia,1503) Bradwardines Arithmetic and Geometry and also Pechams Perspec-tiva, together with Questiones super perspectivam by Henricus or Heinrich


of Hesse (or of Langenstein). In the early years of the university, theteaching of mathematics must have concentrated basically on preparingpupils to study natural philosophy and logic, mainly under the inuence ofthe University of Paris and nominalistic trends.32 In this regard, it is inter-esting to note that Juan de Celaya, a leading member of the terminist andcalculatory group at Paris, teacher of Domingo de Soto, was fromValencia, where his works had a great inuence.33

From the year 1540 onwards, it was compulsory for students of medicineto have arts degrees, as was already the case for students of law and theol-ogy. This must have increased the interest in the study of astronomy andastrology. According to certain documents, in the years 15401550, thestudy of mathematics included arithmetic, geometry, geometrical optics,music, judicial astrology and cosmography (which included astronomy andgeography).34 The upsurge in humanism led to greater interest in astronomyand astrology amongst humanist physicians, who took them as a basis forthe interpretation of Hippocratic texts. The chair of astronomy was occu-pied in the 15551556 academic year by the noteworthy physician andhumanist Pedro Jaime Esteve (ca.15001556). Esteve, in his commentariesto the second book dedicated to the epidemics of the Hippocratic Collectionpointed out medicus astronomiae ignarus non est sectator hipocratis, andshow extensive knowledge of astronomical and astrological literature.35

As regards mathematical subjects, the Constitutions of 1561 mentiononly the teaching of astronomy, and list the following topics: the sphere,the theorica of the planets, tables and the use of astrolabes. The Statutes of1555, however, had already established the need to teach certain principlesof geometry (to enable Aristotles Analytica Priora and Posteriora to beunderstood) and also arithmetic and geometry (in order to grasp naturalphilosophy) and this practice was probably maintained.The teaching of mathematics and related subjects in the Estudio General

in Valencia reached considerable heights during the period when the chairwas occupied by Jeronimo Munoz (?1592), one of the most outstandingscientists of sixteenth century Spain. Munoz began his studies in Valencia,where he graduated as a Bachelor of Arts (1537), and continued studyingin dierent locations in Europe. His own comments tell us that he was astudent of Oronce Fine and Gemma Frisius. He lived in Italy for sometime and taught Hebrew at the University of Ancona. Following his returnto Valencia, he was appointed to the chair of Hebrew in 1563 and in 1564he combined this chair with that of mathematics, a position he held until1578, the year he moved to the University of Salamanca.36

In Valencia, in his chair of mathematics, Munoz taught arithmetic, geom-etry, trigonometry, geometrical optics, astronomy, geography and astrology.Although Munoz published very few works, holograph documents or copies


made by students of his in all these subjects are still to be found in severallibraries across Europe: Salamanca, Barcelona, Madrid, Munich, theVatican, Naples, and Copenhagen.37

Munoz became very well known in Spain as a mathematician, geogra-pher, Hellenist and Hebraist. His fame elsewhere in Europe was duemainly to his study of the supernova of 1572, discussed in his Libro del nu-evo cometa (Valencia, 1573), which he wrote in response to Philip IIsrequest for his opinion on the phenomenon. This book was translated intoFrench by Guy Lefe`vre de la Boderie, a pupil of Guillaume Postel whocollaborated with the Polyglot Bible of Antwerp as a Hebraist.38 He alsobecame known because of the detailed descriptions of his results and con-clusions that were made by such prominent authors as Cornelius Gemmaand Thaddaeus Hagecius. Munoz also corresponded with Hagecius andBartholomaeus Reisacherus of Vienna, another of the authors who dealtwith the supernova. Hagecius also furnished Tycho Brahe with letters heand Reisacherus had received from Munoz about the supernova, whichBrahe copied and used in his discussion of the works by Munoz in theAstronomiae Instauratae Progyimnasmata (1602).39

Munoz was one of the authors who determined the position of the starwith the most precision. Similarly, he was one of the astronomers who wasmost keenly understood the cosmological implications of the phenomenon,namely how difcult it was to maintain the Aristotelian dogma of theincorruptibility of the heavens and to make it compatible with the appear-ance of the nova.These works by Munoz on the nova must be considered in the frame-

work of an ambitious revisionist programme of Aristotelian cosmologyand Ptolemaic astronomy such as can be seen in his commentaries on thesecond book of the Natural History of Pliny and in his additions and com-mentaries on Theon of Alexandrias Commentaries on Ptolemys Almagest.The commentaries on Pliny are dated 1568, apparently the date whenMunoz presented them at the University of Valencia in some extraordinarylessons given by some of the most outstanding professors. In that workMunoz exposes his cosmological ideas, which were similar to that of theStoics. Munoz rejected the idea of a sphere of re, and considered the cos-mos as a continuum of air, which became more rareed the more distant itwas from the central earth, reaching its outer limits; beyond those limitsextended an immense vacuum. With dierent arguments, Munoz alsorejected the existence of the celestial spheres arguing that the planets movenaturally through cosmic air.40

Munozs most extensive and ambitious work on astronomy was histranslation of and commentaries on Theon of Alexandrias Commentaries onPtolemys Almagest, which he began in Valencia in about 1568 and nished


in Salamanca in 1582, although he continued to add notes and data to hiscomments until at least 1589.41 In this work, in his comments and compre-hensive additions to Theons text, Munoz reviewed many aspects ofPtolemaic astronomy, comparing them with the observations, techniquesand calculations of other classic, medieval and renaissance astronomersincluding Copernicus, whom he often quoted. He also provided his owntables and numerous observations made in Valencia and Salamanca, anddescribed in detail a variety of observational instruments for astronomy andtheir respective advantages. Like its model, Ptolemys Almagest, this work isto a large extent a highly technical treatise on mathematical astronomy andMunoz takes great pains to clarify the most dicult sections for the sake ofbeginners (ad tirones), as he himself points out. However, he also includesbroad-based discussions of cosmological matters, ranging from thoseconcerning the position of the earth in the world, with a discussion of theheliocentric theory, to the nature of comets, and in this respect, he sets forthcertain ideas similar to those in his Commentaries on Pliny.Another noteworthy aspect of Munoz teachings is the attention he

devoted to the applications of astronomy, in particular to geography andcartography and to the art of navigation. In both his introductory treatise ofAstronomy and Geography and his additions to Theons Commentary, hepaid particular attention to how geographic coordinates are established,reviewing all known methods and commenting on the instrumentscommonly used for this purpose. Munoz was an expert geographer; he hadestablished the latitudes of certain locations on the Iberian Peninsula withremarkable precision. He also estimated the longitudes of several places,though less successfully. He began the geodesic triangulation of the territoryof Valencia, employing the method described by Gemma Frisius, and ex-plained this procedure in his classes using real examples. The rst knownmap of the Kingdom of Valencia, included by Abraham Ortelio in hisfamous Theatrum Orbis Terrarum, is based to a considerable extent on thework of Jeronimo Munoz.42

Although Munoz was one of the best paid professors at the University ofValencia, his salary was considerably lower than those paid at universities inCastile. The prestige of the University of Salamanca, and its greater proxim-ity to the seat of royal power, was probably one of the reasons why Munozaccepted the offer made to him by this university and moved there in 1578.The University of Salamanca was, from the fteenth century, an active

center of the cultivation of astronomy. In the same way, from the latefteenth century onwards, it was a centre of study and debate about cosmo-graphic matters due to a considerable extent to the inuence of scientichumanism led by Elio Antonio de Nebrija and also the increased importanceof cosmography attained in the enterprises of sea-faring expansion, the


control of the empire and the construction of the State. Nebrija, a teacher ofgrammar, and Nunez de la Yerba, a teacher of medicine, published in Sal-amanca works on cosmography, disseminating in Spain the Geography ofPtolemy and incorporating information on the geographical discoveries. InSalamanca, since 1504, there had also been an extraordinary chair teachingPliny.43

The teaching of cosmography already formed part of the statutes of 1538,which included an earlier project launched by the humanist Fernan Perez deOliva, and in the 1594 statutes, the pertinent regulations were made evenmore specic.44 Morover, from 1508 onwards the nominalist approach wasadopted in the teaching of logic, philosophy and theology at the Universityof Salamanca.45 Professors teaching according to the nominalist approach,included Pedro Margalho, Juan Martnez Silceo and Pedro de Espinosa,noteworthy authors in the realms of both logic and natural philosophy.46

All three also published works on mathematical disciplines. Thus PedroMargalho, a portuguese educated in Paris, published a Phisices compendium(1520) including a Sphaera treatise based on Geminus (via pseudo-Proclus),Aratus, Cleomedes and Ptolemy; a Treatise on proportione; a PhisicesEpitome, dealing with Aristotelian physics; a de intensione et remisioneformarum treatise expounding the doctrines of the calculatores andnominalist of Oxford and Paris. In his Sphaera treatise, Margalho com-pared the medieval doctrines about the distribution of land and water withthe experience of Portuguese seafarers and maintained that these twoelements constituted one and the same surface.47 This is one of the severalinstances that show how some scholastic philosophers followed the pro-gress made in cosmography and geographical knowledge closely.Juan Martnez Siliceo was also educated in Paris, where he published a

treatise on arithmetic entitled Liber Arithmetice practice Astrologis, Phisiciset Calculatoribus admodum utilis...(1513), that was reprinted in Paris (threetimes), Salamanca and Valencia. He also published an edition of RichardSwinesheads Liber Calculationum. Pedro de Espinosa, one of Siliceospupils, penned a Tratactus proportionum (1545), a commentary onSacroboscos Sphaera (1550) and a Philosophia naturalis (1535).48

The statutes of 1538 for the teaching of mathematics established thatthe professor of mathematics will read arithmetic, geometry, astrology,perspective, and cosmography as the students might request. In 1561 newstatutes were drafted, and Copernicus was included among the authorswhose works could be used in astronomy instruction whenever the studentsso voted.The introduction of Copernicuss name in the constitutions was princi-

pally due to the activities of the Aguilera brothers, Juan and Hernando deAguilera.49 Juan de Aguilera, a professor of astrology from 1550 to 1560


and a physician resided in Italy from 1540 to 1550, where he served bothpope Paul III and Julius III. Hernando de Aguilera succeeded his brotherin the chair of astrology. This fact, the incorporation of Copernicuss workas a text that could be used if students voted for it, as an alternative toPtolemy or one of his commentators (Geber or Regiomontanus) has givenrise to the question of whether the work of Copernicus, including his helio-centric theory, was in fact taught. The books in which the visits of therector to the chair are recorded show that Aguilera taught EuclidsElements (books I to VI), the sphere, parts of the Almagest, theories (mod-els) of planets and the Tables of Alfonso X, the astrolabe and its use,cosmography according to Petrus Apianus and Gemma Frisius, and astrol-ogy according to Alcabitius. The name of Copernicus does not appear inthese records.50 Nevertheless, I believe that the possibility cannot be ruledout that on certain occasions Hernando de Aguilera followed CopernicusDe revolutionibus or at least commented on aspects of the work whenexpounding subjects related to the sphere or planetary theories. The infor-mation contained in the books of visits is, in any case, incomplete. Inacademic years 1562/1563 and 1563/1564 for example, there is no referenceto the chair of astrology. The most usual interpretation, in keeping withthe most widespread attitude in Europe towards the work of Copernicus,is that the interest of the Aguilera brothers in De revolutionibus was moreconcerned with the models, data and tables than with the cosmologicalideas it contained. To date however, documentary evidence has notenabled us to answer this question nor to conrm this hypothesis.Two years after Aguileras death, Munoz was appointed to occupy the

chair. The subject matter taught by Munoz in Salamanca was very similarto what he had taught in Valencia. According to the rectors books ofvisitations to chairs, the subjects taught by Munoz in Salamanca between1579 and 1587 included: arithmetic, geometry and perspective according toEuclid, the sphere or introduction to astronomy, models or theories ofplanets, tables and instruments, and astrology, geography, cartography andthe art of navigation.51 Thanks to additional testimonies, we know that healso taught some aspects of the military art (artillery in particular).52 Giventhe similarity in the subjects to be taught in Salamanca, he probably usedthe same texts as in Valencia, and his work on Theon Commentary. Con-cerning the Copernican theory, in his annotations to Theon, Munozdescribes the theory and attempts to refute it with astronomical arguments,inspired by Ptolemy and Theon. Likewise, in his Commentaries on Pliny,and in his introductory treatise to Astronomy and Geography, which heused in teaching in Valencia and also probably in Salamanca, Munozdiscusses and rejects Copernicus theory with astronomical arguments andoccasionally with a mechanical one (the vertical trajectory of falling bodies).


As we have said before, the need to train skilled cosmographers wasone of the main reasons that led Philip II to establish the MathematicsAcademy of Madrid in about 1582. It was with a similar desire to train cos-mographers in about 1590, and in response to the wishes of Philip II tointensify the teaching of mathematics-related subjects, that the teaching ofthese subjects was enlarged in the University of Salamanca where a division(partido) or appointment of an associate teacher of mathematics was setup. In the letter from Philip II granting the economic assistance requestedby the University, the king emphasized how necessary it was to trainwell-educated and skilful professionals able both to teach the subject atuniversity and to be available at seaports or elsewhere, for they are verynecessary and navigation depends upon them.53 The partido or divisionwas assigned to one of Munoz pupils, Gabriel Serrano, until 1592. Thatyear, after the death of Munoz, Serrano himself came into possession of thechair and the physician Antonio Nunez Zamora also a follower ofMunoz, who was later to replace Serrano in the chair was appointed tothe division.54

In 1594, new statutes were drafted for the university. The statutesstipulated that the subjects taught in mathematics should be arithmetic,geometry, Theodosius of Bithynia on spherical triangles, plane and spheri-cal trigonometry (according to Clavius or another modern author),perspective and land surveying. The texts or tables recommended forastronomy were those by Ptolemy, Peurbach, the Tables by Regiomont-anus (of the rst motive) or Reinhold, Clavius, the Alfonsine Tables andthe work of Copernicus, and it was stipulated that Ptolemys Almagestshould be studied in combination with Copernicus De revolutionibus,always beginning with the former. This meant that the choice betweenPtolemy and Copernicus was no longer subject to the vote of the students.Furthermore, astrology continued to be taught and included the study ofthe comets, and it was also compulsory to study Ptolemys Geography,Apianus Cosmografa, cartography, the use of the dierent types of astro-labe, the Radius Astronomicus (or Jacob Sta), the art of navigation andthe military art (artillery and fortication).55 These statutes of the chair ofmathematics must have been drafted by Munoz students, for they werestrictly in line with his teaching. To judge from the visitation records,neither Serrano nor Nunez expounded Copernicus De revolutionibusclosely. These authors in all likelihood followed in their teaching the prac-tice of their master Munoz, expounding and commenting on Ptolemaicastronomy and comparing its quantitative results with the data and tablesof other authors, including Copernicus and Erasmus Reinhold.The University of Alcala was a typical renaissance institution that opened

its doors in the academic year of 15081509. Theology, arts, medicine,


canon law and three levels of grammar were taught there, plus rhetoric and,in the trilingual College, also known as San Jeronimos, Greek, Hebrew,Arabic and Chaldean too. This University had both a mathematics andastronomy chair since its foundation but even today, very little is knownabout the subjects taught there.56 One can assume that there was a stronginuence in mathematics teaching, both from humanist trends and nominal-ist ones. Nebrija was also a professor in Alcala, where he taught rhetoric.Pedro Ciruelo was theology professor in Alcala between 1509 and 1523.Ciruelo studied theology in Paris between 1492 and 1502 where he taughtmathematics and published various works on these themes: a treatise onarithmetic re-edited four times (in Paris), the arithmetic and geometry ofBradwardine and commentaries on the Sphaera of Sacrobosco (with thequestions of Pierre dAilly). In Alcala, Ciruelo published a Cursus quattuormathematicarum artium liberalium (1516) dedicated to arithmetic, geometry,perspective and music. The arithmetic and geometry is principally based onBradwardine, with additions from Ciruelo himself. The perspective is aparaphrase of Pechams work. In his commentary on Aristotles AnalyticaPosteriora (Alcala, 1529), Ciruelo considered mathematics to be the mostperfect case of Aristotelian demonstration and although he ranked theologyhigher than astronomy, he did consider astronomy to be superior to allother sciences due to the dignity of its aim and certainty.57

In the middle of the sixteenth century, the teacher of mathematics inAlcala was Pedro Esquivel (d. ca.1570), who left the chair in 1559 when hewas appointed Palace mathematician and chaplain by Philip II. As a tech-nician, Esquivel carried out several assignments to improve inland water-ways in Castile and Aragon and in 1566 the king commissioned him toconduct a topographical description of Spain. For this purpose, Esquivelused the geodesic triangulation method described by Gemma Frisius, andproduced truly remarkable cartography.58

In 1564 took place a reform of the Constitutions of the University ofAlcala. Regarding the mathematics disciplines was established as mattersto teach: the four liberal (arts): arithmetic, geometry, perspective andmusic, as well as speculative astrology (that is to say, astronomy):sphere, theory of the planets, tables and if there is time surplus (of theyear) astrolabe or some other instruments.59 Furthermore, if there areenough students and they request other lessons as the Almagest ofPtolemy, or Copernicus or some other speculative authors, the lesson willhave to be imparted in other hour so that the ordinary lessons wouldnot lack. That is to say, also in this reform was suggested the work ofCopernicus as alternative to that of Ptolemy for the teaching of astronomyto students. In 1584 a new reform was accomplished, even though the textfor the teaching of the mathematics stayed unchanged.


As far as can be gathered, Gabriel Serrano, Munozs student, was also aprofessor at Alcala, before moving to Salamanca.60 In about 1586 thechair of mathematics and astronomy at Alcala de Henares was occupiedby Diego Perez de Mesa, another of Munozs students.61 In 1591, asmentioned earlier, he competed for the chair at Salamanca left vacant byJeronimo Munoz. He was awarded the chair but did not take possession ofit, deciding to remain in Alcala where he negotiated and obtained a salaryincrease. In 1595, Perez de Mesa, apparently upon the kings commissionor order, moved to Seville to take up the chair created by the Town Coun-cil, at the request of the Cortes (parliament) in Madrid, in collaborationwith the University and the Casa de la Contratacion in Seville. Thecreation of this chair therefore stems from Philip II concern, mentionedearlier, for the training of well-prepared cosmographers, pilots and skilledtechnicians.We do not know what subjects Perez de Mesa taught, in fact at the

University of Alcala, although they may be supposed to be similar, at leastas regards astronomy, to what he taught in Seville, where he also gaveclasses of arithmetic and algebra, practical geometry, astrology and theirapplications in medicine and navigation, giving all classes in Spanish.62

In his Comentarios de Sphera, written for the classes he gave in Seville,Perez de Mesa denes the purpose of cosmography and indicates that thissubject is a science nearly mixed with philosophy and therefore it resolvesmany most wonderful questions of philosophy.63 Like his master JeronimoMunoz, Perez de Mesa considered that astronomers were perfectly entitledto make statements about natural philosophy and he devoted the rst partof his commentary to a discussion of cosmological themes. He denied thatthere was a sphere of re in the concavity of the moon and quoted Coperni-cus and Cardano among other authors in support of this position. He alsodenied the existence of and need for celestial spheres, as well as theincorruptibility of the heavens. To support of this last point, he mentionedthe observations of the supernova made in 1572 by Munoz. He devoted anentire chapter (Chapter 6) to the motion of the earth, although he referredonly to its motion of rotation. For Perez de Mesa, the answer to thisquestion could not be one of absolute certainty but rather of possibility.64

In the new reform of the Constitutions of the University of Alcala thattook place in 1603 the teaching of mathematics was established in threeyears.65 In the rst year, the matters were practical arithmetic (usingGemma Frisius or Oronce Fines books), the rst six books of Euclides,trigonometry, perspective (using Witelo or Alhazens works) and practicalgeometry following Oronce Fine or Fernel. The second year, the sphere ofSacrobosco demonstrating their conclusions with geometry and perspec-tive, as much as is possible, the use of the astrolabe by Gemma Frisius,


the Geography of Ptolemy and the navigation by the compendium whatseem best to the teacher. The third year was intended for the theory ofthe planets according to Peurbach, interpreting each theory (model) byPtolemy; for this year were also indicated the Alfonsine Tables, Deratione temporum and the ecclesiastic computus. This set of matters corre-sponds to those which very probably had been imparted in the two lastdecades of the century Munozs students, Gabriel Serrano and Diego Perezof Mesa. They are similar to those taught by Munoz in Valencia andSalamanca. Nevertheless, it is interesting to emphasize some dierences: inthe rst place, it is notable the absence of all reference to astrology,though it is very probable that, in spite of this, Serrano and Perez of Mesataught this matter, as in fact they did in Salamanca (Serrano) and Sevilla(Perez de Mesa). We recall that Pedro Ciruelo, one of the most outstand-ing teachers of Alcala of the rst decades of the century, was a notabledefender of the astrology against Pico della Mirandola criticism.66 In thecase of optics, while Munoz had followed the Euclidian tradition, in thenew reform it was recommended to follow Witelo or Alhazen, more inagreement with Pedro Ciruelos tradition. In astronomy, the text eliminatesthe reference to Copernicus that was present in the previous reforms. Thiscan be attributed to the growing dreads and anxieties on the part of thephilosophers and theologians of the risks that was implying the work ofthis author; but the same could be said of Salamanca, where the referenceto Copernicus was maintained in the new statutes of this Universitypublished in 1624, that is to say, after the ecclesiastic decree of 1616 inwhich Copernicus De revolutionibus and Zunigas In Job commentaria wereprohibited until corrected.67 Because of this, this issue remains opened tofuture investigations. Finally it is worth emphasizing also the incorporationof the geography and the art of navigation, according to the preoccupationof the Spanish rulers so that the universities provided a good training incosmography.

2. Conclusion

By way of conclusion, it must be said that the mathematical disciplineswere cultivated in Spanish universities in relation to medicine, humanism,natural philosophy and cosmography. In the second half of the sixteenthcentury, those disciplines were taught at the Universities of Valencia,Salamanca, Alcala and Seville by the noteworthy astronomer JeronimoMunoz and his followers. In addition to being an astronomer and mathe-matician, Munoz was a geographer and topographer, Hebraist andHellenist, and was also a follower and sustainer of humanist projects.Despite the presence of a certain demarcation between the subjects of


astronomy and natural philosophy, Munoz and his followers in the univer-sities of Salamanca, Alcala and Sevilla expounded and debated on cosmo-logical matters and considered themselves perfectly entitled to do so. Intheir teachings, they incorporated critically the theoretical and technicalnovelties of their time and placed particular emphasis on the applicationsof the mathematical disciplines. Finally, it has been pointed out that thegreat importance of cosmography in the reign of Philip II, particularly inresponse to the needs of geographers, cartographers and professors of theart of navigation, several of whom began their mathematical education inthe universities, was a driving force behind the study of this subject at uni-versity, together with the study of astronomy. Similarly, university teachersparticipated in various way in cosmographical and geographical activities.Thus, although the different areas of the teaching of mathematics imposedconditions on the discourses and practices of the mathematical disciplines,they did not impede the circulation of persons, knowledge, and practicesamong these areas.

Notes

1 As Maravall (1972) pointed out, when referring to Spanish Monarchy, we have to distinguish be-

tween three dierent levels: each peninsular kingdom, the combination of the kingdoms of Hispanic tra-

dition and the imperial conglomerate that had been built up under the Spanish Crown. With reference

to science and technology, the three levels aected scientic activity: the imperial project, the building

of a modern state and the diversity of the peninsular kingdoms with their own socialpolitical organisa-

tion and their own cultural traditions.2 See Baujouan (1967) and Garca Ballester (2001).3 Nevertheless, as Kagan (1974), p. 79, has noted, it would be dicult to dispute that the Castilian

universities of the sixteenth century were lled with students lacking in professional aspirations.4 See Kagan (1974), p. 232 and passim.5 See below on this literature.6 See especially Vicente, Esteban (1991).7 See Albuquerque (1975) on nautical astronomy. See Haring (1918), Pulido (1950), Lamb (1995),

Lopez Pinero (1979), Esteban Pineiro (1993), Barrera (1999), Sandman (2001), on the Casa de la Con-

tratacion and the dierents posts and oces created. On the Consejo de Indias, see Schafer

(19351947) and Lamb (1995).8 Real cedula, 1527, transcribed in Pulido (1950), pp. 140143.9 On Medina, see Lamb (1972). Recently, Sandman (2001) has examined extensively the question rela-

tive to the teaching in the Casa de la Contratacion.10 Real Cedula, 1553, trancribed in Pulido (1850), pp. 7277.11 From the article on him in the Dictionary of National biography. Borough promoted the translation

into English of Martin Cortes The Arte of Navigation, trans. Richard Eden (London, 1561).12 See Lamb (1995), part.II. There is a modern edition; see Chaves (1983).13 The Libro de Cosmographia and the Suma de cosmograa were not printed. On this books and on

Medinas works in general, see Lamb (1995) part. I and (1972).14 See Navarro et alii (1998) for the editions and translations of Medinas and Cortess books. There is

a modern edition; see Cortes (1990).


15 See Navarro (2003), on Nunes inuence on Cortes. On Nunes, see also Carvalho (1935-), Ventura

(1985), Goncalves (1986), Leitao et alii. (2002).16 The values of the declination of the Sun to calculate the geographical latitude were deducted from

the values of the daily position of the Sun in the ecliptic provided by the tables of ephemeredes, as

those of Abrahan Zacuto or Johannes Stoeer, using the formula: sen d=sen k. sen e (d =declination;k=longitud of the Sun; e=maximum obliquity of the ecliptic). These ephemeris were elaborated fromthe parameters provided by the Tables Alfonses. See Goldstein, Chabas (2000), on Zacuto. On the use

of the Stoeers Ephemerides by Nunes, see Navarro (2000). In this last work I have shown that the

declinations given by Zamorano match those deducted from Stadius Ephemerides but are clearly dier-

ent from those deducted from Leovitius Ephemerides obtained from the Alphonsine Tables. Johannes

Stadius, born 1527, was the rst computer to adopt the Copernican parameters for a major ephemeris.

See Gingerich (1993), pp. 194 and . I have used J. Stadius, Ephemerides...secundum Antwerpiae

longitudinem, ab anno 1554 usque ad annum 1606 (Cologne, 1581).17 See Schafer (19351947); Goodman (1988). See in Edwards (1969) a english translation of Lopez de

Velascos instructions to observe the eclipses in the New World.18 See Vicente and Esteban (1991).19 The Herreras text has 20 pages; printed in 1584, has been reedited by Jose Simon Daz and Luis

Cervera Vera, in facsimile form from a copy preserved in the Bibliothe`que Mazarin of Paris. On the

Academy of mathematics, see also Vicente and Esteban (1991), Esteban and Jalon (1997).20 See a transcription of the royal patent letter in Vicente and Esteban (1991), pp. 115. On Lavanha,

see Sanchez Perez (1934) and Teixeira de Mota (1987).21 See Vicente and Esteban (1991), pp. 90 and .; see also in p. 116, a transcription of the royal patent

letter with the appointment of Onderiz.22 Ms. preserved in the Library of the University of Salamanca, Ms. 2317.23 The edition of 1573 of Nuness work on the subject was entitled De arte atque rationi navigandi. On

Nunes inuence in Portugal and Spain, see Navarro (2003).24 Library of the University of Salamanaca, Ms.2317.25 The information of the activities of the Academy between 1597 and 1600 here described come from

Cristobal de Rojas, in the preface to his Teorica y practica de forticacion (1598) and of Gines Roca-

mora, in the preface to his Sphera del Universo (1599). There is no evidence of the fact that the new

teachers received an ocial appointment. See Vicente, Esteban (1991), pp. 137 and .26 Biblioteca Nacional of Madrid, Ms. 9027.27 The argument was on the demostracion of the proposition 4 of Mechanicorum Liber of Guidobaldo.

See the letters 136 (february, 1597), p. 1719; 139 (october, 1597), pp. 2728; 149 (december, 1598), pp.

6869 and 153 (april, 1599), p. 77, from Neritius to Clavius, in Vol. IV of Clavius Correspondence edi-

ted by Baldini and Napolitani (1992). On Neritius, see ibid, Vol.I, Biograe, pp. 7577.28 See Garcia Tapia (1990), pp. 158159. On Garca de Cespedes, see also Picatoste, Lopez Pinero et

alii (1983), Vol. I, pp. 375376; also Vicente, Esteban (1991) and (2000, 2002a).29 On the amendment, see Vicente, Esteban (1991) and Navarro (2000).30 See Navarro (2000).31 On the University of Valencia, see Felipo (1993), Peset, coord., (1999). Gallego Salvadores (1980),

Febrer (2003).32 In this sense, we can consider signicant the appointment of Tomas Duran to the chair of mathe-

matics of the University of Valencia. Duran published in Valencia mathematical works of Bradwardine,

a prominent member of mertonian calculatores, and in 1509 was proposed for the chair of nomi-

nales of the University of Salamanca created that same year. On Duran, see Robles (1976).33 See Gallego Salvadores (1975), Wallace (1981), Lopez Pinero et alii (1983), vol. I, pp. 203206,

Lopez and Navarro (1995), pp. 8393, Febrer (2003).34 In the inquisitorial process to Jeronimo Conques, graduate in arts in 1545 in the University of

Valencia, he declared that he had studied arithmetic, music, geometry, perspective, judicial astrology

and cosmography. See Ardit Lucas (1969).35 Pedro Jaime Esteve, Hippocrates Coi Medicorum omnium principiis epidemion liber secundus (1551);

Citation on fol.5v. In this work Esteve considers indispensable the astronomy to study the climate in


the dierent regions and to predict the changes of the air. He mentions various observations

accomplished by him in 1549; according to Ximeno (1749), vol. I, p. 112, Esteve elaborated a Libro de

Ephemerides (Book of Ephemerides) for 14881600. On Esteve, see Lopez Pinero et alii (1983), vol. I,

pp. 312314.36 On Munoz, see Navarro (1983), Navarro and Rodrguez (1998), and the preliminary studies in

Munoz (2004).37 See in Navarro and Rodrguez (1998), pp. 205208, a list of Munozs works.38 J.Munoz, Traicte du nouveau comete, Paris, 1574. On Lefe`vre de la Boderie, see Secret (1960). For

the relationship between Lefe`vre and Postel and his contribution to the Polyglot Bible of Antwerp, see

Rekers (1972).39 For Cornelius Gemmas discussion about Munoz study of the nova, see Cornelius Gemma, De

Naturae Divinis Characterismis; seu Raris et admirandis spectaculis in Universo, Libri II (Amberes,

1575), vol. II, pp. 267274. Munoz correspondence with Hagecius and Reisacherus is housed in the

Oesterreichische National Bibliothek, Cod. 10.868, No. 66 and Cod. 10.689, No. 41, fols. 1r-6v. J.L.E.

Dreyer published these letters in his edition of Tychonis Brahe. Opera Omnia (191319), vol. VII, pp.

395403. Munoz (1981) includes a transcription and translation into Spanish and English accompanied

by a facsimile edition of the letter to Reisacherus, according to the copy of Cod.10.689 mentioned ear-

lier. This copy was apparently made by Tycho Brahe; see Navarro and Rodrguez (1998), pp. 207208.40 The holograph manuscript is housed at the Arnamagnaeanske Institute, Copenhagen, AM 8812 4,fols. 147. Published and translated into Spanish by Navarro, Rodrguez (1998).41 The manuscript ends with the words Hieronymus Munnos... translation commmentariorum Theonis

Alexandrini in magnam constructionem CL. Ptolemaei... The holograph is housed at the National

Library in Naples, Ms. VIII, fols.21r-300r. See Navarro, Rodrguez (1998).42 See Navarro, Rodrguez (1998) and the preliminary studies in Munoz (2004).43 See Florez et alii (1990); Navarro (1994); Navarro, Rodrguez (1998).44 See Florez et alii (1990); Navarro (1994, 1995). For the assertions of Perez de Oliva, see Fuertes, ed.

(1984).45 See Munoz (1964, 1967); Beltran de Heredia (1942).46 On Espinosa and Martnez Silceo, see Munoz (1967); on Martnez Silceo, see also Lopez Pinero

etalii (1983), vol. II, pp. 3839; on Margallo, see Florez (1990). See also Navarro (2002a). For this

authors, see also Lohr (1988).47 See Randles (2000).48 See Navarro et alii (1998) for the editions of Marnez Silceo and Espinosa works.49 On Juan and Hernando de Aguilera, see Bustos (1973), and Lopez Pinero et alii(1993), vol. 1, pp.

2830. On the statutes of Salamanca University see, in addition to Bustos, Navarro (1995, 1998).50 The purpose of the inspections (visitas) stipulated in the 1561 statutes was to supervise the conduct

of the professors occupying chairs. For libros de visitas, see Fernandez (1974), and Navarro (1998).51 See Fernandez Alvarez (1974) and Navarro (1998).52 As refered by his student Diego de Alava in the preface to his book El Perfecto capitan instruido en

la disciplina militar y nueva ciencia de la artilleria (Madrid, 1590).53 Letter from Philip II, March 26th 1593, in Esperabe (19141917), Vol. I, pp. 608609; also published

in Bustos (1973).54 See Beltran (19701973), vol. IV, pp. 120 . For some information on Nunez Zamora, see Picatoste

(1981), pp. 223225, and Rodrguez San Pedro (1986), vol. III, p. 71.55 Estatutos hechos por la muy insigne universidad de Salamanca, Salamanca, 1595.56 Gonzalez Navarro (1984); Jimenez, coord.(1996).57 On Nebrija, see Cotarelo (1947), Codoner and Gonzales Iglesias, eds., (1984), Florez et alii (1990)

and LoPez Pinero et alii (1983), Vol. I, pp. 105107; On Ciruelo, see Munoz Delgado(1967), Lopez Pin-

ero et alii (1983), vol.I, pp. 223225; Alabares (1996) and Navarro (2002a, 2002b).58 See Lopez Pinero et alii (1983), Vol. I, pp. 310312. Also, Reparaz (1980), Parker (1992), Esteban

(1996).59 That is, the topics included in Pedro Ciruelos Cursus quattuor mathematicarum artium liberalium

(1516), that are also those ones of the quadrivium, with the exception of astronomy, not comprised in


Ciruelos work, and the distinction between geometry and perspective according to the late Middle

Ages tradition.60 According to Beltran (19701973), Vol. IV, p. 324.61 F. Gil Ayuso, Historia de la Universidad de Alcala, quoted by A. Gonzalez Palencia, ed. (1944), in

the foreword, p. XXXVIII. In the dedicatory to his patron Gaspar de Borja and Velasco, in a Latin

manuscript of Cosmographia (Biblioteca Universitaria de Barcelona, Ms. 446), Perez de Mesa states

that he taught natural philosophy in his youth in Salamanca and Alcala.62 Manuscripts of all these subjects written between 1595 (Astrologia judiciaria) and 1603 (the El arte

de navegar treatise) are housed at the Biblioteca Nacional, Madrid and in the library of Salamanca

University. The manuscript entitled Astrologa judiciaria (Madrid, BN, Ms. 5917), states Judiciary

astrology read in Seville by Diego Perez Mesa, Professor at Alcala de Henares, by order of King Philip

in the year 1595. The treatise on the art of navigation features the date September 6th 1603 at the

end. Ms. 2294 of the Salamanca University Library concerns arithmetic, algebra, astrology and practi-

cal geometry and Perez de Mesa, professor of this city of Seville in the year 1598 is named as the

author on the rst page.63 A copy of Comentarios de Sphera by Perez de Mesa, is housed in the Biblioteca Nacional, Ms. 8882.

It is dated in Seville, September 22nd 1596.64 Perez de Mesa, Comentarios de Sphera, op.cit., fols. 19v-22r. The chapter ends with Copernicus

answers to the objections about the Earths motion.65 The text of the Reforma de Portocarrero in Gonzalez Navarro, ed. (1999), pp. 431626, on pp.

499500.66 See Navarro (2002 a), p.294.67 On Zuniga, see Navarro (1995).

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