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

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.


Introduction 1
Oronce Fine or Finé 2 (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, 3 but also for his work as a cartographer, 4 as a designer and maker of mathematical instruments, 5 as well as an engraver and an editor of scientific books. 6 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) 7 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 1554Borrel , 8 1559. 9 It is rather through his project to promote mathematical teaching in France, 10 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 time 11 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 disciples 12 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). 13 François I founded the institution of the royal lecturers on the suggestion of his librarian Guillaume Budé , 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. 14 The foundation of a chair of mathematics shortly after the creation of the first royal lectureships in Greek and Hebrew 15 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. 16 This is confirmed by the discourse held by Fine and by his successor Petrus Ramus (1515Ramus ( -1572 on the 10 On Fine's role in the development of mathematical teaching in sixteenth-century France, see (Ross 1971, 23-35;Margolin 1976;Tuilier 2006b;Pantin 2009a;Axworthy 2016, 27-37). 11 Bibliographical lists of Fine's works are proposed in (Ross 1971, 398-449;Ross 1974;Pantin 2013a;Axworthy 2016, 407-19). 12 On these authors and on their role in the diffusion of mathematics in the Parisian academic sphere, see (Rey Pastor 1934;Ross 1971, 11-15;Margolin 1976;Pantin 2009aPantin , 2013aOosterhoff 2015Oosterhoff , 2016Oosterhoff , 2018) (Chaps. 2 and 3). 13 Fine wrote this versified eulogy of mathematics, addressed to François I, either before or shortly after his nomination as a royal lecturer around 1530 (Ross 1971, 21;Margolin 1976;Pantin 2009aPantin , 2010Axworthy 2016, 18). 14 The first royal lectureships were dedicated to Greek and Hebrew and were attributed to Pierre Danès and Jacques Toussain for Greek and to François Vatable and Agathius Guidacerius for Hebrew. On the creation of these lectureships and on their first appointees, see (Irigoin 2006;Kessler-Mesguich 2006;Tuilier 2006a). Another institution which was oriented towards the pedagogical needs of humanists and which was founded in France in the same period is the Collège de Guyenne, established in Bordeaux in 1533. 15 On the date of creation of the first royal lectureships, at least on the dates of the first lessons provided in this framework, see (Lefranc 1893, 109;Ross 1971, 21;Tuilier 2006a). On the date of foundation of the first royal chair of mathematics in particular, see (Lefranc 1893, 394;Tuilier 2006a, b;Pantin 2009a;Axworthy 2016, 17-18). 16 On the teaching of mathematics at the University of Paris at the beginning of the sixteenth century, see (Margolin 1976;Tuilier 2006b;Pantin 2009a;Cifoletti 2009;Oosterhoff 2018, 78-79). ignorance of the students and masters of the Parisian Faculty of the Arts in mathematics (Dupèbe 1999, II, 523;Tuilier 2006a;Pantin 2006Pantin , 2009aOosterhoff 2015). 17 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), 18 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, 19 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). 20 Mixing theoretical 17 (Euclid 1536, sig. * 2r): "Dum celebres illas et fidissimas artes, quae solae Mathematicae, hoc est, disciplinae merverunt adpellari: raros admodùm offendi (etiam in numerosa auditorum multitudine) qui satis fido ac liberali animo, tam utile ac jucundum philosophandi genus, à limine (ut aiunt) salutare, ne dicam ad illius penetralia, penitioràque secreta, pervenire dignarentur…. Qui enim ad lauream adspirant philosophicam, jurejurando profitentur arctissimo, sese praenominatos Euclidis libros audivisse. An verò illius elementa, multis ab hinc annis, usque ad nostra viderint, ne dicam intellexerint, tempora (paucis forsitam exceptis, quos aequus amavit Juppiter) non ausim honestè confiteri." See also Ramus (Ramus and Talon 1599, 376): "Mathematicas artes adhuc in publicis Philosophici studii legibus et institutis nullum honorem, nullum separatim locum habuere, ut Mathematum prorsus ignarus, tamen legibus Parisiensis Academi, philosophicam lauream consequatur." (Pantin 2006;Axworthy 2016, 13-14). 18 On the mathematical textbooks read at the faculty of the arts at the beginning of the sixteenth century, see (Chap. 5). 19 This title, which corresponds to the Latin transliteration of a Greek term composed of the suffix προτο-(first) and of the substantive μάθησιϛ (teaching, knowledge), literally means "first teaching" and intends to assert thereby the propaedeutic function of mathematics (Axworthy 2009(Axworthy , 2016. 20 Although the Protomathesis displays a quadripartite structure, it does not perfectly mirror the structure of the quadrivium since Fine then replaced music (or the theory of consonances) with gnomonics. Fine did, however, contribute to the diffusion of the Boethian theory of consonances through the publication of the Epithoma musice instrumentalis ad omnimodam Hemispherii seu Luthine et theoricam et practicam in 1530 (Fine 1530), where this theory of harmony is also taught along its applications to the reading, composing, and playing of lute pieces. Thus, the fact that the Protomathesis does not include any treatise on music could be due to the fact that Fine had already published, or was about to publish, the Epithoma musice instrumentalis when he was assigned to the position of royal lecturer in mathematics and therefore did not consider it necessary to publish another treatise on music, or to publish again the text of the Epithoma within his mathematical compendium. Indeed, as indicated by the date of 1530 on the title pages of the Geometria and of the Cosmographia (Fine 1532, 49r, 101r), the publication process of the Protomathesis started at the latest in 1530. Notwithstanding, the fact that, in the Protomathesis, music is in a sense "replaced" by gnomonics proposes a representation of mathematics which differs from the tradi-and practical knowledge (from Euclidean geometry and the theory of the sphere to the construction and use of surveying instruments and sundials), 21 the publication of the Protomathesis was important for the new image it provided of mathematics, in France and beyond. 22 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, 23 also helped shape the Parisian style of printed scientific books (Pantin 2010, 2013a, Oosterhoff 2016. 24 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. 25 Among the latter, the two first books on which he worked as an editor were the Theorica planetarum of Georg Peurbach (1423Peurbach ( -1461, which was published in Paris in 1515 (Pantin 2009b(Pantin , 2012(Pantin , 2013a, 26 as he was studying at the Collège de Navarre, and the Tractatus de sphaera of Johannes Sacrobosco (died ca. 1256) 27 in 1516 (Pantin, 2009b(Pantin, , 2010(Pantin, , 2013aPettegree andWalsby 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, 28 tional model drawn from the model of the quadrivium (as represented, for example, by the mathematical section of the Margarita philosophica of Gregor Reisch, which Fine worked to edit around 1523) and which aims to promote practical mathematics. 21 On Fine's practical mathematics, see (Métin 2004;Guyot and Métin 2004;Dhombres 2006;Eagleton 2009;Dupré 2009;Brioist 2009a;Mosley 2009;Pantin 2006Pantin , 2009aPantin , 2010Oosterhoff 2014Oosterhoff , 2016Axworthy 2016, 249-300). 22 This work was translated into Italian (along with a treatise on burning mirrors, published in 1551) by Cosimo Bartoli in 1587 as Opere di Orontio Fineo del Delfinato divise in cinque Parti. 23 The Arithmetica practica was reprinted in 1535, 1542, 1544 and 1555; the second book of the Geometria was reprinted in 1544, 1556, 1558, 1584, 1586; the Cosmographia was reprinted twice in 1542 (unabridged and abridged versions), in 1551(unabridged and abridged versions), in , 1552(unabridged and abridged versions), in , 1555(unabridged and abridged versions), in and, in French, in 1551(unabridged and abridged versions), in and 1552. See (Fine 1535(Fine , 1542a(Fine , 1544a, for the bibliographical references of these reprints. 24 On the Parisian context of scientific book printing, see also (Crowther et al. 2015;Valleriani 2017). 25 (Peurbach and Fine 1515;Sacrobosco and Fine 1516;Ricci and Fine 1521;Sacrobosco et al. 1521;Fine 1526Fine , 1527Fine , 1528Fine , 1529Fine , 1532Fine , 1543aFine , b, 1544bFine , 1545Fine , 1553aFine , b, c, 1557Borrhaus 1551;Mizauld 1552). To this, we could add (Reisch and Fine 1535). See also the lists of Fine's works compiled in (Ross 1971, 398-449;Pantin 2013a;Axworthy 2016, 407-11). 26 Fine published in 1525 a revised version of his edition of Peurbach's Theorica (Peurbach and Fine 1525), which only featured the original text (without the commentary of Francesco Capuano de Manfredonia (fl. 15th cent.) (Peurbach 1495), with which it was published in 1515), but with a clearer layout and with the inclusion of marginalia, using for Peurbach's text some of the figures initially drawn for the commentaries (Pantin 2013a). He also published in 1528 a treatise in French entitled La Theorique des cielz, mouvemens, et termes practiques des sept planetes (Fine 1528), which may be regarded as an adaptation or a paraphrase of Peurbach's Theorica planetarum. 27 On the life and works of Sacrobosco, see (Pedersen 1985). 28 On the traditional order of teaching of the various parts of the quadrivium and its reception by Fine, see (Axworthy 2016, 50-53). 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, 29 depicting himself in the place of the astronomer, where the figure of ) 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). 30 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). 31 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, 32 and the second applied to astronomy, commending the usefulness of the science of stars for the contemplation of the divine order. 33 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. 29 An exception to this is his Theorique des cielz (Fine 1528), which was published anonymously and only reprinted with Fine's name postumously, that is in 1557, 1558, 1607, and 1619. 30 On the intellectual filiation between Ptolemy and Fine, see (Pantin 1993(Pantin , 2009bConley 1996, 98-114). 31 Yet, such frontispieces existed for other mathematical disciplines in the contemporary literature, as in the Margarita philosophica of Gregor Reisch,which Fine edited. 32 (Fine 1532, AA8v): "Cùm natura sagax numero mensùque crearit / Singula, ponderibus clauserit inde suis: / Non poteris rerum proprias discernere causas, / Ni teneas numeros, et geometra simul." The frame of the frontispiece features these verses: "Disce prius numeros tellurisque ordine mensus: nam facilem cura haec sternet ad alta viam." This epigram was taken up and slightly modified in ulterior versions of the Arithmetica practica. On the evolution of the epigram in this context, see (Pantin 2009b;Axworthy 2016, 54-56). 33 (Fine 1532, 101v): "Florida divinae quisquis secreta mathesis / Scire cupis, facili mente fruare decet / Nam licet assiduo possis superare labore, / Mens generosa tamen plurima sola capit." In the frame, "Excute sollicito fragiles de pectore curas: et studeas superas arte subire domos." These verses were used again in the unabridged 1542 edition of the Cosmographia, but in the place of the epigram of the 1532 frontispieces.
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). 34 He followed in this regard the discourse held by Johannes Regiomontanus (1436Regiomontanus ( -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. 35 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. 36 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. 34 On the propaedeutic role of mathematics according to Fine, see (Pantin 2009a;Axworthy 2009Axworthy , 2016. On the discussions prior to Fine concerning the relation of astronomy to mathematics in general, but also to the other parts of philosophy involved in the contemplation of the universe, see (Chaps. 2,3 and 4,). 35 (Regiomontanus 2008, 137): "Inter omnes autem hasce disciplinas astronomia instar margaritae non modo sorores suas, reliquas inquam scientias medias, verum etiam omnium disciplinarum matres geometriam et arithmeticam longe antecellit; cujus ortum prae vetustate nimia haud satis comperimus ita ut aeternam aut mundo concreatam non inique putaveris." (Fine 1542a, sig. * 2rv): "Quanquam enim ipsae Mathematicae, omne philosophandi genus adaperiant, et in universum cunctis opitulentur artibus: eò tamen omnes tendere videntur, ut Caeli suscipiendi peculiarem sortitae sint curam. Quam beatissimam contemplationem, Astronomiam vocant." (Fine 1551a, sig. aa2r-aa3v): "Ipsa autem caelestium rerum eruditio, earum disciplinarum beneficio comparatur, quae mathematicae nuncupantur: quarum videlicet essentialis puritas, fida atque inviolabilis certitudo, humana divinis, terrenave caelestibus vel facilè conciliat. Et proinde inter ispius mathematicae partes, ea longè praestantior esse videtur, quae Astronomia dicitur: utpote, in cujus gratiam caeterae omnes videntur excogitatae, et quae caelestia simul et terrestria ratiocinatur corpora." (Fine 1555, sig. 2v): "Omnis itaque philosophia, omnisque certa et inviolabilis doctrina, qualis est Mathematica, eò potissimùm tendere videtur, ut in veram caelestium rerum cognitionem mortales inducat." 36 For this paper, I will mostly refer to the first edition of the Cosmographia, which was first printed 1530 (as indicated by the title page of this particular treatise) though it was published in 1532 with the other parts of the Protomathesis. Because the denomination Cosmographia comes in this edition before that of Sphaera mundi-this is not the case in the later editions-I will use the shortened title Cosmographia.

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). 37 His grandfather and his father, Michel Fine (fl. 1474Fine (fl. -1490 and François Fine (fl. 1494Fine (fl. -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). 38 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 2009aPantin , 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(Pantin , 2009aAxworthy 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 1517 39 ;Bassolis and Fine 1517a, b;Martínez Silíceo and Fine 1519;Ricci and Fine 1521;Fine 1522;Reisch and Fine 1535) 40 and published under his name several mathematical treatises (mostly pertaining to astronomy) (Fine 1526(Fine , 1527(Fine , 1528(Fine , 1529(Fine , 1530, as well as a map of France (Fine 1525). He also practiced during this time as a court astrologer. 41 37 More generally, on Fine's life, education, and career, see (Thévet 1584, 564r-66v;Baldi 1998, 442-55;Thorndike 1941, 284-86;Escallier 1957;Ross 1971, 8-30;Poulle 1978;Aked 1990;Marr 2009;Pantin 2009bPantin , 2013aOosterhoff 2016;Axworthy 2016, 12-22). 38 J. Dupèbe (Dupèbe 1999, II, 522) surmises that he was teaching in 1515 at the Collège de Montaigu, when he edited Peurbach's Theorica planetarum. 39 Fine engraved, for this work, a map of the Holy Land (Conley 1996, 91-98;Pantin 2009b). 40 This edition of Reisch's Margarita philosophica was published in 1535 (Reisch and Fine 1535), but the date of 1523 at the end of the preface indicates that it had been prepared by Fine while he was teaching at the Collège de Navarre. 41 According to certain historical accounts, Fine would have been imprisoned around the years 1524-1525 because of a horoscope that would have either been unfavorable to a member of the court, Louise de Savoie (Lefranc 1893, 178;Gallois 1918), or favorable to the Constable of Bourbon (Charles III of Bourbon), an enemy of the court (Destombes 1971). The first of these two 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(Fine , b, 1545(Fine , 1553a(Fine , b, 1557, geometry (Euclid 1536;Fine 1544bFine , 1556a and the theory of burning mirrors (Fine 1551b).
hypotheses is founded on a letter (written in Lyon on the 3 rd of November, 1526) by Heinrich-Cornelius Agrippa, who claimed that Fine had been imprisoned for having made an unfavorable horoscope to a member of the French court (in order to justify his reluctance to make prognostications to the French court while in Lyon). (Agrippa 1970, II, letter 62, book 4, 844): "Sed et nesciebam me praedariò astrologum conductum, quodque mihi, quod ars illa dictat, monendi dicendique jus relictum non esset, occurritque exemplò Orontius Parrhisiorum insignis mathematicus et astrologus, qui dum veriora, quàm poterat, vaticinaverat, iniquissima captivitate diutinè vexatus est: iamque aiebam apud me, quid, si reliqua misisses prognostica? proculdubio ex fumo in flammam te conjecisses." Yet, although it is generally admitted by historians that Fine spent some time in prison, the reason for this unfortunate event is not fully attested, as other sources suggest that it may have been because of his opposition to the Concordat (Boulay 1665(Boulay -1673Tuilier 2006a) or for having refused, as he worked on the fortifications of Milan, to submit himself to the aforementioned Constable of Bourbon, who would have had him arrested during the construction of a bridge on the Tessino (Escallier 1957, 6-7;Aked 1990). More generally, on this issue, see (Ross 1971, 386-97;Dupèbe 1999, II, 536-38). 42 As indicated by Víctor Navarro Brotóns (Navarro Brotóns 2006), Jerónimo Muñoz, who studied in Valencia for his Bachelor of Arts, referred to Fine as "preceptor noster" in his Astrologicarum et geographicarum institutionum libri sex (Muñoz 2004, 68v).

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. 43 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. 44 In addition to his epigram to the reader, he included several liminary poems at the beginning and at the end of the book, 45 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 Fine 46 and Fine's protector Antoine Silvestre, 47 and Nicolas Petit (1497-1532), 48 addressing Jean Fossier or Jean des Fosses (fifteenth-sixteenth century), 49 one of Fine's disciples (Dupèbe 1999, II, 526 (Élie 1951). 51 On the illustrations in the Venetian editions of Sacrobosco, see (Gingerich 1999;Cosgrove 2007;Barker and Crowther 2013;Oosterhoff 2015). edition of Peurbach's Theorica planetarum. 52 This edition also contains a few tables:  a table displaying the cosmic, chronic, and heliacal rising and setting of the signs; 53  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); 54 a table for the latitudes of the seven climates, coupled with a  table indicating (Sacrobosco and Fine 1516, sig. d1r): "Elevationes poli and Quantitates maximarum dierum." 56 On the mostly practical nature of the material added to the text of Sacrobosco in the sixteenth century, see (Crowther et al. 2015;Oosterhoff 2015;Valleriani 2017 (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 bidimensional 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(Pantin , 2010(Pantin , 2012Oosterhoff 2015. 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 (1495Apian ( -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, 60 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; on the context in which he could have added these annotations, it is possible that such notes were written down in the context of a classroom. On this type of layout and on the practice of notetaking on printed textbooks in Parisian university classrooms at the beginning of the sixteenth century, see (Oosterhoff 2015(Oosterhoff , 2018) (Chap. 9). 60 As suggested by the paper of Isabelle Pantin in this volume (Chap. 9), the fact that an editor indicated his interventions, even only in the colophon, was significant. 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, 61 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). 62 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(Pantin , 2013aOosterhoff 2015Oosterhoff , 2016Oosterhoff , 2018Valleriani 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(Pantin , 2013a. 63 61 On Sacrobosco's Sphaera and on its place in the medieval and Renaissance university curriculum, see (Thorndike 1949, 42-43;Oosterhoff 2015). 62 See (Chap. 9) for this factor as a motivation for the development of the in-octavo tradition. 63 It is interesting to note that, in this respect, Fine published this French paraphrase of Peurbach anonymously. His authorship of this work may be attested by the appearance of the motto "virescit vulnere virtus," which he used on all his works from the publication of his edition of the Ars arithmetica of Juan Martínez Silíceo in 1519 (Martínez Silíceo and Fine 1519), and also by the fact that this work was published again under his name postumously on several occasions. Although the motivations for the anonymous publication of this work remains an open question, one could conjecture that the reason why he published this work anonymously may be related to the fact that he wrote this paraphrase in French, which was a less conventional practice for works traditionally used for academic teaching. His attitude to this question may have changed in the later period, when he wrote and published (or intended to publish) under his name certain treatises in French (Fine 1543a, Fine 1551b, Fine 1556b, though this may be qualified by the fact that some treatises which were first written in French but left at the state of manuscript (perhaps intended to be read by the King or members of his entourage) were later published in Latin (such as Fine 1543b later published in Latin in Fine 1544b). Notwithstanding, the fact that he published a French version of 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). 64 He also stated in the preface of the first edition of the Cosmographia 65 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) 66 and given the role of judicial astrology (which represented the practical part of the science of stars, according to Fine) 67 in the determination of the favorable days for Peurbach's treatise on planetary theory, even anonymously in 1528, could be a sign of his will to change the audience of traditional astronomy, though not necessarily to reach out to a less educated and socially lower class of people, but to a different class of privileged readers such as members of the French court. 64 (Fine 1532, sig. AA3r): "Quod si Deus ipse, ad optatum finem dignetur aliquando perducere: videbis universam Galliam, jam fideliores amplectentem literas, caeteras nationes, non secus ac Lilium spinas, brevi tempore superare. Theologos in primis nativam sacrae scripturae consequi puritatem, tandemque fieri meliores: Philosophos sophistis succedere, et justo deinceps titulo philosophicae dignitatis laurea donari: Medicos humanis passionibus foelicius consulere, nec amplius cum tanto mortalium periculo suas venditare conjecturas: Rerum tandem humanarum judices aequiores, mitioresque succedere (quos omnium artium expedit habere cognitionem) et publicam utilitatem, potius, quàm privatam (ut tenentur) aliquando procurare: et in summa, omnes ad saniora tendere, et Christianam tandem induere pietatem." On Fine's commitment to the teaching of the masters of the University of Paris in his early years, see also (Dupèbe 1999, II, 525-26, 543). 65 (Fine 1532 (Cosmographia, I.1), 102r): "Quam necessaria postmodum Apolineae sit arti, is judicare poterit, quem praesagia Hipocratis legere non pigebit: in quibus coeleste quoddam asserit esse, in quo et ipsum medicum praevidere oportet. Quod Galenus ille medicae artis restaurator in testimonium adducens, omnem substantiam corpoream animatam coelestibus signis et planetis alligari demonstrat. Adde quod viris ecclesiasticis non modo perutilis, verumetiam necessaria videtur Astronomia, idque tanto magis, quanto graviori dignitate fruuntur: ad mobilia festa, caeteraque decus et statum ecclesiae respicientia pensiculatius discutienda. Ob cuius Astronomiae neglectum, ne dicam praelatorum incuriam, a vera sacri Paschatis observatione, et evangelico ritu (horresco referens) tantum plaerunque distamus: ut pudeat hoc commune Christianorum scandalum ulterius aperire." 66 On the uses of Sacrobosco's Sphaera for the domains of medicine and calendar computation, see (Cosgrove 2007;Crowther et al. 2015;Valleriani 2017) (Chap. 5); and on the tradition of computus from the early Middle Ages, see the introduction of Faith Wallis to (Bede 2004, xxxiv-lxiii;Declercq 2000, 49-95). On Sacrobosco's computus (De anni ratione) in particular, see (Moreton 1994). 67 (Fine 1532 (Cosmographia), I.1, 102v): "…universam Astronomiam, veluti quamlibet aliam disciplinam, bifariam discindi, apud omnes, etiam vulgariter eruditos, in confesso est. Aut enim ipsum scire, magisque necessaria consyderat Astronomia, utpote, coelestes globos, sydera, eorum bloodletting and for the administration of remedies, 68 to which should be added the production of medical almanacs (Chap. 5). 69 It is important to note here that, for many students, astrology (both judicial and natural 70 )-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) 71 -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. 72 Because of its relationship to medicine, astrology actually held a privileged place among the mathematical arts in the university curriculum since its foundation. 73 (Axworthy 2016, 201-4) and on the division and hierarchization of the parts of astronomy in the Renaissance Sacrobosco tradition prior and contemporary to Fine,see (Chaps. 4 and 5). 68 On the Galenic theory of critical days and its reception in the Middle Ages and in the Renaissance, see (Azzolini 2005;Pennuto 2008;Cooper 2013). 69 On the production of almanacs, see (Carey 2003(Carey , 2004Eamon 2014;Kremer 2017). 70 On the difference between judicial and natural astrology, see (Vanden Broecke 2003, 18-19). 71 Yet, reservations were expressed since the Late Middle Ages as to whether occult influences should be admitted in addition to the light, heat, and motion of the planets (Vermij 2016), and as to the extent to which the stars influenced the events taking place in the lives of individuals (Vanden Broecke 2016) or as to its potential association with superstition (Vanden Broecke 2003, 9-12). J. Dupèbe (Dupèbe 1999, II, 530-33) mentions in this regard the critical discourse on occult influences held by Gregor Reisch, in his Margarita philosophica (which Fine edited). 72 On astrology as a motivation for the study of astronomy, and of Sacrobosco's Sphaera in particular, in the sixteenth century, see (Pedersen 1978;Eamon 2014;Hübner 2014) (Chaps. 2, 3, 5, 7 and 10), but also (Chap. 9), concerning the inclusion of elements on the practice of domification in the Venetian editions of Sacrobosco. 73 For this reason, astronomy and astrology were often taught in medieval universities, such as in Padua and Bologna, by professors of medicine before the creation of a distinct chairs of mathematics, which were initially designated as lectureships in astrology and/or astronomy before being related to mathematics in general (Grendler 2002, 408, 415-26). See also, on the relation between astronomy and medicine in the medieval university curriculum, (Siraisi 1981, 139-45;Siraisi 1990, 68-69, 128-29, 134-36;Vanden Broecke 2003, 12-16). his studies, 74 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. 75 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(Mizauld , 1551(Mizauld , 1555, 76 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). 77 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.

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 74 As noted by Jean Dupèbe (Dupèbe 1999, II, 526-27) (Dupèbe 1999, II, 528-29). 76 On Antoine Mizauld, his work on iatro-mathematics and the Parisian scene of medical astrology, see (Dupèbe 1999). 77 These works respectively deal with the art of producing almanacs (including their astrological features) and with the division of the celestial houses and of the planetary hours necessary to the casting of horoscopes. Fine also published in 1529 an Almanach novum aimed (as indicated by the title) to help produce elections in the context of medicine, church duties, banking, and many other important functions (Almanach novum insigniora computi et kalendarii succincte complectens ad longos annos duraturum, viris ecclesiasticis, medicis, chirurgicis, trapezitis, quibusvis tandem hominum conditionibus necessarium). On the context of its publication, see (Dupèbe 1999, II, 540-41).
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. 78 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. 79 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(Fine , b, 1551a(Fine , b, 1555, making its focus on the theory of the sphere more obvious (Mosley 2009). 80 What this title also indicates is that this teaching pertained to cosmography, 81 which was, in the sixteenth century, properly developed into a 78 The pedagogical aim of the Protomathesis is first suggested by the context of its publication, which coincides with the beginning of Fine's career as royal lecturer. His new function as royal lecturer is indicated in the title of the work (Orontij Finei Delphinatis, liberalium disciplinarum professoris Regii, Protomathesis). In the preface of L'Esphere du monde, which corresponds to the French translation of the Cosmographia published in 1551, Fine clearly described the pedagogical mission the King François I had assigned to him as a royal lecturer, which he fulfilled, as he writes then, through ordinary lectures and through the publication of written works. (Fine 1551b, sig. A3r): "Et fuz d'autant plus incliné audit estude, que je cogneu le feu Roy vostre pere (auquel Dieu doint repos eternel) outre le bon jugement qu'il avoit de toute chose, comme prince bien né, porter singuliere affection ausdittes mathematiques: desquelles il me ordonna finablement publique interpreteur en l'Université de Paris, ou j'ay fait mon devoir, tant par leçons ordinaires, que par oeuvres escrittes, les remettre sus, & icelles demonstrer l'espace de trent'ans & plus." As seen above, Fine also reiterated the propaedeutic role of mathematics for the higher university faculties in the preface of the Protomathesis, manifesting his will to offer a complete mathematical course for the students and masters of the Faculty of Arts, in addition to the new public to which his teaching was open. 79 See also, for comparison, the place given to the teaching of the sphere in the pedagogical programs of Jacques Lefèvre d'Étaples (Chap. 2), Conrad Tockler (Chap. 5), Pedro Sanchez Ciruelo (Chap. 3), Franco Burgersdijk (Chap. 11), as well as in those of Portuguese and Spanish mathematics professors more generally (Chaps. 7 and 10). 80 The association between the theory of the sphere and cosmography is also indicated by the title of the treatise of Antoine Mizauld, De mundi sphaera, seu cosmographia, libri tres, which was published in 1552 and which Fine contributed to editing. On other works that indicate in their title the connection between cosmography and the theory of the sphere, see (Oosterhoff 2015;Valleriani 2017) (Chap. 7). 81 It may be noted that the Theorique des cielz, which was published four years before the Cosmographia and in which Fine gives the main elements of the general description of the universe (serving him also for the corresponding part of the Cosmographia), he explicitly identified cosmography with the theory of the sphere; (Fine 1528, 3r): "[le mouvement qui est appelle diurnel] appartient traicter en la Cosmographie, ou traicte de l'esphere mondaine: come nous avons faict, et doibt estre presuppose devant ce livre cy." See also (Fine 1528, 33v): "comme nous avons amplement declaré au traicté de la Cosmographie, ou de la Sphere du monde." Here Fine refers to discipline in its own right, bringing together metrical geography and the theory of the sphere. 82 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) 83 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), 84 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). 85 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). 86 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 a prior teaching of cosmography or the theory of the sphere on which he would have previously worked. It is not clear to which treatise he is referring. Perhaps he had already written his Cosmographia in 1528. Indeed, as it was already printed two years before the actual publication of the Protomathesis from the date of the frontispiece of its cosmographical part (Fine 1532, 101r), it is not impossible that he had written it before 1528. But he could also be referring to his edition of Sacrobosco's Sphaera, or less likely to the discourse he has presented in the first pages of the Theorique des cielz, though this may hardly be called a treatise. I would like to thank Isabelle Pantin for her help in verifying the presence of the relevant passages in the 1528 version, which included a visit on her part to the Bibliothèque Nationale de France. 82 On the development of sixteenth-century cosmography and cosmographical culture, see (Mosley 2009;Gaida 2016), as well as (Chap. 2). 83 On this topic, see (Navarro Brotóns 2004;Portuondo 2009;Mosley 2009;Cattaneo 2016;Almeida 2017) (Chaps. 7 and 10). 84 The earlier meanings of the term cosmographia, for instance in the works of Cassiodorus and Bernard Silvestris, are also described in (Cosgrove 2007;Mosley 2009). 85 On the various audiences and applications of the teaching of cosmography in the Iberian peninsula, see (Chap. 7). 86 On the entry of cosmography into the university mathematical curriculum in sixteenth-century Spain, see (Chap. 7). 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. 87 Geography is indeed assigned a comparable epistemological status to astronomy, 88 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. 89 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). 90 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 87 The two orientations of cosmography, and their correspondence with the two main parts of the cosmos, are clearly expressed by Fine in the 1542 edition of the Cosmographia; (Fine 1542b, 5r-v): "Unde κόσμος a Graecis dicitur: et quae de Mundo traditur disciplina, κοσμογραφία (de qua praesentis tractare est instituti) respondenter vocitatur. Est enim Cosmographia, Mundanae structurae generalis ac non injucunda descriptio: prima Astronomiae partem, atque Geographiam, hoc est, caeli terraeque rationem comprehendens." 88 On the epistemological status of geography in Fine's mathematical thought, see (Besse 2009;Axworthy 2016, 329-49). 89 For an overview of Ptolemy's approach to geography, see the introduction of J. Lennart Berggren and Alexander Jones to their annotated translation of the Geography (Ptolemy 2000, 3-54). On the content and the influence of Ptolemy's Geography in the Renaissance, especially on sixteenthcentury treatises on the sphere, see (Besse 2003, 29-30, 111-48;Oosterhoff 2015); in relation to Fine's cartographical work specifically, (Brioist 2009b) and, in a more general perspective, (Šālew and Burnett 2011). 90 (Fine 1551a, sig. aa2r-v): "Inter admiranda naturae sive Dei miracula, duo sunt quae omnium miraculorum superare videntur admirationem: Mundus scilicet, et homo. Quorum partes insigniores sunt rursum duae: utpote, immortalis vel aeterna, et ea quae corruptioni, atque mutationi semper obnoxia est. Mundi nanque pars aeterna, est ipsum caelum, divino lumine Solis illustratum, et suis in primis ornatum corporibus, regulari et indefessa latione circunductis: quae unum atque eundem ordinem perpetuo videntur observare, utpote, quem ex Deo ab ipsa Mundi creatione sunt adepta. Pars vero corruptibilis ipsius Mundi, et quae nunquam in eodem statu permanet, est ipsa elementorum moles, intra caeli cavaturam conglobata: assidua quidem agitatione pertubata, atque alterata, omnium generatorum materia, et alimentum. Haud dissimiliter, homo ex duplici natura compositus esse videtur: aeterna videlicet, hoc est, ipsa anima Deo simili, quam nonnulli substantialem vocant homine: et mortali, utpote corporea, quae ut ex ipsis constare perhibetur elementis, sic et in eadem elementa tandem resolvitur. Homo itaque sic efformatus est, ut utranque suam originem aeternam videlicet, et corruptibilem recognoscere possit et debeat: hoc est, incolere atque gubernare terrena, et simul intelligare et admirari quae coelestia sunt. Nempe cui soli inter animantia, portio mentis ab ipso Deo, caeli et animae, ac omnium eorum quae Mundus comprehendit opifice atque rectore, concessa est." Jean-Marc Besse (Besse 2009) also shows that the duality was interpreted as a double view-point of the mathematician on the cosmos, which cosmography both connects and encompasses. the astronomical circles that divide the celestial sphere and which the cosmographer is required to project onto the terrestrial globe (Cosgrove 2007). 91 Sacrobosco's Sphaera thus became the natural starting point for the mathematical analysis and treatment of the terrestrial space 92 at a global and local level (Besse 2009;Brioist 2009b). 93 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. 94 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 91 As shown by (Chap. 2) through the exemplary case of Lefèvre d'Étaples's 1494 commentary on Sacrobosco, this relation also took place in a reversed order, with the integration of cosmography within the framework of the Sphaera, the addition of many elements from Ptolemy's Geographia, as well as of various tables and computation methods necessary to the cosmographical practice. On the relation between Sacrobosco's Sphaera and cosmography in the sixteenth century, see also (Oosterhoff 2015;Valleriani 2017) (Chaps. 7 and 9). 92 As indicated by Pantin in this volume (Chap. 9), Peter Apian wrote in the address to the reader of his first edition of Sacrobosco's Sphaera (Sacrobosco and Apian 1526, A1v) that this edition was to be regarded as a preamble to his introduction to Ptolemy's Geographia: "…visum est mihi haud inutile fore, si ingenuis adolescentibus primum omnium Astronomiae rudimenta praelegerem, Sphaeram JANI de Sacrobusto accuratissime interpretarer. Futurum tandem existimans ut ex sphaerae circulorumque ejus attenta cognitione, spaciorum terrae coelique absoluta notitia proveniret." 93 As was shown by Besse (Besse 2009), in Fine's Cosmographia, geography (mapping of the earth), chorography (regional mapping), and hydrography (marine cartography) are not distinguished from an epistemological point of view, as they are in the classifications found in contemporary cosmographical and geographical treatises. 94 (Fine 1532, 101r) (Honterus 1535(Honterus , 1440. 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-96 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). 97 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 95 On Apian's contribution to sixteenth-century cosmography, see (Gaida 2016). 96 When Georg Peurbach proposed his own version of the geometrical system intended to model the motions of the planets (theorica planetarum) in the fifteenth century, in order to replace the system previously taught in the universities, he presented it explicitly as a new model (Novae theoricae planetarum) aimed at replacing this old system (Theorica planetarum antiqua). In spite of the popularity of Peurbach's new version, the old theorica continued to be printed, sometimes with the new theorica, up to the sixteenth century. On the history of the Theorica planetarum, see (Pedersen 1981). A parallel may therefore be made between this development and the fact that Sacrobosco's Sphaera continued to be printed up to the seventeenth century, alongside more modern treatises on the theory of the sphere. 97 For a list of the various printed editions of Sacrobosco's Sphere, see the online database coordinated by Matteo Valleriani: https://sphaera.mpiwg-berlin.mpg.de, as well as that of Roberto de Andrade Martins: http://www.ghtc.usp.br/server/Sacrobosco/Sacrobosco-ed.htm (Accessed June 2019) and the bibliographical inventory in (Hamel 2004(Hamel , 2014. Complementary information may be found in (Thorndike 1949;Gingerich 1988;Pedersen 1995;Valleriani 2017). 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, 98 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. 99 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 Cosmographia 100 and was made explicit by Cosimo Bartoli in his Italian translation of the Cosmographia (Fine 1587). 101 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 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, 102 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, 103 book IV for the motion of the sun and its influence on the duration of light and shadows at different latitudes on Earth, 104 and book V for the theory of climates, which is then integrated into the geographical part of the Cosmographia. 105 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, 106 highlighted in particular by the fact that the astronomical section of the work covers four out of five books, 107 and also by the fact that, within this part, the disposition and structure of Sacrobosco's Sphaera remains overall clearly identifiable (Mosley 2009). 108 More precisely, the topics common to the first book of the Sphaera and to the first book of the Cosmographia 109 are the distinction between the elementary and the heavenly regions, along with their respective divisions, 110 the motions and the sphericity of the heavens, 111 the immobility, sphericity, and centrality of the earth in the middle of the universe. 112 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 abovedescribed content. 113 In the second book of the Cosmographia, Fine followed the model of the Sphaera by presenting the various astronomical circles that divide the worldly sphere 114 -107 The four first books occupy 38 of the 53 folios of the Cosmographia. 108 The commentary on the Sphaera of André do Avelar (Chap. 10) (Fine 1532, I.4, 104v-05v): "Quaenam coelestium orbium figura, atque motus qualitas" [in marg. Quod coelum sit sphaericum Prima ratio, à commoditate. Quae dicantur isoperimetrae. Secunda ratio à necessitate. Quod coelum circulariter moveatur. Motus universalis totius coeli. Motus secundus orbibus peculiaris, priori contrarius.] and (I.5, 105r-v): "De generali eorundem coelestium motuum expressione. [in marg. Primus motus quem diurnum appellamus, unde proveniat. De proprio singulorum orbium motu]." 112 (Sacrobosco and Fine 1516, I, sig. a3v): "Quod terra sit rotunda;" (sig. a4r): "Quod acqua sit rotunda. Quod terra sit centrum mundi" and "De immobilitate terrae." See also (Fine 1532, I.6, 106r-07v): "De quiete, loco, et figura ipsius Terrae. [in marg. Quod terra non movetur circulariter. Terra non movetur motu recto, secundum totam. Quod terra est in medio totius Universi. Telluris et aquae superficies unica. Quod idem globus, à septentrione ad austrum sit rotundus. Quod globus ex Tellure et acqua resultans respectu universi, imperceptibilis sit quantitatis. Terra mundi centrum esse videtur.]" 113 The relation between the Theorique des cielz and the Cosmographia is confirmed by the fact that these works contain the same diagrams for the spheres of the elements (Fine 1528, 2v;Fine 1532, 103r) and for the celestial spheres (Fine 1528, 3r;Fine 1532, 104r). 114 (Sacrobosco and Fine 1516, II, sig. a4v): "Capitulum secundum de circulis ex quibus sphaerae materia componitur: et illa super coelesti quae per istam imaginatur componi intelligitur." See also (Fine 1532, II.1, 108r): "Liber secundus cosmographiae, sive Mundi Sphaerae, De principalioribus circulis in eadem Sphaera prudenter imaginatis." namely, the equinoctial or celestial equator (along with the poles of the world), 115 the colures, 116 the meridians and the horizons, 117 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, 118 to which adds the division of the heavens into five zones. 119 All these circles were considered again in the first chapter of the fifth book, through their projection onto the terrestrial globe. 120 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 115 (Sacrobosco and Fine 1516, II, sig. a4v Fine to present the distinction between cosmic, chronic, and heliacal risings, 121 as well as the distinction between right and oblique ascensions. 122 Book IV, which deals with the motion of the sun and its effect on the duration of daylight, tackles the inequality of natural days 123 and the difference between artificial day and night. 124 121 (Sacrobosco and Fine 1516, III, sig. b3v-b4r) Book V, which deals with the second main division of cosmography (geography), considers the distinction of the climates. 125 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). 126 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 ter-125 (Sacrobosco and Fine 1516, III, sig. c4r-d1v): "De divisione climatum [in marg. Divisio climatum. Diffinitio climatum. Primum clima. Secundum clima. Tertium clima. Quartum clima. Quintum clima. Sextum clima. Septimum clima. Quid latitudo climatis. Quid longitudo climatis]." See also (Fine 1532, V.2, 143v-44v): "De Parallelis Climatum distinctoribus: quonam item pacto, dato lucis arcu singulorum parallelorum, polares investigentur altitudines. [in marg. De parallelis climatum distinctoribus. Climatum diffinitio. Distributio climatum. De climatum magnitudine.]" 126 On the evolution of illustration practices in Renaissance astronomical treatises, including editions and commentary on Sacrobosco, see (Gingerich 1999;Pantin 2001;Cosgrove 2007;Barker and Crowther 2013;Oosterhoff 2015).  (Sacrobosco et al. 1488, sig. a8v:). HAB Wolfenbüttel: 16.1 Astron restrial globe and of mathematical instruments, 127 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). 128 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 127 This richness of detail appears indeed also in his representation of instruments, for instance in the demonstration of the difference between umbra recta and umbra versa, where the illustration includes some form of contextualization by the presence of a human figure or a landscape in the background (Fine 1532, resp. fol. 110v, 113r, 138r). On this aspect in Fine's astronomical frontispieces, see (Pantin 2009b) and, on the visual changes made to Sacrobosco's Sphaera through the integration of cosmographical elements, see (Chap. 2). 128 As shown by (Chap. 9), 15 of Fine's 1532 Cosmographia were taken up in the edition of Sacrobosco published in Wittenberg in 1538. From (Sacrobosco and Fine 1524, sig. a2v). Courtesy of the Library of the Max Planck Institute for the History of Science, Berlin 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).
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(Pantin , 2013aMosley 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(Fine , 1531b(Fine , 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 2009aPantin , 2010Pantin , 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. 129 He also left aside the distinction between the division of the sphere 129 (Fine 1532 (Geometria libri duo), 53v): "solidas figuras, primum sese offert Sphaera, omnium regularissima: hoc modo diffinienda. Sphaera est corpus solidum, regulare, unica superficie terminatum: in cujus medio punctum assignatur, centrum eiusdem adpellatum, à quo ad ipsam orbicularem et terminativam superficiem, omnes quae ducuntur rectae lineae sunt invicem aequales…. Imaginatur autem describi Sphaera, ex completo semicirculi circunductu: cum videlicet semicirculi diametro manente fixo, ejusdem circuli plana superficies abstractivè circunducitur, quatenus unde ferri ceperat revertatur." (Sacrobosco and Fine 1516, sig. a2r): "Sphera igitur ab Euclide sic describitur. Sphera est transitus circumferentie dimidii circuli que fixa diametro quousque ad locum suum redeat circumducitur. Id est Sphera est tale rotundum et solidum quod describitur ab arcu semicirculi circumducto. Sphaera etiam a Theodosio sic describitur. Sphaera est solidum quoddam una superficie contentum in cuius medio est punctus a quo omnes lineae ductae ad circumferentiam sunt aequales. Et ille punctus dicitur centrum sphaerae." It should be noted that, contrary to the order found in Sacrobosco, Fine first presented the definition of the sphere by Theodosius, which defines the sphere by its spatial properties, and then the definition of Euclid, which on the contrary defines the sphere by its mode of generation. Such order is also found in later commentar-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 heavens 130 and when appealing to the distinction between right and oblique horizons. 131 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. 132 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. 133 In the Cosmographia, Fine also left aside the section dealing with eclipses presented by Sacrobosco in the fourth chapter, whose absence in ies on Euclid's Elements, for instance, in those of François de Foix-Candale or Henry Billingsley, in which was clearly enunciated the distinction and hierarchization between essential definitions and genetic definitions of geometrical objects, adequately illustrated by the difference between Theodosius and Euclid's respective definitions of the sphere (Euclid 1566, 140r-v;Euclid 1570, 315v). This distinction is also made in the medieval commentaries of Sacrobosco, for instance, in Pierre d'Ailly's Quaestiones (Sacrobosco et al. 1531, 146v), who argued that the Euclidean definition was less proper than the definition of Theodosius to define the sphere, since it defined it by one its accidental features (the motion by which it is produced) rather than by one of its essential features (its spatial properties). 130 (Fine 1532 (Cosmographia), I.3, 103v-104v): "De coelestium orbium numero, atque positione." 131 (Fine 1532, II.6, 112v) (Fine 1542b, 5v): "Est enim Cosmographia, Mundanae structurae generalis ac non injucunda descriptio: prima Astronomiae partem." 133 Peurbach's treatise starts directly with the sun's theorica; (Peurbach and Fine 1515, 2v): De Sole. "Sol habet tres orbes a se invicem omni quamque divisos atque sibi contiguos…." 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. 134 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), 135 stating that these imaginary representations are not only fictitious and useless, but also entirely alien to the contemplation of the mathematician. 136 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 sphere 137 and thus clearly positioned himself on this aspect against the form and style of Sacrobosco's teaching of astronomy. 138 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  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. 140 He furthermore added various chapters and subsections containing practical material (which, however, maintain an overall theoretical scope), 141 pertaining, for instance, to the positioning of stars, to the measurement of the altitude of the sun, as well as to judicial astrology, 142 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. 143 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. 144 In chapter II.4, Fine presented the means to find the declination of the ecliptic for any degree of the equinoctial. 145 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. 146 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) 147 and the circles of hours and their role in the constitution of sundials. 148 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. 149 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, 150 along with the corresponding table. 151 Similar material is given in the chapters III.4 and III.5 for the computation of the arcs of 146 (Fine 1532, II.4, 110v-11v) (Fine 1532, II.9, 115v): "De circulis horarum distinctoribus. Non prorsus aspernandam consequenter judicamus horarium circulorum designationem: ab ipsis enim universa, tum horarum, tum solarium horologiorum ratio potissimum deducitur." 149 (Fine 1532, II.10, 116v-17v): "Quibus circulis 12 coeli partes (quas vocant domos) separentur: atque de circulo positionis appellato. De circulis tandem coelestium domiciliorum distinctoribus, variae inter Astronomos reperiuntur opiniones; Figura generalis 12 coelestium domiciliorum secundum judiciarios Astrologos." Though he recognised the benefits of the division of Regiomontanus, he recommends his readers to follow Campanus: (Fine 1532, 117v): "Si meo tamen stare velis judicio a via Campani non discedes: multo siquidem fideliora, cum probatissimus Astrologis poteris elicere judicia." On the different modes of distinction of the sphere in 12 astrological houses, see (Bezza 2014). 150 (Fine 1532, III.3, 120v): "Quaenam ascensionis atque descensionis accidentia in recto contingant sphaerae situ, necnon de rectarum ascensionum supputatione;" (121v): [in marg. Canon supputationis rectarum ascensionum cuiuslibet arcus Eclipticae]. 151 (Fine 1532, III.3, 122v): "Tabula rectarum ascensionum singulorum arcuum Eclipticae, ab Ariete gradatim initiatorum, per Authorem supputata." ascension of the ecliptic in the oblique sphere, calculated for the latitude of Paris, 152 which includes a table of the ascensional differences between the arcs of the zodiacal signs in the right and oblique spheres, 153 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, 154 as well as corresponding tables for the total ascensions of each sign for the same latitudes. 155 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. 156 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, 157 and of the beginnings of the remaining astrological houses, according to the methods of Campanus and of Regiomontanus. 158 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. 159 159 Compare, for example, the following passages: (Fine 1532, IV.1, 131r): "Ut autem universa dierum naturalium discrimina, et mediocrum ad veros dies, aut econtra, reductionem concipere facile possis: theoricam motus ipsius Solis, ad salvandam, supputandamque motus eiusdem Solis circa Mundi centrum observatam irregularitatem subtiliter excogitatam, hoc loco perstringere libet…. Imaginantur itaque prudentiores Astronomi, solarem orbem in tres adinvicem contiguos orbes separari…" See also (Peurbach and Fine 1515, 2v): De Sole. "Sol habet tres orbes a se invicem omni quamque divisos atque sibi contiguos…;" (Fine 1532, 131v): "Fingunt praeterea, circa idem centrum eccentrici, circulum quendam, partem Eclipticae itidem eccentricum nominatum, cujus circunferentia per centrum Solis transire diffinitur…. In hunc porro circulum eccentricum procedentium ex Mundi centro rectarum linearum, maxima est in qua centrum eccentrici…et lon-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. 160 In this part, he used the illustration of the sun's motion which he engraved for his Theorique des cielz (Fine 1528;Pantin 2010). 161 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, 162 providing a table of the lengths of artificial days for the latitude of Paris, 163 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. 164 gior ob id vocatur longitudo, apogium sive augem eiusdem indicans eccentrici: minima vero reli-qua…, quae longitudo brevior respondenter appellatur, et perigium, hoc est, punctum apogio, vel augi denotat oppositum…." See also (Peurbach and Fine 1515, 8v-9v): "Imaginamur autem in sole eccentricum circulum per lineam a centro eccentrici usque ad centrum solare euntem: super centro eccentrici regulariter motam una revolutione facta describi: qui semper est pars superficiei eclyptice orbis signorum octave sphaere. Aux solis in prima significatione sive longitudo longior est punctus circumferentie eccentrici maxime a centro mundi remoto. Et determinatur per lineam a centro mundi per centrum eccentrici utrinque ductam: quae linea augis dicitur. Oppositum augis sive longitudo propior est punctus circumferentie eccentrici maxime centro mundi propinquus et semper augi diametraliter opponitur. Longitudo media est punctus circumferentie inter augem et oppositum augis…;" and (Fine 1532, 131v): "Orbis vero medius, deferens Solem appellatus, circa proprium centrum et axem, secundum ordinem signorum (praeter diurnum, et supradictorum orbium motum) regulariter circumfertur: ita ut Sol de circumferentia proprij eccentrici 59 minuta, et 8 fere secunda partis quotidie perambulet." See also (Peurbach and Fine 1515, 7v): "Sed orbis solare corpus deferens motu proprio super suo centro secundum eccentrici regulariter secundum successionem signorum quottidie lix minutis et octo secundis fere de partibus circunferentie per centrum corpis solaris una revolutione completa descripte movetur." 160 (Fine 1532, IV.1, 131r): "Ut autem universa dierum naturalium discrimina, et mediocrum ad veros dies, aut econtra, reductionem concipere facile possis: theoricam motus ipsius Solis, ad salvandam, supputandamque motus eiusdem Solis circa Mundi centrum observatam irregularitatem subtiliter excogitatam, hoc loco perstringere libet…." This section is maintained in the 1542 unabridged version (Fine 1542a, 48v-50v), where it is described as a Digressio notanda, but not in the later versions. 161 Compare (Fine 1528, 16r) and (Fine 1532, 131v (Fine 1532, IV.2, 135v): [in marg. Alia maximarum vel minimarum dierum supputandi ratio. De supputando maximo lucis arcu, seu continuatae lucis tempore, ad quamvis poli sublimitatem complemento maximae Solis declinatio majorem…]; (136r): "Tabula maximarum dierum artificialium, ab Aequatore circulo, ad polum usque arcticum, gradatum per authorem supputata." 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. 165 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. 166 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, 167 which he complemented with a corresponding table indicating the lengths of shadows according to the altitude of the sun. 168 He also taught the means to calculate the elevation of the sun at a given place, 169 along with a table indicating the altitude of the sun for the latitude of Paris at each hour of the day throughout the year. 170 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 165 (Fine 1532, IV.3, 136r): "De Horis tam aequalibus quàm etiam inaequalibus." (137r-v): [in marg. Diurnae vel nocturnae et inaequalis horae quantitatem invenire…. De mutua horarum conversione. Inaequales horas, ad aequales reducere…. Aequalium ad inaequales conversio…. Ut vulgares horae convertantur in astronomicas]. 166 (Fine 1532, IV.3, 136v): [in marg. Planetam quamlibet horarem temporali diei, vel noctis artificialis dominantem invenire]. and Planetam dominans hora prima. Successio planetarum, pro reliquis horis à prima, tam diei quàm noctis artificialis (table). 167 (Fine 1532, IV.4, 138r-v): "De utraque umbra, recta scilicet et versa, earumque differentis et calculo: unà cum Solarium altitudinum supputatione [in marg. Umbrae diffinitio. Umbra recta. Umbra versa];" (140v): "…quadratum…geometricum…, planisphaerio atque caeteris instrumentis inscribi solitum: quo duce, per alterutrius umbrae intersectionem, rerum altitudines, planicies, et profunditates, hoc est, omnem longitudinem elevatam, jacentem, vel depressam proportionaliter emetimur." See also (Fine 1551a, 43r): "Hinc tractum esse videtur quadratum illud geometricum, quod tum in quadrantibus, tum in Astrolabiorum dorso figuratur." 168 (Fine 1532, IV.4, 137v): (138v-39v) [in marg. Ex Solis altitudine data utriusque umbrae perscrutari longitudinem. Idem per umbrarum absolvere tabulam…]; (139r): "Tabulae utriusque umbrae, rectae scilicet et versae, in partibus qualium umbrosum est 12: ad singulos gradus Solaris altitudinis, per Authorem exactè supputata." 169 (Fine 1532, IV.4, 139v): [in marg. Eandem Solis altitudinem generaliter supputare…. Ut meridiana Solis altitudo colligatur]. 170 (Fine 1532, IV.4, 140r): "Tabula elevationum solis, seu locorum eiusdem in Ecliptica, qualibet hora diei artificialis, Ad poli sublimitatem 48 graduum et 40 minutorum, per Authorem fideliter supputata." 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. 171 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. 172 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. 173 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 equator 174 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 corre-171 (Fine 1532, V.1, 141v): "Inter majores itaque circulos, quos in coelesti sphaera constituimus…. Unà cum singulis datorum quoruncumque locorum parallelis, per ipsa quidem loca liberè, gradatimve ab Aequatore distributis. Ut quemadmodum eorundem coelestium circulorum officio, syderum venamur habitudines: haud dissimiliter per eos, quos super ipso terrestri globo designamus, locorum positiones, atque distantias obtinere valeamus." 172 (Fine 1532, IV.4, 143r): "Tabula demonstrans rationes aequatoris, seu magni cujusvis, ad singulos parallelos, ab eodem Aequatore, versus utrumque polorum ipsius Mundi, gradatim distributos. Primo, in partibus, qualium Aequatoris quadrans perhibetur esse 90. Secundo, in partibus, qualium unus gradus ejusdem Aequatoris est 60." 173 (Fine 1532, V.2, 143v-44v): "De parallelis Climatum distinctoribus: quonam item pacto dato lucis arcu, singulorum parallelorum polares investigentur altitudines;" (143v): "Quanquàm autem haec Climatum excogitatio à vulgaribus Geographis in septenarium redacta sit numerum: nihilominus tamen ab Aequatore versus utrunque Polum, et usque ad eo parallelos, ubi Sol ad diei naturalis quantitatem semel in anno sinè nocte luscescit, 24 sunt annumeranda;" (144r-v): [in marg. Propter quid 7 tantummodo à vulgaribus Geographis, sint ordinata climata. De peculiari climatum nomenclatura. Exemplaris 7 vulgarium climatum descriptio. Qualis vera climatum distributio. Quot sint climata, secundum veram imaginationem]. On the transformation of the theory of climates in the sixteenth century, see (Cattaneo 2009; Leitão and Almeida 2012; Almeida 2017; Valleriani 2017) (Chap. 7). It should be noted, however, that the only diagram that accompanies this chapter presents the ancient division of the climates, and not the modern division, as the later editions from 1551 on would do. Interestingly, the representation of the terrestrial globe in these earlier diagrams that present the seven climate division is more realistic (representating the seas and continents) than in the later diagram that presents the 24 climate theory, which is, on the other hand, closer in style to the more abstract diagram presenting the climates in the Venetian editions of Sacrobosco (for instance, Sacrobosco et al. 1488, BB10r). 174 (Fine 1532, V.2, 144v) sponding maximum length of artificial days. 175 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. 176 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. 177 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. 178 In the fifth chapter, Fine taught the means to measure the distance between two places from their respective longitudes and latitudes. 179 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. 180 In the chapter V.7, which is also the very last 175 (Fine 1532, V.2, 145r): "Tabula polarium altitudinum seu distantiarum ab Aequatore singulorum parallelorum, pro maximarum dierum artificialium quantitate ab eodem Aequatore distributorum: per authorem fideliter supputata." 176 (Fine 1532, V.3, 145v): "De longitudine, atque latitudine locorum: qualiter praeterea tam longitudo, quàm etiam latitudo sit investiganda;" (146r-v): [in marg. Qualiter ex eadem eclipsi Lunari duorum locotum longitudinalis elicitur differentia. Quidnam sit dati cuiuslibet loci latitudo. Latitudinis duorum locorum differentia. Data altitudine Solis meridiana unà cum eius declinatione, latitudinem loci concludere. Idem per stellas fixas quae oriuntur et occidunt respondenter absolvere. Eandem loci latitudinem, per stellas semper apparentes colligere]. 177 (Fine 1532, V.3, 147r-48v): "Tabula longitudinum ab occidente, atque latitudinum ab Aequatore, insigniorum locorum, civitatum et oppidorum, per saniores nostrae melioris Europae regiones constitutorum, Ab Authore recenter verificata." For the list of coordinates Fine takes up from Ptolemy, at least for his map of France, see (Gallois 1918;Dainville 1970;Broc 1983;Conley 1996, 81-110;Pelletier 2009, 13). 178 (Fine 1532, V.4, 148v): "Quantum itineris respondeat uni gradui, vel ipsi toto maximo terrestri circulo: ut etiam locorum itinerariae metiri debeant profectiones;" (149r): [in marg. Quantum itineris capiat unus gradus Meridiani per similium segmentorum observatam respondentiam elicere…. Quantum uni gradui respondeat in Terra, secundum Ptolemaeum. Quantum universus terrestris ambitus per eundem Ptolemeum. Quod directa locorum itinera fieri debeant super magni circuli segmentum demonstratio]. 179 (Fine 1532, V.5, 149v) (Fine 1532, V.6, 152r): "De numero, situ, atque ordine ventorum, ad hydrographiae cognitionem potissimùm spectantium." (152r-53r): [in marg. De numero, ordine, atque positione ventorum 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). 181 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 (1488Münster ( -1552 (Münster and McLean 2016;Mosley 2009). 182 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. 183 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 secundum veteres naturas atque philosophos…. De numero, ordine, atque positione ventorum secundum recentiores hydrographos. Qualia ventorum iuxta navigantes hodiernos nomina. Ventorum lineamenta, qualiter in Cartis hydrographicis describenda. De succendi ventorum hydrographica]. 181 (Fine 1532, V.7, 154r-55r): "Qualiter tandem oblatae cujuscunque regionis, vel partis habitabilis Orbis Chorographia, ex praedictis colligenda sit: quonam item modo hemisphaerica parallelorum atque Meridianorum contextura, ad positionem locorum necessaria in plano rationabiliter extendatur;" (154r): [in marg. Chorographiae gallicae in aliarum exemplum descriptio. Chorographiae ex curvis lineis contextae figurationis exemplum. Ut hemisphaerica parallelorum atque Meridianorum delineanda sit contextura]. 182 On the relation between the teaching of the Sphaera and the exposition of the new geographical discoveries, see (Chap. 7). 183 The fact that, contrary to Apian (Apian 1540, 31v-44v) or even Lefèvre (Sacrobosco and Fine 1538, 11v-13r), he restricted the list of geographical coordinates to European cities was perhaps due to the fact that he wanted to provide the information that he considered most relevant to his addressed readership. 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 sixteenthcentury editions of Sacrobosco's treatise (Crowther et al. 2015). 184 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 position 185 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, 186 as well as by the preface of the Geometria libri duo, where Fine asserts their necessity for the astronomical teaching that follows. 187 This also corroborates the idea presented by Fine in the prefaces of his later editions of the Cosmographia that the 184 See also, for the specific case of the Spanish and Portuguese editions and commentaries on the Sphaera, (Chaps. 7 and 10). 185 Its overarching importance in the Protomathesis is also marked by the fact that it is the only one of the four treatises to present a properly elaborate frontispiece, containing an allegorical representation of the discipline with Fine in the role of the astronomer, and that, moreover, this frontispiece appears both to introduce the corresponding treatise (the Cosmographia) and to introduce the whole compendium. On this frontispiece, see (Pantin 2009b). 186 (Fine 1532 (Cosmographia), I.1, 102r): "Divinam coelestis rationis doctricem Astronomiam, post Arithmeticae atque Geometriae praemissa rudimenta, consequenter adgressuri;" (II.1, 108r): "Quinam sint majores, quive minores in sphaera circuli, decimo capite libri Geometriae nostrae sufficienter annotavimus;" (II.2, 109r): "quod tametsi quilibet in sphaera circulus (uti capite primo libri tertij nostrae docuimus Arithmeticae) in 12 partes invicem aequales a Mathematicis dividatur;" ibid.: "De signorum tandem in gradus sub divisione, et graduum in minuta, et deinceps minutorum in succedentia partium discrimina, praeallegato capite primo libri tertij nostrae Arithmeticae sufficienter tractavimus," etc. 187 (Fine 1532 (Geometria libri duo), 50r): "Non incommodum judicavimus, studiose lector, post Arithmeticae praxim, insignora Geometriae tradere rudimenta. utpote, quae non modo succedentibus nostris geographicis vel astronomicis operibus, passim sese offerunt accommoda: verumetiam universo mathematicarum studio videntur admodum necessaria." 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. 188 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).

Cosmology in the Sphaera and in the Cosmographia 189
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). 190 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). 191 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. 192 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(Pantin , 2012(Pantin , 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 2013Malpangotto , 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), 193 in which an argumentation is presented against the existence of star-190 See, for example, the commentaries of Robert Anglicus, Michael Scot or Cecco d'Ascoli in (Thorndike 1949). 191 Capuano also produced an often reprinted commentary on Peurbach's Theorica planetarum (Peurbach et al. 1495) and which Fine used for his 1515 edition of Peurbach (Pantin 2012). 192 On the medieval and early Renaissance discussions on homocentric spheres, see (Duhem 1913-1959Carmody 1951;Kren 1968;Goldstein 1980;Grant 1994, 563-66;Lerner 1996, I, 99-110;Barker 1999Barker , 2011Shank 2002;Omodeo 2014, 77-79). 193 Agostino Ricci was a converted Jew, a follower of Abraham Zacut, and a Christian kabbalist. He was for that matter the brother of Paolo Ricci, a professor of philosophy at the university of Pavia and later a physician at the court of Emperor Maximilian I, who translated the kabbalistic work less spheres (such as the ninth sphere admitted by Sacrobosco) (Johnson 1946;Pantin 1995, 442;Nothaft 2017) 194 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. 195 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,196 but also Robert Recorde (ca. 1512Recorde (ca. -1558 (Recorde 1556, 10, 278-79) and Christopher Marlowe (1564Marlowe ( -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 Sha'are Orah by Joseph Gikatilla under the title De porta lucis (Gikatilla and Ricci 1516). This translation was the main source of Johann Reuchlin's De Arte Cabbalistica (Reuchlin 1517). On Agostino and Paolo Ricci, see (Goodrick-Clarke 2008). 194 As pointed out in the preface of this edition, Fine would have been incited to edit Ricci's De motu by members of the audience to the lectures he gave around 1520 on the motions of the eighth sphere. (Ricci and Fine 1521, 1v-2r) 195 Fine's diagram of the eight-sphere model is presented in his Theorique des ciels (Fine 1528 and later editions), in all his editions of the Cosmographia, including the frontispiece of the Protomathesis, which was taken up in the unabridged 1542 edition (Fine 1542a); in the frontispiece of the commentary on the Sphaera by Lefèvre d'étaples (Sacrobosco and Lefèvre 1527), and in the In eos quos de Mundi sphaera conscripsit (Fine 1553b). It is also taken up in Charles de Bovelles's Geometrie practique (Bovelles 1551, 53r), for which Fine redrew the illustrations and which he contributed to editing, as well as insure it to be securely printed, as is indicated in the preface of the first edition (dating from 1542): (Bovelles 1551, 2r): "…Orontius Regius Mathematicus…. Duo protinus ingenuè spopondit. se quidem cum primis daturum operam, ut aereis tipis invulgata, plurimis esse visui: figurarum quoque quas ibidem frequentius inscripsi, futurum ligneis in tabellis pictorem. Necnon (quod praecipuum est) adversum mendas observaturum vigiles praeli excubias." 196 (Agrippa 1531, F5v): "Augustinus Ritius, mihi in Italia summa familiaritate devinctus." spheres. 197 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,198 such as Pierre d'Ailly (Sacrobosco et al. 1498;Duhem 1916, 168-71), Prosdocimo de' Beldomandi (ca. 1380-1428) (Sacrobosco et al. 1531;Duhem 1913-1959Markowski 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. 199 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.
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. 220 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. 190ca. 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(Dobrzycki [1965Neugebauer 1975, 633;Grant 1994, 315-16;Nothaft 2017). 221 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), 222 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. 223 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. 1320Oresme (ca. -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. 224 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 motions 225 (and which is a key-argument for dividing the heavens into different ethereal spheres), 226 nor the visible motions of the stars, 221 On the reception of the theory of trepidation in Renaissance France, see (Pantin 1995, 435-56). 222 In these cases, mainly the circularity of celestial motions and the attribution of a particular sphere for each motion. 223 On the debates concerning the ontological status of eccentric spheres and epicycles prior to Fine, see (Chaps. 5 and 6). 224 As shown by Pantin (Pantin 1995, 436-37, 440), the attribution of supplementary motions to the eighth sphere, and thus of additional spheres above it, was detrimental also to the computation of calendars, as it required one to accurately determine the duration of the solar year, as well as to calculate the motions of planets, whose positions were determined in reference to the positions of the zodiacal signs. 225 (Aristotle, De caelo I.2, 269a2-9, II.8, 289b31-32;Metaphysics V.6, 1016a5-6). 226 (Fine 1532 (Cosmographia, I.3), 103v): "Saniores tamen in hoc convenerunt, quod septem sunt orbes planetarum, id est, errantium syderum, utpote Saturni, Jovis, Martis, Solis, Veneris, Mercurij, et Lunae: una cum orbe stellarum fixarum, hoc est, fixam et invariatam inter sese distantiam observantium, quod Firmamentum a fixione syderum appellare solemus. Perceptum est etenim, septem errantia sydera varijs et inaequalibus circunduci motibus, a peculiari stellarum fixarum latione compel us to believe that there are more than eight celestial spheres in the heavens. 227 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. 228 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. 229 Appealing to the authority of Plato (ca. 427-347 BCE), of Aristotle, of Ptolemy, and of Averroës-followed, he claimed, by most mathematicians 230 -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." 231 Moreover, Fine said (again paraphrasing Ricci) that the more recent astronomers who defended the ten-sphere distinctis. At cum stellae non moveantur nisi ad motum orbis (secundo Coeli) necessum est ipsum coelum in tot orbes particulares separari, quot sunt diversi motus astrorum simplices. Si nanque coelum esset continuum, unico simplici motu circunvoleretur (quinto Metaphysicae) quoniam impossibile est idem corpus simplex pluribus moveri simplicibus motibus (primo Coeli)." 227 (Fine 1532, 103v): "Octo igitur praecipui sunt ponendi coelestes orbes: praedictorum videlicet septem planetarum, et Firmamentum omnium aliorum maximum, tot, tamque decoris syderibus ornatum. Supra quem fixarum stellarum orbem, nec syderum claritate, nec aliqua convincente ratione, coelum aliquod mobile cogimur assignare." 228 On the function and the status of the Empyrean heaven in medieval theology and cosmology, see (Grant 1994, 371-89;Bezza 2014). 229 (Fine 1532, 103v): "Admittimus tamen (si universa coelorum non sufficiat machina) coelum empyreum nominatum, felicem beatorum sedem, ne videamur a Theologorum opinione dissentire: id tamen ab omnibus, etiam philosophis, quiescere dicitur." 230 (Fine 1532, I.3, 103v-04r): "Octonario igitur, cum veteribus, et quidem probatissimis Caldeorum, Aegyptiorum, et Graecorum (qui circa motus astrorum philosophati sunt) erimus contenti mobilium coelorum numero. Nec plures divinus ille Plato in de Repub. Epinomides, et de Thimaeo, Aristoteles secundo Coeli, et ejus Commentator Averrois, et Ptolemaeus primo et septimo magnae constructionis possuisse videntur: imo nec universa mathematicorum schola, paucis admodum exceptis." On this issue, it is interesting to note that the historical studies written on this topic from the nineteenth century show, on the contrary, that most precopernican astronomers postulated the existence of at least nine spheres, if not ten, to account for the anomalies which were observed in the motion of the fixed stars (Duhem 1913-1959Grant 1994, 315-18;Lerner 1996, 205). 231 (Fine 1532, I.3, 103v-04r): "quorum aliqui novem, plaeriquae decem, contra tot gravissimos authores imaginati sunt, et stabilium coelestium orbium numerum, nulla cogente ratione, violarunt." See also (Ricci and Fine 1521, 19r): "Stultum igitur per Herculem, atque temerarium esset, contra tot gravissimos tum philosophiae tum mathematicae authores novam sententiam proferre: Et statutum jam per tot temporum curricula coelorum numerum, nulla cogente ratione violare." system "wrongly attributed such an extravagance to Ptolemy, to King Alfonso and to Johannes Regiomontanus." 232 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. 233 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. 234 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.
As shown by discussions that were raised on this issue since antiquity, 250 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, desdits 12 signes du zodiac: pareillement lesdits luminaires et planetes, en tournant environ et au tout de la terre par le premier et universel mouvement de tout le monde, d'orient par le mydi en occident, dedans le temps et espace de 24 heures (que nous appellons jour naturel) causent de rechef tous les jours de l'an divers et particuliers effects selon qu'ils sont en orient ou occident, et notablement eslevez ou deprimez sous la terre, et qu'ils ont plus forte ou debile irradiation sur lesdites choses de ça bas." 248 (Fine 1553a, 4r): "Nam quemadmodum sydera (potissimum errantia) Zodiacum ipsum gradatim perambulando, pro diversa suorum radiorum in haec inferiora projectione, proprias vires imprimere, naturalesve potestates multifariam exercere videntur: Non aliter ad universalem motum caeli (quem diurnum, sive primum appellant) absque intermissione revoluta, diversas irradiationes, tum ascendendo super horizontem, aut sub illo descendendo, tum ipsum praeterlabendo meridianum, intra diem naturalem censentur contrahere: et horum propterea inferiorum qualitates, sive naturas, pendenter immutare." 249 On the relation between astrology and cosmography in the sixteenth century (as considered through the case of academic and courtly environment of Louvain), see (Vanden Broecke 2003, 113-18 and 129-33). 250 On these earlier discussions, see (Duhem 1913-1959. and considered that the influence of the signs of the zodiac was stronger when the visible and the invisible signs were superposed (Chap. 4). 251 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. 252 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. 253 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. 254 Ricci referred to this discourse in the above-mentioned section of his De motu (Ricci and Fine 1521, 18r, 21v), 255 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. 256 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) 257 and came up later on several occasions, notably in the second edition of the Cosmographia (unabridged version), 258 but also in the 1532 edition of the Cosmographia when dealing with the sun's theorica. 259 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, 260 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." 261 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.
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. 262 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. 263 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. 264 This discourse of Ptolemy 262 This is the great question raised by P. Duhem, in Σῴζειν τὰ φαινόμενα (Duhem 1908), in which he deploys an instrumentalist interpretation of precopernican astronomy. 263 This has been shown by several historians of astronomy in the last decades, among which (Lloyd 1978;Jardine 1979Jardine , 1982Westman 1980;Barker and Goldstein 1998;Morelon 1999;Barker 2011). 264 (Ptolemy 1984, XIII.2, 600): "Now let no one, considering the complicated nature of our devices, judge such hypotheses to be over-elaborated. For it is not appropriate to compare human [constructions] with divine, nor to form one's beliefs about such great things on the basis of very 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," 265 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," 266 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 dissimilar analogies. For what [could one compare] more dissimilar than the eternal and unchanging with the ever-changing, or that which can be hindered by anything with that which cannot be hindered even by itself? Rather, one should try, as far as possible, to fit the simpler hypotheses to the heavenly motions, but if this does not succeed, [one should apply hypotheses] which do fit. For provided that each of the phenomena is duly saved by the hypotheses, why should anyone think it strange that such complications can characterise the motions of the heavens when their nature is such as to afford no hindrance, but of a kind to yield and give way to the natural motions of each part, even if [the motions] are opposed to one another? Thus, quite simply, all the elements can easily pass through and be seen through all other elements, and this ease of transit applies not only to the individual circles, but to the spheres themselves and the axes of revolution. We see that in the models constructed on earth the fitting together of these [elements] to represent the different motions is laborious, and difficult to achieve in such a way that the motions do not hinder each other, while in the heavens no obstruction whatever is caused by such combinations. Rather, we should not judge 'simplicity' in heavenly things from what appears to be simple on earth, especially when the same thing is not equally simple for all even here. For if we were to judge by those criteria, nothing that occurs in the heavens would appear simple, not even the unchanging nature of the first motion, since this very quality of eternal unchangingness is for us not [merely] difficult, but completely impossible. Instead [we should judge 'simplicity'] from the unchangingness of the nature of things in the heaven and their motions. In this way all [motions] will appear simple, and more so than what is thought 'simple' on earth, since one can conceive of no labour or difficulty attached to their revolutions." 265 (Fine 1532, 102v): "…consyderat Astronomia, utpote, coelestes globos, sydera, eorum motus, et passiones, ac ejuscemodi." 266 (Fine 1532, 102r): "Nam subjectum Astronomiae est ipsum coeleste corpus, inter omnia corpora praestantissimum, omni prorsus alteratione privatum, supremo et nobiliori loco, motuque circulari omnium motuum priori ac perfectiori decoratum." 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. 267 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).

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, 268 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.
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