Oronce Fine and Sacrobosco: From the Edition of the Tractatus de sphaera (1516) to the Cosmographia (1532)

Abstract

This paper considers the contribution of the French mathematician Oronce Fine to the diffusion and transformation of Johannes de Sacrobosco’s Tractatus de sphaera by considering his 1516 edition of the Sphaera and his Cosmographia, sive sphaera mundi (in Protomathesis, 1532). The article first describes Fine’s life and career, as well as his work as editor of the Sphaera. In a second part, it considers what Fine, in the Cosmographia, has drawn and left aside from the Sphaera, revealing the consequent transformations to the teaching of Sacrobosco’s theory of the sphere and its adaptation to the cultural and intellectual environment in which Fine evolved. A last part considers the treatment, in the Cosmographia, of the cosmological representations transmitted by Sacrobosco and by subsequent interpreters of Ptolemaic astronomy concerning the number of celestial spheres and its relation to judicial astrology.

1 Introduction

The author would like to thank Roberto de Andrade Martins, Peter Barker, Jamie Brannon, Marius Buning, Kathleen Crowther, Charlotte Girout, Thomas Horst, Tayra Lanuza Navarro, Elio Nenci, Richard Oosterhoff, Isabelle Pantin, and Matteo Valleriani, for the insightful comments and exchanges on the theses of this paper.

Oronce Fine or Finé¹ (1494–1555), a French mathematician from the Dauphiné, is chiefly known to historians of science for having been the first to teach mathematics as a royal lecturer within the institution founded by François I in March 1530,² but also for his work as a cartographer,³ as a designer and maker of mathematical instruments,⁴ as well as an engraver and an editor of scientific books.⁵

If it should be admitted that Fine played an important role in the development of mathematics in sixteenth-century France, it is not primarily by the content of his works or of his teaching, which historians of mathematics have not regarded as significant to the advancement of the mathematics of his time (Ross 1975; Poulle 1978)⁶ and which was criticized in Fine’s lifetime by Pedro Nuñez (1502–1578) and Jean Borrel (Johannes Buteus) (1492–1564/72) in books aimed to expose the mistakes contained in his treatises (Nuñez 1546; Borrel 1554,⁷1559).⁸ It is rather through his project to promote mathematical teaching in France,⁹ a project he carried out—besides his life-long career as a royal lecturer in mathematics—by writing and publishing a substantial number of mathematical treatises, through which he contributed to the dissemination and expansion of the mathematical culture of his time¹⁰ in the continuity of the endeavor led in this direction by Jacques Lefèvre d’Étaples (ca. 1450–1536) (Chap. 2), Pedro Sanchez Ciruelo (1470–1548) (Chap. 3), and their disciples¹¹ in Paris from the end of the fifteenth century on, as well as by assiduously calling upon the perfection and value of mathematics, as he did in particular in his Epistre exhortative touchant la perfection et commodite des ars liberaulx mathematiques (Fine 1531a).¹²

François I founded the institution of the royal lecturers on the suggestion of his librarian Guillaume Budé (1467–1540), in order to create a college where humanists would teach subjects such as Greek and Hebrew which were neglected by the university curriculum, but which were considered necessary to the study and interpretation of ancient pagan and Christian authors.¹³ The foundation of a chair of mathematics shortly after the creation of the first royal lectureships in Greek and Hebrew¹⁴ indicates that the teaching of mathematics provided at the University of Paris was judged insufficient, in spite of the efforts made in the previous decades by the circles of Lefèvre and Ciruelo to change this situation.¹⁵ This is confirmed by the discourse held by Fine and by his successor Petrus Ramus (1515–1572) on the ignorance of the students and masters of the Parisian Faculty of the Arts in mathematics (Dupèbe 1999, II, 523; Tuilier 2006a; Pantin 2006, 2009a; Oosterhoff 2015).¹⁶

As the first royal lecturer in mathematics, Fine’s main assignment was to offer greater visibility to the mathematical arts in France and to reform the traditional mathematical curriculum by introducing, in addition to the mathematical content generally taught within the universities (Boethian theories of numbers and of consonances, Euclidean plane geometry, and Ptolemaic geocentric astronomy),¹⁷ more practical branches of mathematics, through which the utility of mathematics would be more easily displayed, both for the disciplines taught in the higher faculties (medicine, law, and theology) and for the moral and material aspects of human life.

Fine’s mathematical teaching program as a royal lecturer was made public through the Protomathesis,¹⁸ a quadripartite mathematical compendium published in 1532, shortly after his assignment to this function. This monumental work provided a teaching on practical arithmetic (De arithmetica practica libri IIII), on theoretical and practical geometry (De geometria libri II), on cosmography (De Cosmographia, sive sphaera mundi libri V), as well as on gnomonics (or the art of sundials) (De solaribus horologiis et quadrantibus libri IIII).¹⁹ Mixing theoretical and practical knowledge (from Euclidean geometry and the theory of the sphere to the construction and use of surveying instruments and sundials),²⁰ the publication of the Protomathesis was important for the new image it provided of mathematics, in France and beyond.²¹ As shown by Isabelle Pantin, the publication of this work, as well as of the many separate and successive editions of the different treatises that compose it,²² also helped shape the Parisian style of printed scientific books (Pantin 2010, 2013a, Oosterhoff 2016).²³

Astronomy held a central role in Fine’s mathematical work. This is manifested by the dominant number of treatises on astronomy, astrology and astronomical instruments, amid the numerous works he wrote, published and edited.²⁴ Among the latter, the two first books on which he worked as an editor were the Theorica planetarum of Georg Peurbach (1423–1461), which was published in Paris in 1515 (Pantin 2009b, 2012, 2013a),²⁵ as he was studying at the Collège de Navarre, and the Tractatus de sphaera of Johannes Sacrobosco (died ca. 1256)²⁶ in 1516 (Pantin, 2009b, 2010, 2013a; Pettegree and Walsby 2012, 1020–21).

A central role was also attributed to astronomy within Fine’s mathematical teaching program, since the Cosmographia, sive mundi sphaera (of which four out of five books deal with spherical astronomy) stands, among the different treatises that compose the Protomathesis, simultaneously as the culminating point of the quadrivium,²⁷ following arithmetic and geometry (music having been left aside), and as the condition of the application of mathematics to a more practical and specialized type of knowledge, then mainly represented by cartography (through the fifth book of the Cosmographia) and gnomonics (the last part of the Protomathesis).

The importance of astronomy in Fine’s mathematical work is also displayed by the manner in which he represented himself in the frontispieces of his editions and treatises pertaining to astronomy,²⁸ depicting himself in the place of the astronomer, where the figure of Ptolemy (ca. 85–ca. 165) was often positioned in the frontispieces of late fifteenth-century astronomical textbooks—that is, beneath the sphere of the universe (sometimes represented as an armillary sphere), holding a book (open or closed) and/or an astronomical instrument, the muse Urania placed besides him (Pantin 1993; Conley 1996, 98–115; Pantin 2009b; Barker and Crowther 2013).²⁹ Although he perpetuated in this way a preexisting visual tradition, it is notable (as Pantin points out) that Fine, who usually engraved the visual material contained in his editions and treatises, did not produce comparable illustrations for his works on other mathematical disciplines (Pantin 2009b).³⁰ It also seems significant that the astronomical frontispiece which is associated with the first edition of his Cosmographia—where Fine represented himself sitting under a celestial sphere while holding both an astronomical instrument and an open book—was used twice in the Protomathesis, once at the head of the entire compendium (after the general index) and once at the head of the Cosmographia, though with two different epigrams: one applied to mathematics in general, and more specifically to arithmetic and to the role of the knowledge of numbers and measure for the knowledge of the creation and of its components,³¹ and the second applied to astronomy, commending the usefulness of the science of stars for the contemplation of the divine order.³² The use of this frontispiece to introduce the whole compendium, along with the accompanying epigram explaining the importance of mathematics for the knowledge of the causes of the wordly substances, also confirms the central and overarching role of astronomy within the quadrivium according to Fine.

Moreover, although Fine greatly emphasized the need to develop, in France, the teaching of all mathematical disciplines, theoretical and practical, he also regularly asserted in his prefaces the predominant importance of astronomy over the other parts of the quadrivium with respect to the primary purpose of mathematics according to the ancient model of education—that is, to open the path to wisdom and to the knowledge of the divine order that governs the universe, in conformity with Platonic epistemology (Barker and Crowther 2013; Axworthy 2016, 151–64).³³ He followed in this regard the discourse held by Johannes Regiomontanus (1436–1476) in the inaugural oration of his lessons at the University of Padua in 1464 (Pantin 2009a), where astronomy is described both as the crowning of the quadrivium and as the reason for which the other parts of mathematics should be studied.³⁴

With respect to the scope of the present volume, the aim of this paper is to examine the significance and transformation of Sacrobosco’s astronomical teaching, both in Fine’s work as an editor of the Tractatus de sphaera and as the author of the Protomathesis and, more precisely, of the Cosmographia, sive sphaera mundi.³⁵ Following a brief biographical outline, I will first describe Fine’s work for the 1516 edition of the Sphaera to examine afterwards how the Cosmographia, his major astronomical work, relates to the content of Sacrobosco’s treatise. I will conclude this paper by considering Fine’s conceptions (as they appear in the Cosmographia) on the cosmological representation transmitted by Sacrobosco and by subsequent interpretations of Ptolemaic astronomy concerning the number of celestial spheres, as an illustration of the uses and transformations of cosmological knowledge in the tradition of Sacrobosco’s teaching of spherical astronomy.

2 Fine’s Life and Career

Fine was born in 1494 in Briançon to a family of high social rank, which constituted a stimulating environment for the development of scientific interests during the first years of his life (Thomé de Maisonneufve 1922; Thomé de Maisonneufve 1924, 5–10; Ross 1971, 8–11).³⁶ His grandfather and his father, Michel Fine (fl. 1474–1490) and François Fine (fl. 1494–1499), were both physicians (Wickerheimer 1979, 154, 553). The former wrote a treatise on the plague, whose posthumous publication Oronce contributed to in 1522 (Fine 1522; Dupèbe 1999, II, 521). The latter is said to have built an equatorium, which William Gilliszoon of Wissekerke (ca. 1444) described in his Liber desideratus super celestium motuum indagatione sine calculo (Wissekerke 1538, sig. A2r–v; Poulle 1961). When his father died, he was sent to Paris and entrusted to Antoine Silvestre, a family friend from Briançon who taught arts at the Collège de Montaigu and theology at the Collège de Navarre (Launoy 1677, 646–47; Élie 1951), where Fine studied. Fine obtained his Master of Arts in 1516 at the Collège de Navarre (Launoy 1677, 678), where he started the same year to teach mathematics both privately and publicly at least until 1527 (Dupèbe 1999, II, 533).³⁷ He began in parallel to study at the faculty of medicine, obtaining his bachelor’s degree in 1522 (Concasty 1964, 50b, 54a–b; Dupèbe 1999, II, 526–27). From 1528 on, he taught mathematics at the Collège de Maître Gervais (Dupèbe 1999, II, 540–41; Boudet 2007; Pantin 2009a, 2013a).

François I’s choice to assign Oronce Fine to the first royal chair of mathematics was likely influenced by the support Fine received from humanists close to the royal court (Dupèbe 1999, II, 530, 533, 538), as well as by his multifaceted mathematical activity in the years 1515–1530 (Pantin 2006, 2009a; Axworthy 2016, 14–17). These were the years during which he worked on his first editions and illustrations of mathematical and non-mathematical works (Peurbach and Fine 1515; Sacrobosco and Fine 1516; Le Huen 1517³⁸; Bassolis and Fine 1517a, b; Martínez Silíceo and Fine 1519; Ricci and Fine 1521; Fine 1522; Reisch and Fine 1535)³⁹ and published under his name several mathematical treatises (mostly pertaining to astronomy) (Fine 1526, 1527, 1528, 1529, 1530), as well as a map of France (Fine 1525). He also practiced during this time as a court astrologer.⁴⁰

Fine remained a royal lecturer until his death in 1555. During the 25 years he taught mathematics in the name of the King of France, which would have represented an exceptionally long career for such a position at the time, he published many other treatises, among which several reprints of the first three parts of his Protomathesis, as well as new treatises dealing with astronomy and its applications (Fine 1543a, b, 1545, 1553a, b, 1557), geometry (Euclid 1536; Fine 1544b, 1556a, b) and the theory of burning mirrors (Fine 1551b).

Thanks to his high-quality editions, geographical maps, and mathematical treatises, Fine rapidly gained an international reputation, notably in Italy and in England (Johnson 1946; Heninger 1977a, b; Feingold 1984, 59, 116, 118; Tredwell 2005, 185; Eagleton 2009; Mosley 2009; Leitão 2009; Wagner 2010; Rampling 2012; Valleriani 2013, 76–77; Axworthy 2016, 22–27; Valleriani 2017, 430). The Italian translation of the Protomathesis by Cosimo Bartoli (1503–1572), published with the translation of the De speculo ustorio by Ercole Bottrigari (1531–1612) (Fine 1587) as well as the English translation of his Canons des ephemerides by Humphrey Baker (fl. 1562–1587) (Fine 1558), testify to the international and long-lasting influence of Fine’s mathematical teaching throughout the sixteenth century. His Cosmographia notably appears in the Bibliotheca selecta compiled by the Jesuit Antonio Possevino among the sources relevant to the study of astronomy (Possevino 1593, 201; Margolin 1976; Mosley 2009). His astronomical and arithmetical works were included in the programs of the Jesuit College of Messina (Sasaki 2003, 21; Gatto 2006), of the University of Pisa (Schmitt 1974, 1975) (Chap. 10), of the University of Cambridge (Johnson 1946; Feingold 1984, 39), and also very likely of the Spanish University of Valencia (Navarro Brotóns 2006).⁴¹ At the University of Oxford, the statutes of 1565 recommended astronomy lecturers to teach Fine’s Cosmographia as a possible alternative to Sacrobosco’s Sphaera (Goulding 2010, 88–89).

3 Fine’s Edition of Sacrobosco’s Sphaera and It’s Significance for His Pedagogical and Scientific Project

Sacrobosco’s Sphaera was, as indicated above, the second treatise on which Fine worked as an editor. It was first published in 1516 in Paris by Regnault Chaudière (died 1554), when Fine was about twenty years old and was teaching at the Collège de Navarre.⁴² As indicated by the title and the colophon, the work he performed for this edition of the Sphaera consisted in applying corrections, engraving woodblocks, and adding marginal indications.⁴³ In addition to his epigram to the reader, he included several liminary poems at the beginning and at the end of the book,⁴⁴ written by himself and by some of his colleagues or condisciples, such as the poet Hugues d’Ambert or Hugues de Colonges (fifteenth–sixteenth century), who was then addressing both Fine⁴⁵ and Fine’s protector Antoine Silvestre,⁴⁶ and Nicolas Petit (1497–1532),⁴⁷ addressing Jean Fossier or Jean des Fosses (fifteenth–sixteenth century),⁴⁸ one of Fine’s disciples (Dupèbe 1999, II, 526).⁴⁹ The engravings provided by Fine for this edition were mostly modeled on the figures found in the editions of Sacrobosco’s Sphaera published in Venice from the end of the fifteenth century onwards (Cosgrove 2007; Pantin 2010) (Chap. 9),⁵⁰ though he integrated, for the representation of the motion of the sun, engravings he had produced for his edition of Peurbach’s Theorica planetarum.⁵¹ This edition also contains a few tables: a table displaying the cosmic, chronic, and heliacal rising and setting of the signs;⁵² a table for the rising and setting of the signs in the right sphere (indicating the durations of the rising and setting of each sign and the quantities of the corresponding arcs of equinoctial);⁵³ a table for the latitudes of the seven climates, coupled with a table indicating the duration of the longest artificial days for these latitudes.⁵⁴ The inclusion of these tables, though quite elementary, demonstrates Fine’s will to add to the Sphaera complementary elements of a practical nature,⁵⁵ in line with the material added by Lefèvre d’Étaples in his 1495 commentary on Sacrobosco (Sacrobosco et al. 1495; Oosterhoff 2015) (Chap. 2). This also anticipated his later contribution to the diffusion and transformation of spherical astronomy in the tradition of Sacrobosco’s Sphaera in the Cosmographia. The printed marginalia mark out the different topics dealt with by Sacrobosco, the authors referred to such as Virgil (Publius Vergilius Maro, 70–19 BCE), Ovid (Publius Ovidius Naso, 43 BCE–ca. 18), Lucan (Marcus Annaeus Lucanus, 39–65), al-Farghānī (Abū al-ʿAbbās Aḥmad ibn Muḥammad ibn Kathīr al-Farghānī, ca. 805–870), as well as the distinction and hierarchical status of the various arguments provided,⁵⁶ giving this edition the style and the structure of university textbooks while making the text easier to read and consult.⁵⁷ The large white spaces on the exterior margins also facilitated note-taking, as shown by extant exemplaries containing substantial hand-written marginal notes.⁵⁸

Fine’s contribution to the 1521 edition of Lefèvre d’Étaples’s commentary on Sacrobosco’s Sphaera (Sacrobosco et al. 1521) is mainly indicated by the inclusion of the frontispiece he had drawn for his edition of Peurbach’s Theorica planetarum published in 1515 (Pantin 1993; Conley 1996, 98–105) and, starting with the 1527 edition (Sacrobosco et al. 1527), through the introduction of a new frontispiece in which he represented himself resting on the ground while contemplating a bi-dimensional worldly sphere situated above him (Pantin 2012). He probably also added the marginal annotations, re-engraved some of the woodblocks, and changed the design of the tables and layout of the text (Pantin 2009b, 2010, 2012; Oosterhoff 2015, 2016) (Chap. 2). However, the content of the text, of the tables, and most of the illustrations are drawn from the earlier editions of Lefèvre’s commentary on Sacrobosco.

Although these editorial interventions in Sacrobosco’s Sphaera may be regarded as minor, notably as they are not related to the content of the text, they would nevertheless have a certain impact on the reading and the reception of the work by its readers. As Isabelle Pantin has shown in this volume (Chap. 9), such interventions in the layout, the illustrations, and the editing of the text may be held as innovations, just as the commentary or the inclusion of new textual material. This is all the more significant in the case of authors such as Fine or Peter Apian (1495–1552), who were also involved in the technical aspects of the production of the book, either as engravers, as editors, or as printers, as they could then control the production of the works in order to suit their own agendas, in particular when they themselves taught mathematics, since they could aim to satisfy certain conditions required by their teaching practice through their editorial interventions.

The fact that Fine intended this work as a university textbook is not only suggested by the layout of the text, but also by the fact that he produced it while he was a professor of the Arts Faculty, which he made explicit in the title of his address to the reader (Artium liberalium professor) and, more generally, by the established place of Sacrobosco’s Sphaera among the works studied in the mathematical curriculum of the Parisian University.

At that point in Fine’s career, especially after the work he had done on Peurbach’s Theorica planetarum published a year before, this edition of Sacrobosco’s Sphaera represented a meaningful move to confirm his competence as an editor of scientific books, since he then likely intended to maintain this activity as an auxiliary source of income while teaching mathematics at the Collège de Navarre. It would certainly enable him to gain visibility in this function,⁵⁹ as Sacrobosco’s Sphaera was a highly demanded work and therefore an easily marketable product, especially as it was part of the standard mathematical curriculum of the university (Crowther et al. 2015; Oosterhoff 2015; Valleriani 2017). The teaching of Sacrobosco’s theory of the sphere actually represented one of the most important parts of the mathematical program of university faculties of arts since the Late Middle Ages,⁶⁰ given that it was used to introduce students to astronomy, as well as to the reading of the De caelo of Aristotle (384–322 BCE) (Valleriani 2017), offering them a general description of the structure of the cosmos and of the motions of the stars, as well as the geometrical tools required to apprehend them (Pantin 1995, 31–36; Oosterhoff 2015) (Chap. 2). The teaching of Ptolemy’s Almagest, as well as the more in-depth study of planetary motions provided by the medieval Theorica and later by the new Theorica provided by Peurbach, were considered too advanced for beginners and were therefore taught at a later stage of the mathematical curriculum (Pantin 1995, 29–31; Barker 2011; Crowther et al. 2015; Oosterhoff 2015; Valleriani 2017) (Chap. 6).

For that matter, Fine’s edition offered students a stand-alone version of Sacrobosco’s text, devoid of any commentary, printed in an easily transportable format (in-quarto), and not bound to other astronomical treatises within large compendia, as was the case for many editions of Sacrobosco published at the end of the fifteenth century (Oosterhoff 2015).⁶¹ It would therefore have been more affordable for university students and easier to bring to class.

Hence, through this edition and the other works he edited during this period, Fine contributed to the stylistic reform of scientific and pedagogical texts instigated by Lefèvre d’Étaples and his disciples at the end of the fifteenth century and which enabled Paris to have a central role in the production of printed textbooks and scientific works (Pantin 2009a, 2013a; Oosterhoff 2015, 2016, 2018, ch. 4; Valleriani 2017). As shown by Isabelle Pantin, the explicit manner in which he indicated early on his role in the various editions he worked on (including in his edition of Sacrobosco’s Sphaera), and the evolution of this practice of identification (Conley 1996; Pantin 2009b), reveals his pride and desire to assert himself as an active promoter of the mathematical culture of his time (Pantin 2010, 2013a).⁶²

In this regard, the fact of providing a new edition of Sacrobosco, especially one that was more accessible to college students, would have been, for Fine, a means to demonstrate his commitment to the pedagogical model of the university, though he later went on to criticize the pedagogical methods used in the Faculty of the Arts for the teaching of mathematics (Axworthy 2016, 30–33). For that matter, despite Fine’s later project to renew the mathematical teaching provided within the Parisian academic sphere, he did remain faithful, at least in the first years of his career as a royal lecturer, to the curricular model of the university, asserting, in the preface of the Protomathesis, the importance and propaedeutic value of mathematics for the three superior faculties of the university: medicine, law, and theology (Axworthy 2016, 186–87).⁶³ He also stated in the preface of the first edition of the Cosmographia⁶⁴ the necessity of astronomy for the students of medicine and theology, given its importance for the computation of calendars and for the determination of the dates of Easter and other mobile religious celebrations (Axworthy 2016, 172–74)⁶⁵ and given the role of judicial astrology (which represented the practical part of the science of stars, according to Fine)⁶⁶ in the determination of the favorable days for bloodletting and for the administration of remedies,⁶⁷ to which should be added the production of medical almanacs (Chap. 5).⁶⁸

It is important to note here that, for many students, astrology (both judicial and natural⁶⁹)—because of its uses in medicine, and also because of its place in Renaissance society and courtly life, as it was held (at least in principle) to guide decisions in all aspects of individual and communal human life (Azzolini 2005; Carey 2010; Eamon 2014)⁷⁰—was often an incentive to study astronomy, and in particular Sacrobosco’s Sphaera, which taught how to determine the positions of the zodiacal signs from different latitudes.⁷¹ Because of its relationship to medicine, astrology actually held a privileged place among the mathematical arts in the university curriculum since its foundation.⁷² The fact that Fine had been trained in medicine, although he does not appear to have practiced as a physician after his studies,⁷³ would have made him clearly aware of the importance of astrology for the medical art, as would his long-lasting friendship with Antoine Mizauld (1510–1578).⁷⁴ Mizauld was a physician and a professor of medicine in Paris as well as an astrologer, and published several works on iatro-mathematics (Mizauld 1550, 1551, 1555),⁷⁵ some of which Fine contributed to (at least as the author of some of the liminary texts).

It would therefore be reasonable to think that Fine also viewed his editorial work on Sacrobosco’s Sphaera as a contribution to the training of astrologers, to help them learn how to calculate the positions of planets in relation to the zodiacal signs and the celestial houses (Valleriani 2017), an activity in which he himself engaged as a court astrologer and which he later promoted through the publication of the Canons des ephemerides (Fine 1543a) and the De duodecim caeli domiciliis (Fine 1553a).⁷⁶ As we will see, the importance of judicial astrology as a motivation to study astronomy was also set forth in the Cosmographia, when Fine discussed the general structure of the cosmos and the number of celestial spheres, claiming the necessity to reject certain cosmological models in order to safeguard the validity of judicial astrology.

4 From the Sphaera of Sacrobosco to the Cosmographia, sive Sphaera mundi

While in the 1516 edition of Sacrobosco’s Sphaera, Fine most likely intended to address a public of students, in the Cosmographia, he addressed a slightly different readership—namely, the audience of the royal lecturers. This new public still included university students, but was also composed of humanists, curious notables, and members of the court, and was in principle open to anyone, especially as the teaching of the royal lecturers did not lead to any degree (Pantin 2006). This gave Fine and the other royal lecturers the flexibility to propose a teaching program that was relatively different from that which was provided at the Faculty of the Arts (when it was provided at all). In Fine’s case, this reformed mathematical teaching program was communicated, as noted above, through the publication of the Protomathesis in 1532.⁷⁷ It is within this work that appeared the first edition of the Cosmographia, which refers for a considerable part of its content and structure to the textual paradigm of the Tractatus de sphaera.⁷⁸

The fact that Fine’s Cosmographia relates to Sacrobosco’s Sphaera in terms of its content, composition, and finality is suggested by its title: Cosmographia, sive sphaera mundi, a title which evolved from 1542 into Sphaera mundi, sive cosmographia (Fine 1542a, b, 1551a, b, 1555), making its focus on the theory of the sphere more obvious (Mosley 2009).⁷⁹ What this title also indicates is that this teaching pertained to cosmography,⁸⁰ which was, in the sixteenth century, properly developed into a discipline in its own right, bringing together metrical geography and the theory of the sphere.⁸¹ It was thus not only conceived as a practice relating to the production of the mappa mundi, or to the cartographical description of the contours of the terrestrial world (for which reason the title of cosmographer, in the Iberian peninsula, was attributed to those in charge of casting navigational charts and of constructing mathematical instruments)⁸² or as a synonym of geography, following the designation of Ptolemy’s Geographia as Cosmographia by its fifteenth-century translator, Jacopo d’Angelo (1360–1410) (Broc 1980; Milanesi 1994; Cosgrove 2007; Besse 2009; Mosley 2009; Tessicini 2011),⁸³ but also as relating to spherical astronomy. The shifts in meaning of the term cosmography (from maps to treatises, from applied practical knowledge to academic teaching, or from the consideration of the terrestrial globe to that of the universe in its totality) display the various orientations and also the tensions inherent to cosmographical knowledge in the sixteenth century (Cosgrove 2007).⁸⁴ These show also that cosmography, as well as geography, was a knowledge in transformation, not only with regard to its content (by integrating the new geographical discoveries), but also with regards to its status, since neither geography nor cosmography was acknowledged as a proper discipline or object of teaching (distinct from natural philosophy, astronomy, or natural history) before the Renaissance (Besse 2003, 10).⁸⁵ It remains that, within the sixteenth-century treatises on cosmography, the different definitions of cosmography remained connected, since, as shown by Fine’s Cosmographia in particular, it was also defined as the means to describe the universe in its entirety and in its various parts, though what cosmographical treatises offered was less a visual description of the world and of its two main regions than a method or set of principles necessary to produce such a description (Besse 2009; Mosley 2009).

In this framework, the mapping of the universe in its two main parts, celestial and terrestrial, required the projection of the circles which divide the celestial sphere in spherical astronomy onto the terrestrial globe, establishing a correspondence between the systems of longitudinal and latitudinal positioning of celestial objects and terrestrial places (Broc 1980, 66–68; Milanesi 1994; Besse 2003, 36–37, 46–48; Besse 2009; Mosley 2009). Through the mathematical correspondence this establishes between the celestial sphere and the terrestrial globe, cosmography is presented as an essentially dual teaching.⁸⁶ Geography is indeed assigned a comparable epistemological status to astronomy,⁸⁷ leaving aside the more qualitative approach of Strabo (ca. 63 BCE–ca. 24) and Pomponius Mela (died ca. 45) in favor of the mathematical mode of description of the earthly contours followed by Ptolemy in the Geography.⁸⁸ In Fine’s prefaces to the 1551 edition of the Cosmographia, this double orientation of cosmography is justified by the double function of man—on one hand, called on to inhabit the earth; on the other, invited to contemplate the heavens (Besse 2009; Mosley 2009; Axworthy 2016, 154–59).⁸⁹

Through the development of cosmography as a discipline, the teaching of Sacrobosco’s theory of the sphere was absorbed into a larger framework. The Tractatus de sphaera provided in this regard one of the most adequate teachings of the astronomical circles that divide the celestial sphere and which the cosmographer is required to project onto the terrestrial globe (Cosgrove 2007).⁹⁰ Sacrobosco’s Sphaera thus became the natural starting point for the mathematical analysis and treatment of the terrestrial space⁹¹ at a global and local level (Besse 2009; Brioist 2009b).⁹² This relation between the theory of the sphere and geography is clearly expressed by the content and division of Fine’s Cosmographia, in which the first four books are dedicated to the description of the celestial region of the cosmos (leaving aside, however, the trajectories of the moon and the five planets) and the fifth book to the description of the earth.⁹³

The topics tackled in the Cosmographia which properly deal with spherical astronomy are in large part the same as those in Sacrobosco’s Sphaera, although Fine does not explicitly acknowledge this inheritance, which reveals the traditional and omnipresent character of this textual model in sixteenth-century treatises on the sphere (Mosley 2009; Oosterhoff 2015; Valleriani 2017). Sacrobosco’s name is indeed never mentioned, as Fine rather refers in general terms to previous authors, thereby acknowledging the existence of an established tradition. He does not either tacitly take up any parts of Sacrobosco’s text, as was sometimes done in early modern books pertaining to the theory of the sphere (Valleriani 2017, 428). In most pre- and early-modern treatises on spherical astronomy and on cosmography, the imprint of the canonic model of Sacrobosco’s Sphaera remains underlyingly present, as shown also by other cosmographies written and published before Fine’s Cosmographia, such as the Cosmographiae introductio of Martin Waldseemüller (1470–1520) and Matthias Ringmann (1482–1511) (Waldseemüller and Ringmann 1907), the Liber cosmographicus of Peter Apian (Apian 1524),⁹⁴ or the Rudimentorum cosmographiae of Johannes Honterus (1498–1549) (Honterus 1535, 1440–63).

The fact that Fine never mentions Sacrobosco’s name in this context, even as an authority among others, may seem paradoxical given the importance of the Tractatus de sphaera in Fine’s early career as a master of the Faculty of the Arts and as an editor of scientific books, but also given the clearly identifiable imprint of Sacrobosco’s treatise on the structure of the Cosmographia, to which could be added its significance for the development of early modern cosmography more generally. The absence of any explicit mention of Sacrobosco in the Cosmographia, along with the fact that he does not (even tacitly) quote Sacrobosco’s text, may be due to Fine’s will to detach himself in name and in principle from what could by then be considered as “the old sphere” in order to promote his own version of “the new sphere”—to take up the distinction between the theorica vetus and the theorica nova—⁹⁵ while surreptitiously basing the latter on the former. The will to revise the doctrine of the sphere devised by Sacrobosco’s Sphaera transpires in particular through Fine’s criticisms of the literary parts of the traditional teaching on the sphere, which is one of the distinctive marks of Sacrobosco’s treatise, as will be shown later.

The fact of following the textual content and design of the Sphaera without mentioning the name of Sacrobosco is not unheard of over its period of diffusion in print. Matteo Valleriani (Valleriani 2017, 427–28) established that, among the nearly 400 different printed treatises that may be counted as belonging to the tradition of the Tractatus de sphaera between 1472 and 1697, a certain number of works relate to Sacrobosco’s treatise by their structure and by their visual material without mentioning Sacrobosco’s name (Chap. 1).⁹⁶ Even among the works that quote Sacrobosco’s text (entirely or partially), such as the Elementa sphaerae mundi sive cosmographiae in usum Scholae Mathematicae Basilensiis of Peter Ryff (1552–1629) (Ryff 1598) analyzed by Matteo Valleriani (Valleriani 2017) or the treatise on the sphere of André do Avelar (Avelar 1593), considered in this volume by Roberto de Andrade Martins (Chap. 10), there are cases of treatises that do not feature his name. Moreover, sixteenth-century cosmographies, and in particular those mentioned above, do not mention Sacrobosco’s name as the main source of their doctrine of spherical astronomy either, in spite of their reappropriation of parts of the Sphaera’s content, design and images.

Although Fine’s Cosmographia does not explicitly relate to Sacrobosco’s Sphaera and does therefore not feature any parts of Sacrobosco’s text, but rather aims to offer a new teaching on the worldly sphere,⁹⁷ the first edition of the Cosmographia maintains the style of a commentary on a canonic text, as would a commentary on the Sphaera, each chapter starting by enunciating a general teaching on the topic at stake and offering, in a separate section, a commentary on this teaching printed in a smaller font (Fig. 8.1). In the main part, each element of teaching is indicated by a letter in superscript, which allows us to identify the commentary in the second part, marked by the corresponding letter in the margin. This commentary-type exposition disappeared in the subsequent editions and translations of the Cosmographia, apart from the unabridged version of 1542 (Fine 1542a), where the main text is enriched with portions of the initial commentary.⁹⁸ The fact that this textual disposition was intended as a form of commentary is confirmed by the subtitle on the main title-page of the Cosmographia⁹⁹ and was made explicit by Cosimo Bartoli in his Italian translation of the Cosmographia (Fine 1587).¹⁰⁰ This textual layout, which clearly confirms the pedagogical aim of this work, did not commonly appear in contemporary cosmographical treatises.

Fig. 8.1

The commentary-like layout of the Cosmographia. The main teaching is clearly separated from the commentary and referred to in its different sections by letters placed in the margins. From (Fine 1532). Augsburg, Staats- und Stadtbibliothek—2 Math 30, fol. 112v, urn:nbn:de:bvb:12-bsb11199761-8

With regard to the division and ordering of its content, Fine’s Cosmographia follows quite closely the structure and thematic division of the Sphaera, and this more than other early sixteenth-century cosmographical treatises, including those mentioned above, which put a greater emphasis on geography, and especially on descriptive geography and on the study of populations (Mosley 2009). Indeed, in the cosmographies of Waldseemüller/Ringmann (Waldseemüller and Ringmann 1907), Apian (Apian 1524), or Honterus (Honterus 1535), the topics dealt with by Sacrobosco in the first three chapters are treated only partially and/or superficially, for instance as a preliminary introduction to its geographical part. In comparison, the topics dealt with by Sacrobosco in these first three books are extensively dealt with in Fine’s Cosmographia and in practically the same order. Admittedly, some topics which were tackled separately by Sacrobosco, and which were marked as distinct sections in previous printed editions (also in Sacrobosco and Fine 1516), were sometimes brought together in one chapter (as were the tropics, the polar circles, and the five zones); and notions (such as natural days) which, on the contrary, were not dedicated a specific exposition in Sacrobosco’s text, constitute the subject of a separate chapter in Fine’s work. This denotes a will, on Fine’s part, to reorganize and clarify the content of the traditional teaching on the sphere and to make it more accessible to readers less familiar with it.

The main structure of the Cosmographia is also slightly different from that of the Sphaera. Although the topics considered by Sacrobosco in the first two books (on the general structure of the world and on the circles dividing the celestial sphere) are respectively dealt with in the first two books of the Cosmographia,¹⁰¹ the topics discussed in Sacrobosco’s third book are distributed in two books (book III and IV) and extend to a part of the third book (book V), therefore occupying the last three books of Fine’s treatise—that is, book III for the rising and setting of the signs,¹⁰² book IV for the motion of the sun and its influence on the duration of light and shadows at different latitudes on Earth,¹⁰³ and book V for the theory of climates, which is then integrated into the geographical part of the Cosmographia.¹⁰⁴ Moreover, certain complementary chapters pertaining to the more modern teaching of the sphere are occasionally inserted between some of the more traditional chapters and various elements of a more practical nature are added in the last four books. Yet, in spite of these differences, Fine’s Cosmographia stands out among the sixteenth-century cosmography treatises by its strong focus on the theory of the sphere,¹⁰⁵ highlighted in particular by the fact that the astronomical section of the work covers four out of five books,¹⁰⁶ and also by the fact that, within this part, the disposition and structure of Sacrobosco’s Sphaera remains overall clearly identifiable (Mosley 2009).¹⁰⁷

More precisely, the topics common to the first book of the Sphaera and to the first book of the Cosmographia¹⁰⁸ are the distinction between the elementary and the heavenly regions, along with their respective divisions,¹⁰⁹ the motions and the sphericity of the heavens,¹¹⁰ the immobility, sphericity, and centrality of the earth in the middle of the universe.¹¹¹ It may be noted here that, in the Theorique des cielz (Fine 1528), Fine added to his exposition of Peurbach’s planetary theory a preliminary exposition on the general structure of the universe, which presents the above-described content.¹¹²

In the second book of the Cosmographia, Fine followed the model of the Sphaera by presenting the various astronomical circles that divide the worldly sphere¹¹³—namely, the equinoctial or celestial equator (along with the poles of the world),¹¹⁴ the colures,¹¹⁵ the meridians and the horizons,¹¹⁶ the tropics, the polar circles, as well as the zodiac, the ecliptic and the various modes of division and representation of the zodiacal signs in the sphere,¹¹⁷ to which adds the division of the heavens into five zones.¹¹⁸ All these circles were considered again in the first chapter of the fifth book, through their projection onto the terrestrial globe.¹¹⁹

The topics dealt with by Sacrobosco in the third chapter of the Sphaera are those with which Fine dealt most extensively, as they cover books III and IV, and a part of book V. The third book, which deals with the risings and settings of the signs, allows Fine to present the distinction between cosmic, chronic, and heliacal risings,¹²⁰ as well as the distinction between right and oblique ascensions.¹²¹ Book IV, which deals with the motion of the sun and its effect on the duration of daylight, tackles the inequality of natural days¹²² and the difference between artificial day and night.¹²³ Book V, which deals with the second main division of cosmography (geography), considers the distinction of the climates.¹²⁴ The remaining topics dealt with in book V, as well as some topics considered in books II to IV, do not belong, strictly speaking, to the list of topics considered by Sacrobosco and will be presented later.

The relation between the Cosmographia and the Sphaera is also made clear by the use of illustrations similar to those found in prior editions of Sacrobosco, notably in the editions printed in Venice in the late fifteenth century and in the early sixteenth century, to which relate some of the engravings Fine produced for his own edition of the Sphaera and for the Cosmographia, though the style of the drawing is perceptibly different (Pantin 2010).¹²⁵ Compare, for example, the illustrations in the first book of Sacrobosco’s Sphaera in the 1488 edition—as indicated in (Chap. 9), it was the first printed edition of Sacrobosco that included the complete set of “Venetian Sacrobosco figures”—with those in Fine’s edition and in the Cosmographia for the revolution of the heavens (Figs. 8.2, 8.3 and 8.4). In the Cosmographia, the engravings are both richer in detail when it comes to the representation of the terrestrial globe and of mathematical instruments,¹²⁶ and more abstract when representing the geometrical configuration of parts of the celestial sphere and the modes of computation of the positions of stars (Pantin 2010; Mosley 2009).¹²⁷ It is interesting to note here that there is a great difference between these works in the depiction of the sphere of earth when representing the mutual disposition of the elementary spheres (Figs. 8.5, 8.6 and 8.7), as, in Fine’s edition of Sacrobosco (Fig. 8.6), contrary to the Venetian edition (Fig. 8.5), the spheres of earth and of water are drawn in the form of a single orb. The relative similarity with the corresponding illustration in the Cosmographia (Fig. 8.7), which is also found in the Theorique des cielz (Fine 1528, 3r), indicates that Fine then wished to introduce an updated knowledge of the relation between the spheres of earth and water. However, the representation of the nesting orbs that divide the world according to the number of planets and according to the different motions of the fixed stars is different in Fine’s edition of the Sphaera (Sacrobosco and Fine 1516, sig. a2r) and the Cosmographia (Fig. 8.7), since for the former (which, on this topic, closely follows the Venetian editions of the Sphaera), there are nine spheres, as was taught by Sacrobosco, while for the latter there are only eight celestial spheres, according to what Fine taught in this context (as will be shown in the last section).

Fig. 8.2

De caeli revolutione. The representation of the circular motion of the stars in the Venetian incunabula editions. From (Sacrobosco et al. 1488, sig. a8v:). HAB Wolfenbüttel: 16.1 Astron

Fig. 8.3

De caeli revolutione. The representation of the circular motion of the stars in Fine’s edition of the Sphaera. From (Sacrobosco and Fine 1524, sig. a3r). Courtesy of the Library of the Max Planck Institute for the History of Science, Berlin

Fig. 8.4

De caeli revolutione. The representation of the circular motion of the stars in the Cosmographia. From (Fine 1532). Augsburg, Staats- und Stadtbibliothek—2 Math 30, 105r, urn:nbn:de:bvb:12-bsb11199761-8

Fig. 8.5

Quae forma sit mundi. The disposition of the elementary spheres according to the Venetian incunabula editions. From (Sacrobosco et al. 1488, sig. a8r). HAB Wolfenbüttel: 16.1 Astron

Fig. 8.6

Divisio sphaerae mundi elementaris regio. The disposition of the elementary spheres according to Fine’s edition of the Sphaera. From (Sacrobosco and Fine 1524, sig. a2v). Courtesy of the Library of the Max Planck Institute for the History of Science, Berlin

Fig. 8.7

De coelestium orbium numero, atque positione. The disposition of the elementary and celestial spheres according to the Cosmographia. From (Fine 1532). Augsburg, Staats- und Stadtbibliothek—2 Math 30, 104r, urn:nbn:de:bvb:12-bsb11199761-8

If the structure of Sacrobosco’s treatise may be easily recognized behind the list of topics and thematic divisions of the Cosmographia, this treatise was still an occasion for Fine to provide an expanded and up-to-date teaching on the various topics dealt with in the Sphaera (Pantin 2010, 2013a; Mosley 2009)—in this, it was not so different from the various sixteenth-century editions of Sacrobosco that expanded the original text by collating it with complementary material (Crowther et al. 2015; Valleriani 2017; Pantin 2013b). It was also an occasion for Fine to offer a teaching on his practice of cartography, materialized by his own terrestrial or regional maps (Fine 1525, 1531b, 1536), some of which were produced before he was enrolled as a royal lecturer and which undoubtedly contributed to his recognition by the French court (Conley 1996, 115–32; Dupèbe 1999, II, 530, 541; Brioist 2009b; Pantin 2009a, 2010, 2013a).

In wanting to provide a modernized teaching of the sphere, Fine chose to change or leave aside several chapters he very likely judged obsolete or irrelevant to the learning required in this framework, or which simply repeated elements already taught in previous parts of the Protomathesis, as was the case for the preliminary definitions of the geometrical sphere by Euclid (3^rd century BCE) and Theodosius (ca. 160–ca. 100 BCE), which Sacrobosco included at the beginning of the first book and which Fine had presented beforehand in the geometrical part of the Geometria libri duo.¹²⁸ He also left aside the distinction between the division of the sphere according to substance (secundum substantiam) and according to accident (secundum accidens), which was probably due to the fact that such a distinction would be tacitly expounded afterwards when dealing with the composition of the heavens¹²⁹ and when appealing to the distinction between right and oblique horizons.¹³⁰

As in the other cosmographical treatises mentioned above, the part concerning the theory of the planets was completely left aside, with the exception of the theory of the sun, which was then integrated into the chapter on natural and artificial days. The fact that the motion of the planets is not dealt with in this context was very likely due to its belonging to a different section of the teaching of astronomy within the traditional mathematical curriculum (Barker 2011; Valleriani 2017). Fine, in the division of astronomy he presents in the preface of the first edition of his Cosmographia, implicitly points out that, although the theory of the sphere and the theory of planetary motions both belong to the same type of knowledge (mathematical or theoretical astronomy), they represent distinct subdivisions of this teaching, the theory of the sphere representing the first part and the theory of planetary motions, the second part.¹³¹ Thus, for Fine (in accordance with the common model for the teaching of astronomy in medieval and Renaissance faculties of the arts), if spherical astronomy could be taught independently from planetary theory, the teaching of planetary motions required at least a basic understanding of the theory of the sphere, as shown by his Theorique des cielz, in which he included a general description of the cosmos, which was absent from the original text of Peurbach’s Theorica planetarum.¹³² In the Cosmographia, Fine also left aside the section dealing with eclipses presented by Sacrobosco in the fourth chapter, whose absence in this context is less expected, as eclipses related to the theory of the motion of the sun and were necessary at the time for the calculation of longitudes, as indicated in the third chapter of book V.¹³³

Fine also expressed strong reservations in the second book concerning the various geometrical representations or divisions of the twelve zodiacal signs which are described in the Sphaera (Sacrobosco and Fine 1516, sig. b1v-b2r),¹³⁴ stating that these imaginary representations are not only fictitious and useless, but also entirely alien to the contemplation of the mathematician.¹³⁵ He furthermore manifested the will, in the third book, to distance himself from the literary approach adopted in the Sphaera, where ancient Roman authors (Virgil, Ovid, Lucan) are regularly quoted. As already mentioned, he criticized this approach as violating the mathematical purity of the teaching of the sphere¹³⁶ and thus clearly positioned himself on this aspect against the form and style of Sacrobosco’s teaching of astronomy.¹³⁷ The authors he referred to (ancient and modern) in this context are indeed mostly mathematicians, astronomers, and philosophers: Aristotle, Euclid, Eratosthenes (ca. 276–ca. 195 BCE), Ptolemy, al-Farghānī,¹³⁸ al-Battānī (850–929), Averroës (1126–1198), Abraham ibn Ezra (1089–ca. 1167), Campanus of Novara (ca. 1220–1296), Alfonso X of Castile (1221–1284), Nicolas of Cusa (1401–1464), Georg Peurbach, Johannes Regiomontanus, Johannes Werner (1468–1522), Albert Pigghe (1490–1542), and Agostino Ricci (1512–1564). The absence of Sacrobosco’s name, within this relatively long list of authorities, stands out all the more.

Besides the integration of new authorities, Fine made many more additions. He first of all quasi-systematically added examples to illustrate the meaning of the theoretical and practical teaching he provided. These examples, which are present throughout the five books of the Cosmographia, often consist of commented figures, showing, for example, the position of a star in relation to the equinoctial or the ecliptic in order to explain its mode of calculation. When a computational procedure is taught, numerical examples are used.

Fine also completed or reassessed some of the notions dealt with by Sacrobosco in the Sphaera by comparing, for example, the notions of declination and right ascension with the notions of latitude and longitude of stars and their respective relations to the celestial equator and to the ecliptic in chapter 3 of book II.¹³⁹ He furthermore added various chapters and subsections containing practical material (which, however, maintain an overall theoretical scope),¹⁴⁰ pertaining, for instance, to the positioning of stars, to the measurement of the altitude of the sun, as well as to judicial astrology,¹⁴¹ as Jacques Lefèvre d’Étaples had done in his commentary on Sacrobosco’s Sphaera (which Fine contributed to reprint in 1521) (Chap. 2). These additional elements did not necessarily bring forth new knowledge and, for some, were mainly meant to offer complementary or up-to-date information required by the practice of astronomers and cartographers on the basis of concepts enunciated in Sacrobosco’s Sphaera.¹⁴² For instance, a few of the tables provided by Fine are similar to those found in Lefèvre’s commentary on Sacrobosco, although the format and the compiled data (which Fine claims to have computed himself) are slightly different.¹⁴³

In chapter II.4, Fine presented the means to find the declination of the ecliptic for any degree of the equinoctial.¹⁴⁴ He then also described the mode of fabrication and the use of the quadrant to measure the altitude of the sun and provided a table for the declination of the sun, along with the instructions on how to compute and use it.¹⁴⁵ After dealing with the celestial circles taught in Sacrobosco’s second book, he introduced, in chapters 8 and 9 of book II, circles relating to the horizontal positioning of the viewer—namely, the perpendicular and parallel circles that divide the sphere according to the horizon (verticals and circles of altitude)¹⁴⁶ and the circles of hours and their role in the constitution of sundials.¹⁴⁷ Chapter 10, which is the last of book II, deals with the astrological division of the heavenly sphere into twelve houses, where Fine compared the modes of distinction of the celestial houses adopted respectively by Campanus and by Regiomontanus, as well as its application to the astrological chart.¹⁴⁸

In chapter 3 of the third book, Fine introduced instructions on how to compute the ascensions of the parts of the ecliptic in the right sphere for each degree of the equator from the vernal equinox,¹⁴⁹ along with the corresponding table.¹⁵⁰ Similar material is given in the chapters III.4 and III.5 for the computation of the arcs of ascension of the ecliptic in the oblique sphere, calculated for the latitude of Paris,¹⁵¹ which includes a table of the ascensional differences between the arcs of the zodiacal signs in the right and oblique spheres,¹⁵² as well as tables for the oblique ascensions of the signs for the latitudes of 48°40′ on the northern and southern hemispheres for each degree of the equator,¹⁵³ as well as corresponding tables for the total ascensions of each sign for the same latitudes.¹⁵⁴ Fine then also provided a table for the oriental and occidental latitudes of the rising and setting of the arcs of the ecliptic (and thus of the sun) for the latitude of Paris.¹⁵⁵ In this fifth chapter of book III, he also added a part on the determination of the rising degree of the ecliptic, also called the ascendant or the horoscope,¹⁵⁶ and of the beginnings of the remaining astrological houses, according to the methods of Campanus and of Regiomontanus.¹⁵⁷

In the fourth book, when dealing with the inequality of natural days, Fine replaced Sacrobosco’s discourse on the annual spiral motion performed by the sun along the ecliptic in the course of a year by the theory of the sun’s motion found in Peurbach’s Theorica planetarum.¹⁵⁸ The aim of this teaching, as Fine put it, is to help understand the difference between natural days and show how to obtain the true motion of the sun from its mean motion and vice versa.¹⁵⁹ In this part, he used the illustration of the sun’s motion which he engraved for his Theorique des cielz (Fine 1528; Pantin 2010).¹⁶⁰ However, this part is not a full substitution, since in the next chapter (chapter IV.2) he resorted to Sacrobosco’s description of the daily parallel circles obtained from the annual spiralling of the sun from one tropic to the other to describe the differences between day and night in the right and oblique spheres (Fine 1532, 133v–34r). In this chapter, Fine taught the means by which astronomers determine the lengths of artificial days and nights for each degree of the ecliptic for any latitude,¹⁶¹ providing a table of the lengths of artificial days for the latitude of Paris,¹⁶² as well as the means to calculate the duration of the longest artificial day for each degree of latitude from the equator to the North Pole, along with a corresponding table.¹⁶³

The third chapter of Book IV considers the distinction between the equal and the unequal hours that divide artificial days and nights according to the latitude of the viewer and shows how to calculate the length of unequal hours for the latitude of Paris, as well as how to reduce unequal hours to equal hours and vice versa.¹⁶⁴ Fine also explained at this occasion the correspondence between the planets (and their rising in the first hour of the artificial day) and the names of the days of the week (Saturn on Saturday, the sun on Sunday, etc.), which he represented through a little table also indicating the planets ruling the first hour of the night, as well as the means to determine the planets ruling the other planetary hours for any day of the week.¹⁶⁵

In chapter IV.4, which deals with the altitude of the sun and with the shadows produced by the sun for different parts of the world, Fine explained the distinction between umbra versa and umbra recta, and introduced the means to compute the lengths of shadows with the help of the geometrical square (quadratum geometricum) placed on the back of planispheres or astrolabes,¹⁶⁶ which he complemented with a corresponding table indicating the lengths of shadows according to the altitude of the sun.¹⁶⁷ He also taught the means to calculate the elevation of the sun at a given place,¹⁶⁸ along with a table indicating the altitude of the sun for the latitude of Paris at each hour of the day throughout the year.¹⁶⁹

In book V of the Cosmographia, as noted above, Fine taught the principles of geography, hydrography, and chorography in the tradition of Ptolemy’s Geography (Brioist 2009b) and according to the previously mentioned cosmographical projection onto the terrestrial globe of the celestial circles and zones described in the second book of Sacrobosco’s Sphaera. This book also includes the theory of the climates and the ancient and modern distinctions of winds. Chapter V.1 considers in particular the various circles projected onto the terrestrial globe with the addition of the 89 parallels that divide each hemisphere of the terrestrial globe horizontally at one degree intervals from the equator and which enable, by their intersection with the meridians, to determine the longitudinal and latitudinal coordinates of the various places on earth.¹⁷⁰ For this, Fine provided a table indicating the circumference of a quadrant for each parallel in degrees of the equator and the quantity of their respective longitudinal degrees in minutes and seconds of arc.¹⁷¹ Chapter V.2 properly deals with the theory of the climates, presenting the twenty-four parallels that enable one to distinguish them. In this context, Fine criticized the distinction of seven climates found in Sacrobosco’s Sphaera and in earlier teachings on the sphere, which he attributed to the limited knowledge of their authors concerning the boundaries of the habitable world.¹⁷² He then gave instructions on how to calculate the height of the pole from the given length of the artificial day for each degree starting from the equator¹⁷³ and a table indicating the distance from the equator of each parallel delimiting the beginning and the end of a climate zone, as well as the corresponding maximum length of artificial days.¹⁷⁴ This table also indicates, for comparison, the situation of the seven climates of the earlier tradition of the sphere (7 vulgaria climata).

In the third chapter of book V, Fine taught the geographical notions of longitude and latitude of terrestrial places, as well as the notions of longitudinal and latitudinal differences, along with their means of calculation through lunar eclipses.¹⁷⁵ He then provided a table of the longitudes and latitudes of various cities in France, Germany, the Italian peninsula, Spain, Sicily, Sardinia, Corsica, Ireland, Scotland, and England.¹⁷⁶ Chapter V.4 teaches how to measure the distances between places and the correspondence between the distances measured in terrestrial units of lengths (in paces, miles, leagues, and stadia) and in degrees of great circles, using the method given by Ptolemy in his Geography.¹⁷⁷ In the fifth chapter, Fine taught the means to measure the distance between two places from their respective longitudes and latitudes.¹⁷⁸ The following chapter deals with the hydrographical distinction and classification of winds, ancient and modern, into twelve and thirty-two different winds, respectively, which Fine represented through compass roses, as well as through a small table for the ancient distinction comparing the Latin and Greek names of the twelve winds.¹⁷⁹ In the chapter V.7, which is also the very last chapter of the Cosmographia, Fine presented the cartographical procedures necessary to produce regional maps (chorography), which he illustrated through a delineation of the French borders on a coordinate grid indicating the location of Paris, as well as various projection techniques drawn from Ptolemy’s Geography for the mapping of an eighth or a half of the terrestrial globe (Brioist 2009b).¹⁸⁰

As noted by Adam Mosley, Fine’s geographical doctrine in the Cosmographia does not convey the will to present the new world discoveries, as did, on the contrary, the cosmographies of Apian (Apian 1540), Waldseemüller (Waldseemüller and Ringmann 1907) or Sebastian Münster (1488–1552) (Münster and McLean 2016; Mosley 2009).¹⁸¹ This is all the more peculiar in light of the fact that in 1531 Fine himself cast a Nova et integra universi orbis descriptio (Fine 1531b)—in parallel to preparing the Protomathesis for publication—which offered a cartographical representation of the known world, that is, featuring the new geographical discoveries, according to a bi-cordiform projection. Generally speaking, for a treatise of cosmography which dedicates a separate book to geography and which furthermore signals in this geographical part the heritage of Ptolemy’s Geography, the Cosmographia contains very little concrete information on the actual locations and contours of the various terrestrial regions, mainly offering methodological elements for the practice of cartography and a list of geographical coordinates for European cities situated in Europe.¹⁸² It therefore seems that Fine’s doctrine on the topic, as is the case for the rest of the material examined here, mostly aimed to correct and complete the theory of the sphere transmitted by Sacrobosco and its tradition, teaching in a rather theoretical manner the principles and methods used by cosmographers to determine the location of a place on earth and to represent terrestrial regions or the entire terrestrial globe on a map.

In the Cosmographia, Fine therefore followed in its broad outline the structure of the teaching provided by Sacrobosco, taking up the design and style of the Sphaera while leaving aside its text, to provide an updated and enriched theory of the sphere. In doing so, he perfectly illustrated the openness and yet the stability of the design of the Tractatus de sphaera in the sixteenth century, as described by Richard Oosterhoff and Matteo Valleriani (Oosterhoff 2015; Valleriani 2017). Indeed, Fine’s treatise introduced elements of knowledge exterior to Sacrobosco’s teaching and at times drawn from different disciplines, but which were traditionally related to the early modern teaching on the sphere and which thus fit appropriately with the thematic structure of the Sphaera. The knowledge Fine added to Sacrobosco’s teaching of the sphere in the Cosmographia is drawn from practical astronomy (astronomical computation procedures), instrument-making, judicial astrology, metrical geography, and cartography. Hence, if this list of additional elements is compared with the list compiled by Valleriani (2017) of the different disciplines that were associated with Sacrobosco’s text by various sixteenth-century editors or commentators, Fine’s Cosmographia appears to be quite similar in its form and intention to later sixteenth-century editions of Sacrobosco’s treatise (Crowther et al. 2015).¹⁸³

Moreover, the position of the Cosmographia within the Protomathesis may itself be conceived as the association of the traditional teaching of the sphere with disciplines and knowledge exterior to it. Indeed, within the Protomathesis, which was conceived from the start as a unified compendium (Pantin 2010), each part being required to make sense of the others, the Cosmographia holds both a central position¹⁸⁴ and a role of connecting link. Practical arithmetic and geometry (theoretical and practical), which precede the Cosmographia among the various parts of the Protomathesis, are both presented as necessary for cosmography, as it requires the methods of computation, the geometrical concepts and the measurement techniques and instruments provided by practical arithmetic and geometry. This is confirmed by the numerous references made to both treatises in the Cosmographia,¹⁸⁵ as well as by the preface of the Geometria libri duo, where Fine asserts their necessity for the astronomical teaching that follows.¹⁸⁶ This also corroborates the idea presented by Fine in the prefaces of his later editions of the Cosmographia that the learning and mastering of astronomy is the reason why the other parts of the quadrivium (that is, arithmetic, geometry, and music) should be studied. Moreover, the Cosmographia is set forth as the condition and therefore as the necessary introduction to the last part of the Protomathesis—that is, the De solaribus horologiis, et quadrantibus, which deals with the art of sundials.¹⁸⁷ Within this general teaching program, the theory of the sphere is both clearly distinguished from geography and presented as the condition of the apprehension and representation of the terrestrial globe and of its regions. It comes across, furthermore, as a necessary introduction to the astrological interpretation of the celestial influences. The situation of the Cosmographia within the Protomathesis thus allows us to regard the teaching of practical arithmetic, geometry, and dialing—in addition to the teaching of geography, cartography, and astrology—as a body of knowledge added to and associated with the traditional teaching of the theory of the sphere or as a set of complementary notions relevant to its study and application in a variety of contexts (Valleriani 2017). This again allows us to relate the composition of the Cosmographia, especially in its first edition, to the early modern practice of expanding the text of Sacrobosco, as was done in its later sixteenth-century editions and commentaries (Pantin 2013b; Crowther et al. 2015; Oosterhoff 2015; Valleriani 2017); (Chap. 5).

Fine’s Cosmographia, in spite of the fact that it never mentions Sacrobosco’s name, was in any case associated with the tradition of Sacrobosco’s theory of the sphere by later generations, as it was proposed as a possible alternative to Sacrobosco’s Sphaera in certain university teaching programs, but also because some of its illustrations were taken up in later editions of the Sphaera, such as certain in-octavo editions presented by Pantin in this volume (Chap. 9).

5 Cosmology in the Sphaera and in the Cosmographia

This section is in large part derived from the analysis developed in (Axworthy 2016, 211–38). To avoid repetitions, I will not refer again to this prior study in this section.

Beyond the similarities and differences between Fine’s Cosmographia and Sacrobosco’s Sphaera with respect to their content and thematic structure, an important point of comparison is their respective representations of the cosmos and their cosmological principles.

The Sphaera of Sacrobosco was at the time, and up to the seventeenth century, still considered a valid source to teach and learn about the general structure of the cosmos in academic and non-academic circles (Crowther et al. 2015), even if the cosmological stances of the Sphaera were discussed and confronted with alternative theses in many of its commentaries, and this from an early time (Thorndike 1949; Oosterhoff 2015) (Chaps. 6 and 7).¹⁸⁸ Such discussions may be found, for instance, in the Quaestiones of Pierre d’Ailly (1350–1420) (Sacrobosco et al. 1498) and in the extensive commentary by Francesco Capuano de Manfredonia (Sacrobosco et al. 1499; Shank 2009) (Chap. 4).¹⁸⁹

These discussions were not marked in the first decades of the sixteenth century by the strong debates raised in the late sixteenth and early seventeenth century on the cosmological models devised by Nicolaus Copernicus (1473–1543), Tycho Brahe (1546–1601), Johannes Kepler (1571–1630), and Giordano Bruno (1548–1600), but nevertheless acknowledged alternative models, such as those postulating the existence of homocentric planetary spheres, in place of the epicycles and eccentric spheres of Ptolemy.¹⁹⁰ In doing so, they contributed to opening the path to a progressive distancing from Sacrobosco’s Ptolemaic universe, qualifying to a certain degree its value for the study of cosmology. This likely helped emphasize its importance for the more practical aspects of astronomy and for its applications to cartography, navigation, and judicial astrology, as shown by the great number of editions containing complementary technical information regarding such domains in the second half of the sixteenth century (Crowther et al. 2015; Oosterhoff 2015; Valleriani 2017). It nevertheless remained for certain later commentators, such as Christoph Clavius (1537–1612) and Francesco Giuntini (1523–1590), an appropriate place to discuss cosmological ideas (Sacrobosco and Clavius 1570; Sacrobosco and Giuntini 1577; Lattis 1994; Pantin 2013b).

Although Fine, through his editorial work on both Sacrobosco’s Sphaera and on Peurbach’s Theorica planetarum in the years 1515–1516 (Pantin 2010, 2012, 2013a), contributed to the diffusion of the Ptolemaic geocentric cosmological model, he also illustrated, through his subsequent astronomical publications, his awareness of the problems raised by the cosmological models promoted by Sacrobosco and by subsequent followers of Ptolemaic cosmology, such as Peurbach (Aiton 1987; Barker 2011; Malpangotto 2013, 2016), with respect to the principles of natural philosophy.

In 1521, Fine edited the De motu octavae sphaerae of Agostino Ricci (Ricci and Fine 1521),¹⁹¹ in which an argumentation is presented against the existence of starless spheres (such as the ninth sphere admitted by Sacrobosco) (Johnson 1946; Pantin 1995, 442; Nothaft 2017)¹⁹² and which he followed in its broad outline in the Cosmographia, but also in the other astronomical works he published after 1521, at least through the visual representation of the cosmos he proposed in this context.¹⁹³ As indicated by Francis R. Johnson (Johnson 1946), the discourse of Ricci and Fine on this issue influenced Heinrich-Cornelius Agrippa (Agrippa 1531, F5v–F6r), a friend of Agostino Ricci,¹⁹⁴ but also Robert Recorde (ca. 1512–1558) (Recorde 1556, 10, 278–79) and Christopher Marlowe (1564–1593) (Marlowe 1604). Ricci and Fine’s opinion was also referred to in a critical manner by Giuntini and Clavius in their respective commentaries on Sacrobosco’s Sphaera (Sacrobosco and Clavius 1570, 68–69; Sacrobosco and Giuntini 1582, 17–19).

What Fine’s Cosmographia provided in this respect in the continuation of Ricci’s De motu was a reassessment of the cosmological model presented in the Sphaera and in later accounts of geocentric cosmology concerning the number of celestial spheres.¹⁹⁵ In discussing the order of the cosmos in the first book of the Cosmographia, Fine did not aim to present the various opinions regarding the representation and division of the universe, as did certain previous commentators of Sacrobosco,¹⁹⁶ such as Pierre d’Ailly (Sacrobosco et al. 1498; Duhem 1916, 168–71), Prosdocimo de’ Beldomandi (ca. 1380–1428) (Sacrobosco et al. 1531; Duhem 1913–1959, IV, 294–96; Markowski 1981), Francesco Capuano (Sacrobosco et al. 1508a; Shank 2009) (Chap. 4) in the fourteenth and fifteenth centuries, or Clavius at a later period (Sacrobosco and Clavius 1570), but rather to teach the version he judged to be physically true, as in the case of the mutual disposition of the spheres of water and earth, which he considered to form together a single orb instead of distinct concentric spheres, as was proposed in Sacrobosco’s Sphaera in accordance with Aristotle’s doctrine of the elements.¹⁹⁷ Yet on the issue of the number of celestial spheres, in addition to asserting his personal opinion, his intention was to offer a clear rebuttal of the theses he considered false.

It must be noted that Fine, in the Cosmographia as well as in the first pages of the Theorique des cielz (though in a much more concise version), was overall in line with the cosmological model transmitted by Sacrobosco, which is ultimately drawn in its broad outline from Aristotle’s De caelo, with the integration of Christian elements. Fine indeed defended in the first book of the Cosmographia the dual opposition between elementary and celestial regions, which compose together the whole universe, the elementary region representing the corruptible part of the universe and the celestial region, its incorruptible part.¹⁹⁸ He accepted the simplicity of the elements, as well as their quadripartition and mutual disposition within the sublunary region,¹⁹⁹ though he added to this (as was often done) their combination with the four sensible qualities (which Fine also represented as a diagram),²⁰⁰ as well as the Aristotelian tripartition of the region of air (Heninger 1977b 32–33, 106–07; Cosgrove 2007).²⁰¹ He also took up Sacrobosco’s argument of divine providence as an explanation for why the sphere of earth is not entirely covered by the sphere of water, judging insufficient the physical arguments traditionally brought forth, such as the absorbent power of earth or the influence of the stars.²⁰² He admitted, moreover, that the heavens are made of ether,²⁰³ as well as their essential incorruptibility and circular motion around the earth.²⁰⁴ He also accepted the sphericity and finiteness of the universe, taking up two of the three arguments given by Sacrobosco to prove that the sky is spherical—namely, the arguments of commodity and of necessity, leaving aside the theological argument of the similitude between the universe and the divine archetype or creator.²⁰⁵ Although he mentioned the argument Sacrobosco drew from al-Farghānī in favor of the sphericity of the heavens (according to which the sun would be closer when situated above us and further away when situated nearer to the horizon if the heavens were not spherical), he did not mention the theory of the refraction caused by the vapors of the atmosphere to explain why the sun actually seems closer to us near the horizon, perhaps because he was aware of the falseness of this theory (Pantin 2001).²⁰⁶

He also accepted the division of the heavenly realm into concentric contiguous spheres²⁰⁷ according to the different motions that take place in it, as well as the double motion of the heavens, from east to west (for the daily motion of the entire universe) and from west to east (for the particular spheres of the planets and of the firmament).²⁰⁸ He also accepted the order of the planets taken up by Sacrobosco from Ptolemy, despite the uncertainties and debates relating to the order of the sun, Mercury, and Venus (given the quasi-equality of their period of revolution), just adding to this the numbers given by Ptolemy and al-Battānī for the revolution of the fixed stars.²⁰⁹ In this context, he included a table to display the physical qualities of the planets.²¹⁰

If Fine assumed that the sphere of earth and the sphere of water together form a unified globe, he admitted, through arguments found in Sacrobosco’s Sphaera, that the resulting globe is situated at the center of the universe,²¹¹ is deprived of motion,²¹² possesses a spherical form,²¹³ and is of imperceptible magnitude in comparison to the universe, being therefore assimilable to a geometrical point.²¹⁴

As indicated above, the two chapters in which Fine deviated the most from the cosmological model transmitted by Sacrobosco and by later geocentric cosmological accounts are chapters 3 and 5 of Book I, where he respectively dealt with the number of celestial spheres²¹⁵ and with the mode of transmission of the diurnal motion (and of motion in general) from the primum mobile (or the first moved sphere) to the inferior spheres.²¹⁶

In these two chapters, Fine rejected the systems that admit the existence of one or several mobile spheres deprived of stars above the sphere of the fixed stars on account of their incompatibility with the principles of natural philosophy established by Aristotle (Johnson 1946; Heninger 1977b, 38–39; Pantin 1995, 442; Cosgrove 2007; Nothaft 2017). Such cosmological models, which were by far the most popular in the Middle Ages and in the Renaissance, included therefore not only the representation of the cosmos adopted by Sacrobosco, which postulated the existence of nine concentric contiguous orbs or spheres (with one starless sphere above the Firmament as the primum mobile), but also the models that admitted ten spheres, as illustrated, for example, by the representation of the universe provided in Apian’s Liber cosmographicus (Chap. 5) (with two starless spheres above the fixed stars).²¹⁷ Fine took up this discourse in the unabridged version of the second edition of the Cosmographia dating from 1542 in a shorter and slightly modified version (Fine 1542a, 3r, 5r), but not in the abridged version, nor in the later editions. However, he maintained in all these versions the diagram representing the heavens as composed of only eight spheres.

The number of distinct celestial spheres or heavens was an important issue in the development of premodern cosmology, especially in the context of treatises on the Sphaera, since the attribution of a distinct encompassing sphere for each distinct celestial motion stemmed from the will to account for the appearances while offering a cosmological model that conformed to the principles of natural philosophy. In this sense, it held a comparable status to the problem of the reality of Ptolemy’s epicycles and eccentric circles in the tradition of the Theorica planetarum, although it did not raise as many difficulties as the latter.²¹⁸ As will be shown later, Fine addressed both issues, although he mainly focused on starless spheres.

In this framework, the necessity to postulate the existence of one or several spheres above the Firmament, which was initially held to surmount the seven spheres of the planets and to enclose the whole universe, was due to the fact that the fixed stars (according to the observations of astronomers since Hipparchus (ca. 190–ca. 120 BCE)) were seen to move according to two distinct motions taking place in opposite directions and that (according to Aristotelian physics) it would be impossible for one single material body, especially a body made of pure and immutable matter, to properly move according to two distinct and opposite motions. In other words, as the sphere of the fixed stars was at first attributed the diurnal motion—that is, the east-to-west motion of the entire universe in twenty-four hours, since it is the most exterior of all celestial spheres and thus the only sphere able to carry the whole universe in its motion at once—it could not also be properly and simultaneously attributed the motion of precession of the equinoxes, according to which the fixed stars appeared to move of approximately one degree from west to east every century on the poles of the ecliptic, completing their revolution in thirty-six thousand years (according to Ptolemy, on the basis of Hipparchus’s discovery) (Neugebauer 1975, 54, 160, 292–97). For this reason, astronomers and natural philosophers admitted the existence of a sphere deprived of stars above the Firmament, which would be the proper cause of the diurnal motion of the universe. The eastward motion of precession of the equinoxes could therefore be properly attributed to the sphere of the fixed stars, as in the cosmological model depicted by Sacrobosco.

The later admission of a tenth sphere, and hence of an additional starless sphere above the Firmament, was based on the observation of a motion distinct from the diurnal motion and from the precession of the equinoxes in the trajectory of the fixed stars. Indeed, while the fixed stars were already seen to revolve (along with the planets) from east to west in twenty-four hours, and from west to east uniformly by one degree every century according to the precession of the equinoxes, the equinoxes (the first degrees of Aries and of Libra) were also observed to move back and forth in small circles over a period of seven thousand years, resulting in a complete revolution of the eighth sphere in forty-nine thousand years. This motion of oscillation of the equinoxes, which is described in a work entitled De motu octavae sphaerae and regarded as a Latin translation of a treatise by the Arab mathematician Thâbit ibn Qurra (Thâbit ibn Qurra 1960; Neugebauer and Thâbit ibn Qurra 1962), was commonly called by the Latin astronomers trepidation (trepidatio), or motion of access and recess (accessus et recessus) (Neugebauer 1975, 298, 598, 631–34; Nothaft 2017). While trepidation was initially admitted as a correction of the motion of precession of the equinoxes, it came to be considered in the Latin world as a motion independent of the latter, requiring it to be accounted for by a separate sphere, distinct from the ninth, to which the precession of the equinoxes had been previously attributed (Dobrzycki 2010 [1965]; Neugebauer 1975, 633; Grant 1994, 315–16; Nothaft 2017).²¹⁹

Although the addition of a ninth and a tenth sphere, just as the admission of partial orbs, stemmed from the will on the part of astronomers to account for the apparent motions of the stars through a system that would conform to the principles of Aristotelian natural philosophy (Morelon 1999) (at least to some of them),²²⁰ certain reservations were raised with regards to the ontological status of these spheres, as the fact that they did not carry any star made it difficult to prove their physical reality, just as it was to prove the existence of epicycles and of the eccentric spheres.²²¹ One of Sacrobosco’s early commentators, Robert Anglicus, objected to the existence of a ninth sphere in so far as it contradicted the principle that nature does not do anything in vain, since (as established by Aristotle) the orb only exists in order to carry a star (Thorndike 1949, 147). In line with this objection, some philosophers and astronomers added that if these were only fictional devices aimed to account for the apparent motions of the fixed stars, as did Averroës, Nicole Oresme (ca. 1320–1382), as well as Agostino Ricci they should be banished from the astronomical representation of the cosmos, given that there are other, more simple ways to account for the motion of the fixed stars (Ricci and Fine 1521; Oresme 1968, 488–91; Grant 1994, 319; Lerner 1996, I, 201–08).

The arguments Fine set forth in his Cosmographia against the existence of starless spheres (and therefore against the systems postulating more than eight celestial spheres) are founded on Aristotelian physical principles, but also on the assertion of its incompatibility with judicial astrology.²²² Following Agostino Ricci’s De motu octavae sphaerae (Ricci and Fine 1521), Fine intended to show in chapter I.3 that neither the principle established by Aristotle that one simple body cannot be properly attributed two different motions²²³ (and which is a key-argument for dividing the heavens into different ethereal spheres),²²⁴ nor the visible motions of the stars, compel us to believe that there are more than eight celestial spheres in the heavens.²²⁵ The only additional sphere he was willing to place above the sphere of the fixed stars is the Empyrean heaven, which had been admitted by his predecessors for theological reasons, as the abode of divine and holy beings.²²⁶ The Empyrean heaven would probably have been admissible for Fine because it was generally understood as deprived of motion, and therefore would not have compelled him to postulate any other motion than those attributable to the stars and to the whole universe.²²⁷

Appealing to the authority of Plato (ca. 427–347 BCE), of Aristotle, of Ptolemy, and of Averroës—followed, he claimed, by most mathematicians²²⁸—Fine described the celestial models which admit more than eight material spheres as dreams or fictions, saying (through a discourse that is very close to Ricci’s words) that “those who, against so many renowned authors, have imagined (for some of them) nine and (for most of them) ten spheres, violated, without being forced to it by any compelling reason, the number of solid celestial orbs.”²²⁹ Moreover, Fine said (again paraphrasing Ricci) that the more recent astronomers who defended the ten-sphere system “wrongly attributed such an extravagance to Ptolemy, to King Alfonso and to Johannes Regiomontanus.”²³⁰

As Ricci had done when he invoked the authority of Averroës in order to disprove the existence of starless spheres, Fine first appealed to the physical principle of simplicity or economy, stating that, in nature, a system that is simpler or that appeals to a smaller number of causes will always be chosen over a system that is more complex and that admits a greater number of causes than required.²³¹ In this context, the precession of the equinoxes would be accounted for through the distinction between the motion of the part and the motion of the whole.²³² By admitting that the precession of the equinoxes properly belongs to the eighth sphere, Fine supposed that the diurnal motion may be assigned to the latter only in so far as it partakes, just as any other celestial spheres, in the motion proper to the whole universe.

To explain how the universal motion can be transmitted to all the celestial spheres without having to be properly attributed to the last or highest sphere (as this sphere would otherwise need to transfer the diurnal motion to the inferior spheres by dragging the sphere immediately inferior and contiguous to it in its motion, which would then transmit it to the sphere immediately inferior to it, and so on), Fine adopted the causal model defended by Ricci, and before him by Averroës in his commentary on Aristotle’s De caelo, according to which the sphere of the fixed stars would move the whole world and each of its parts by transmitting to the inferior spheres the vital virtue necessary for them to start moving spontaneously, according to their individual trajectory.²³³ To defend this thesis, Fine appealed (as did Ricci) to the traditional comparison between the macrocosm (the universe) and the microcosm (the human being or the living being) to show that the whole body of the universe, just as the body of man or of the animal, can be said to move according to one motion, while its inner parts (the spheres, or the limbs of the living body) move according to distinct motions.²³⁴ These would nevertheless also be moved with the whole body according to its proper motion. In this framework, since the whole universe, rather than a separate sphere, is said to properly move from east to west, the eighth sphere may be considered as properly moved according to the precession of the equinoxes.

Within such a system, the attribution of the diurnal motion to the whole aggregation of the spheres, and not to one particular sphere, certainly renders obsolete the mechanical model commonly presented during the Middle Ages, notably in Sacrobosco’s Sphaera, to account for the transmission of the diurnal motion to the totality of the particular spheres—that is, through the raptus exerted by the higher sphere on the sphere immediately inferior to it and thereby on the rest of the spheres successively (Lerner 1996, 179–80, 188, 195–201; Grant 1997). Nevertheless, the concept of a first moved sphere, or primum mobile, to which the diurnal motion is properly attributed and which would transmit to the other spheres the power to move according to its own motion,²³⁵ is not entirely done away with, since this model still requires a sphere that receives the vital virtue of the whole universe in the first place and transmits it to the other spheres (including the spheres of the elements).²³⁶ The primum mobile, just as the heart, Fine says, is the first organ to receive the vital virtue of the living body, through which it will be transmitted to the remaining limbs or organs. Here, the primum mobile would still correspond to the highest sphere, because it is the most perfect sphere, being endowed with the most uniform motion and being the closest to the Empyrean heaven and to God (Ricci and Fine 1521, 10v; Grant 1994, 322–23; Lerner 1996, 204–05).²³⁷ Hence, Fine declares that it is “absurd and directly alien to philosophy, against nature and the order of things, to imagine new heavens on the Firmament and to dream of superfluous mobile circles without being compelled to it by reason or persuaded by experience.”²³⁸

In the cosmological model then depicted by Fine, no place is left for the trepidation or the oscillation of the equinoxes observed in the trajectory of the fixed stars, which Fine, like Ricci, simply rejected as physically impossible. As he presented the modern determination of the motions of the eighth sphere,²³⁹ he declared, explicitly referring to Ricci’s De motu (and agreeing with him on the rate of precession, following al-Battānī),²⁴⁰ that trepidation should not only be rejected because it contradicts the physical principle of uniformity of celestial motions, which (given the difformity of such a motion) cannot be rationally seen as going back and forth, but also because its admission would overthrow judicial astrology—that is, would call into question the validity of the astrological art and the predictions of its practitioners.²⁴¹

We do not want to ‘titubate’ (titubare)²⁴² any longer on this inconceivable motion (as when we follow the opinion of other astronomers), since we declare and openly acknowledge, driven by reason, that this opinion is the weakest, not to say the falsest of all, and was fallaciously imagined by the most pernicious and most ignorant disciples [of astronomy], causing the greatest damage to human beings by overthrowing judicial astronomy. For I know that there is nobody (who is not entirely deprived of philosophical knowledge) who denies that this highly irregular motion of celestial bodies is repugnant to all, as Agostino Ricci demonstrates in his small treatise.²⁴³

This argument is far from secondary here, given that judicial astrology, as the part of astronomy whose aim is to determine the influence of celestial motions on the events of the sublunary world, was, as mentioned earlier, regarded at the time as an important part of the activity of astronomers (and of mathematicians in general) within the community; it represented, for most people, one of the main incentives to study astronomy and mathematics in general. The fact that the Cosmographia contains several sections on astrology and on the casting of horoscopes confirms that this was not, to Fine, an anecdotic part of the astronomer’s activity. For that matter, Fine openly admitted the influence of celestial motions on sublunary events, considering the motion and light of stars to be the intermediaries by which the virtues of the celestial world are diffused into the sublunary world, as set forth in the chapter II.10 of the Cosmographia²⁴⁴ and in his works pertaining more specifically to astrology, such as the Canons des ephemerides²⁴⁵ and the De duodecim caeli domiciliis.²⁴⁶ All this hints to the expectations of students as well as of the audience of the royal lecturers with regards to the purpose of astronomy as it had been the case for the students of the Faculty of the Arts with respect to the study of Sacrobosco’s Sphaera.²⁴⁷

One of the main problems posed to astrologers by the admission of starless spheres was the complexity this added to the computation of the positions of planets with respect to the zodiacal signs. This matter already raised concerns in relation to the admission of the precession of the equinoxes, since the fact of attributing to the visible stars a proper motion from west to east along the poles of the ecliptic, in addition to the diurnal motion of the universe (as was taught in the Tractatus de sphaera), called into question the possibility of using the firmament as an immobile system of reference to calculate the motions of the planets and to determine their positions and conjunctions in relation to the signs of the zodiac. Therefore, astronomers posited the existence of another set of constellations on the ninth sphere, which would be the invisible images of the constellations of the eighth sphere and which served as a system of reference to determine the positions of the planets and the proper motion of the visible constellations.

As shown by discussions that were raised on this issue since antiquity,²⁴⁸ this practice was not only a problem in view of the invisibility of the constellations of the ninth sphere, which made it difficult to use these constellations as a system of reference in order to determine the motion of the planets and of the visible constellations, but also because these invisible stars were attributed an influence on the inferior bodies or at least an impact on the influence of the planets according to their position relatively to these invisible signs. Astrologers, therefore, had to take into account the influence of these invisible signs in their astrological predictions, which would introduce additional complications in the establishment of astrological charts. For example, certain medieval astrologers, such as Pietro d’Abano (ca. 1257–1316) in the Conciliator differentiarum philosophorum (Pietro d’Abano 1520, 14r), attributed an influence to both the visible signs and the invisible signs, and considered that the influence of the signs of the zodiac was stronger when the visible and the invisible signs were superposed (Chap. 4).²⁴⁹ But as the signs of the ninth sphere are invisible, it would be very difficult in practice to decide when the superposition takes place. Hence, for Fine, the fact of rejecting the existence of starless spheres (and of partially rejecting the cosmological system adopted by Sacrobosco) was not only a question of safeguarding Aristotelian cosmology, but also a question of guaranteeing the validity of the calculations and of the predictions of astrologers.

Although this argument only makes a small appearance in Fine’s rebuttal of the existence of starless spheres, it would be highly significant in this context and may have even been his main incentive to reject starless spheres.²⁵⁰ This could very likely have been the case for Ricci himself, who was a court astrologer at Casale Monferrato and, as mentioned above, a friend of Cornelius Agrippa (Johnson 1946; Goodrick-Clarke 2008). Indeed, in his De motu, Ricci clearly questions the ability of the more recent astrologers to offer a solid prognostic on the basis of the nine- or ten-sphere system, denouncing thereby the damage these systems caused to the practice of judicial astrology.²⁵¹ This argumentation seems to have been influenced by the Disputationes adversus astrologiam divinatricem of Giovanni Pico della Mirandola (1463–1494), who, in book VIII of this treatise, used the uncertainty concerning the numbers of the celestial spheres and the problems raised by the admission of immobile and invisible signs as additional arguments to dismiss judicial astrology.²⁵² Ricci referred to this discourse in the above-mentioned section of his De motu (Ricci and Fine 1521, 18r, 21v),²⁵³ yet not in order to dismiss the validity of astrology, but rather to dismiss the systems postulating nine or ten spheres.

Yet whether or not this was the main incentive for Fine’s rejection of starless spheres, this discussion was also an occasion for him to reassess the traditional representation of the cosmos in its relation to the principles of Aristotelian natural philosophy (which were followed by Sacrobosco with regard to the general structure of the cosmos and the nature of the elements) as well as of the role of the astronomer in determining the physical order of the universe.

With respect to the function of the starless spheres in the astronomer’s apprehension of the celestial order, an apparent tension however emerges in Fine’s discourse in the Cosmographia, since although he partly based his dismissal of starless spheres on their incompatibility with the principles of natural philosophy, he concluded chapter 5 (immediately after asserting the absurdity of trepidation) by conceding the usefulness of the astronomer’s fictions to account for the irregularities of the visible motion of the stars.

So all the things which the wisest astronomers have thought up above the eight sphere were only the imagination of immobile circles, through which they were able to regulate the motion of the Firmament and of the other orbs which are inferior to it. The same judgement should be passed on the particular orbs of the errant stars—that is, the epicycles and the eccentric spheres, and their very particular motions—as well as on similar inventions. These were subtly invented for the sole purpose of saving the apparent variety of each motion and to render the quantity of their irregular motions computable by the power of geometry.²⁵⁴

This discourse held an important place in Fine’s thought, as it first appeared in the Theorique des cielz at the beginning of the chapter on the motion of the eighth sphere (Fine 1528, 33r–v)²⁵⁵ and came up later on several occasions, notably in the second edition of the Cosmographia (unabridged version),²⁵⁶ but also in the 1532 edition of the Cosmographia when dealing with the sun’s theorica.²⁵⁷ It was also put forward in an unfinished manuscript draft of a work entitled Speculum astronomicum (Fine n.d.), which was intended to present the theoretical principles, the mode of fabrication, and the use of an instrument to determine planetary positions, and which directly referred to the Cosmographia. As Fine indicated in this text, starless spheres would have the same status as Ptolemy’s epicycles and eccentric spheres,²⁵⁸ all corresponding to abstract geometrical devices used by astronomers to account for the apparently irregular motions.

What one generally needs to know first is that all the things which the wisest astronomers have imagined concerning the number, the figure and the various motions of the celestial orbs have only been so in order to calculate the apparent irregularity of the celestial motions. And nobody would think (aside from he who is entirely deprived of philosophical knowledge) that each of these things really exists, since they were only invented through a geometrical and purely imaginary theory so that the true motions of the stars could be obtained. Indeed, the particular orbs of the heavens, which move around the center of the world (as may be seen from the Theory of the planets) would be about twenty-six, that is, leaving aside the epicycles and small orbs situated around them, which are adapted to the diversity of the motions. We have shown sufficiently clearly in the first book of our prior Cosmographia, and we will reveal it elsewhere in a more complete treatise (if God allows it), how absurd and directly alien to philosophers it is to admit their existence. We have to concede, therefore, whether we want it or not, that the divine and incomprehensible wisdom kept to itself the eternally admirable quality of the celestial motions, but has, in its merciful benevolence, granted to men the ability to apprehend and eventually calculate the quantity of these motions through a geometrical and abstract discourse.”²⁵⁹

This passage was entirely crossed out in the manuscript, but was initially placed within the first proposition of the treatise and shows Fine’s eagerness at this point to discuss the ontological status of the constructions necessary for astronomers to calculate and predict the apparent positions and trajectories of stars. As he explains it then, if starless spheres (as well as partial epicycles and eccentric spheres) are not physically real, they would be regarded as useful by astronomers for determining the positions and trajectories of the stars and planets, given that the human mind was not endowed by God with the ability to apprehend the true quality of the celestial order, though it was made able to access the true motions of the stars—that is, their visible position from the earth at any moment of their cycle—by means of geometrical and abstract devices.

Therefore, if, on one hand, Fine invited astronomers in the Cosmographia to dismiss starless spheres because they contradict the principles of natural philosophy and because they introduce unnecessary complexity into the practice of astrologers, he acknowledged, on the other hand (in this text as in the Cosmographia), the utility of such fictions, alongside epicycles and eccentric spheres, to account for the visible celestial motions, since the true causes and structure of these motions would remain incomprehensible and thus hidden to the human mind.

Yet from what Fine said in the Cosmographia in reference to Ricci’s De motu, this would not justify the admission of starless spheres beyond the starry heaven, since he held them not only as contradictory to the principles of natural philosophy, but also as mathematically irrelevant, since they would not be necessary to account for the variety of the motions of the fixed stars. Furthermore, he considered starless spheres as certainly more problematic than partial orbs in regard to the validity of judicial astrology, given that the knowledge of the motion of the fixed stars is more important to determine the true positions of planets in relation to the zodiacal signs than the causes of their stations and retrogradations.

Hence, although the astronomer would not be able to fully access the true order of the heavens, he should, to the extent of his abilities, still attempt to determine the nature of celestial substances as much as it is possible by always choosing the hypothesis that is most simple and best conforms to the principles of natural philosophy. This is why, in the Theorique des cielz, although Fine taught Peurbach’s theory of planetary motions and expounded the motion of trepidation, referring in this process to separate spheres for both precession and trepidation (as Peurbach had done), he represented the heavens as divided into only eight spheres at the beginning of the treatise (Fine 1528, 3r), just as he did in the frontispiece of the 1527 edition of Lefèvre’s commentary on the Sphaera and in the various editions of the Cosmographia.

What Fine then seemed to condemn in the Cosmographia and in the Speculum astronomicum is not so much the use of geometrical fictions for calculational purposes, but rather the fact of admitting them as physically real and also of appealing to them, even as calculational devices, when these are not necessary to account for the apparent motions of the heavens, especially when they have a role to play in the determination of the planets’s influence in the framework of judicial astrology, as in the case of the ninth sphere.

This discussion, therefore, brings forth the vexed issue of the reality and of the function premodern astronomers attributed to partial orbs and starless spheres in the description of the celestial order, be it according to the model defended by Sacrobosco or that transmitted by Peurbach.²⁶⁰ Now, if, in view of the incompatibility of the Ptolemaic astronomical model with the principles of Aristotelian natural philosophy, certain astronomers and philosophers restricted these models to mere calculational devices, which would only be fit to predict the apparent positions of the stars from the earth and to cast tables of ephemerides, this cannot be straightforwardly interpreted as a sign that they did not attribute to astronomy the right and the duty to investigate and to describe the true order of the heavens to the extent that it is humanly possible.²⁶¹ This is marked in particular by Ptolemy’s will to maintain the circularity of celestial motions and by his physical account of partial orbs in the Planetary hypotheses (Goldstein 1967; Morelon 1999), which was known to the Latin medieval and Renaissance astronomers through derived Arabic cosmological accounts (Lerner 1996, I, 94–99) (Chap. 6). As shown in particular by Peter Barker, there are also various examples, in the Middle Ages and in the precopernican Renaissance, of astronomers attributing physical reality to partial orbs and starless spheres (Barker 2011) (Chap. 6). This may also be confirmed by the fact that, in Almagest XIII.2, Ptolemy dismissed the opinions of those who rejected certain astronomical models (notably his own) on account of their complexity by qualifying the ability of the human understanding to decide on the degree of simplicity that is appropriate to divine realities (as celestial bodies and motions were considered to be) on the basis of what is simple in the elementary world.²⁶² This discourse of Ptolemy did not intend to fully validate the physical reality of his cosmological system, but it certainly aimed to restrict attacks on its physical possibility.

For Fine as well, the fact that it is not possible for the human mind to grasp the true quality of celestial motions in no manner means that the astronomer should not, as much as possible, attempt to investigate and describe the true order of the cosmos. This is not only indicated by his assertion of the physical impossibility of starless spheres and by his will to maintain an eight-sphere system in the general structure of the heavens (even in his adaptation of Peurbach’s Theorica), but is also suggested by his definition of astronomy in the preface of the 1532 edition of the Cosmographia, where astronomy (then specifically identified as theoretical or mathematical astronomy) is said to study the “celestial globes, stars, their motions, their accidents and things of the kind,”²⁶³ and more generally “the celestial body itself, the most illustrious of all bodies, which is absolutely deprived of alteration, is situated in the highest place, is the noblest and is endowed with circular motion, that is, with the first and most perfect of all motions,”²⁶⁴ a description which is derived from the Aristotle’s De caelo and which states the physical qualities of the celestial body. Moreoever, even if Fine’s rejection of starless spheres was primarily motivated by the will to safeguard astrology, this motivation never seems entirely separate from the will to determine the real position of the fixed stars in relation to the earth and the planets, since the very influence of the planets (itself considered physically real)—in other words the action operated by their light, heat, and motion (plus occult influences, when they were admitted) on the events occurring in the elementary world—is determined by their disposition in relation to the zodiacal signs.²⁶⁵

The discourse presented by Fine in the first edition of the Cosmographia concerning the number of celestial spheres thus shows that, in the context of a teaching on the sphere tacitly based on Sacrobosco’s Sphaera, which he approached with the intention of offering a properly up-to-date and complete teaching of its theoretical and practical aspects, he considered it important to establish the representation of the cosmos on a physically acceptable foundation, in particular as it played a crucial role for him in asserting the validity of astrology.

Rejecting in this manner the general structure of the universe and the type of celestial causality (raptus) defended by Sacrobosco, among many others, Fine’s Cosmographia demonstrates again the openness of the textual design of the Tractatus de sphaera, which allowed the transformation of various part of its content, notably concerning its cosmological stances, without disturbing the general structure of its teaching on the theory of the sphere (Oosterhoff 2015; Valleriani 2017).

6 Conclusion

The relation between Oronce Fine’s astronomical work and Sacrobosco’s Sphaera, from the edition of the Sphaera in 1516 to the Cosmographia in 1532, instantiates the royal lecturer’s various talents as a mathematician, a professor of mathematics, a cartographer, a maker of scientific instruments, an editor, and an engraver. Through his early contributions to the diffusion of the Sphaera in a format accessible to students, Fine demonstrated his active commitment to the mathematical curriculum of the Faculty of the Arts in his first teaching years at the Collège de Navarre. As he started to teach mathematics as a royal lecturer about fifteen years later, he offered his new audience a renovated teaching of spherical astronomy rooted in Sacrobosco’s Sphaera, in which he included the practical notions necessary to its application in judicial astrology, cartography, nautical geography, and dialing. Given the strong emphasis Fine placed on practical knowledge in the Cosmographia, this work constituted an important element of his project to transform the traditional mathematical curriculum.

In its relation to the Sphaera, the Cosmographia cannot be straightforwardly considered a commentary since it does not feature Sacrobosco’s text. However, it integrated many aspects of Sacrobosco’s teaching in its content and format, for which Fine’s treatise could be considered an appropriate alternative to the Sphaera in certain teaching programs of sixteenth-century institutions. For that matter, if Fine innovated on his 1516 edition of Sacrobosco’s Sphaera with regard to the layout of the text and by the inclusion of a few tables,²⁶⁶ the Cosmographia offered a complete renovation of the content of the traditional teaching of spherical astronomy from which the very name of Sacrobosco is entirely absent.

Within the Protomathesis, the theory of the sphere appears as a core teaching, demonstrating the necessity of arithmetic and geometry to study astronomy while providing the necessary principles for the study and practice of cartography, judicial astrology, and gnomonics. The relation of the Cosmographia to the other parts of Fine’s mathematical teaching, as well as his recurrent assertion of the higher necessity of astronomy for the contemplation of the Creation and of the Creator himself, thus allowed him to present this discipline as the crowning of the traditional quadrivium, as well as the condition of its fruitfulness.

The Cosmographia was also the occasion for Fine to express his opinion on the order of the cosmos and on the relation between mathematics and natural philosophy, as was the teaching of Sacrobosco’s Sphaera for several of its commentators. Showing the importance of offering a correct cosmological system in order to safeguard astrology, given its dependence on the knowledge of the true relation of the planets to the zodiacal signs, Fine also addressed the expectation of many students of the Faculty of the Arts at the time with regard to the applications of astronomy and of Sacrobosco’s Sphaera, notably for the practice of medicine.

Fine’s Cosmographia, which represented a means of disseminating Sacrobosco’s teaching on the sphere that differed in its form and intention from proper editions and commentaries on the Sphaera, also gives us an illustration of the various manners in which the content of this thirteenth-century elementary treatise was adapted to the needs of the sixteenth-century reader and how it contributed to the transformation and to the promotion of mathematical teaching in Renaissance France. While the name of Sacrobosco did not explicitly appear in works such as Fine’s Cosmographia—which in Fine’s case reveals an ambiguous relation to a source he himself edited and taught during his years at the Collège de Navarre—, the Sphaera remained a clearly identifiable and stable source at the time for the study and the application of the theory of the sphere, through which it was able to maintain, in European academic and non-academic scientific culture, the status of universal reference for the introductory teaching of astronomy until the seventeenth century.