Regiomontanus on , Physical Orbs, and Astronomical Fictionalism: Goldsteinian Themes in the “Defense of Theon

against George of Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 Trebizond”

Michael H. Shank University of Wisconsin-Madison To honor Bernard Goldstein, this article highlights in the “Defense of Theon against ” by Regiomontanus (1436–1476) themes that resonate with leading strands of Goldstein’s scholarship. I argue that, in this poorly-known work, Regiomontanus’s mastery of Ptolemy’s mathematical , his interest in making astronomy physical, and his homocentric ideals stand in unresolved tension. Each of these themes resonates with Gold- stein’s fundamental work on the , the Planetary Hypotheses, and al-Bitruji’s Principles of Astronomy. I flesh out these tensions with an intriguing interpretation of the history of astronomy, in which Regiomontanus contrasts the two-dimensional “eccentric astronomy” attributed to the Almagest and the Arabs with the three- dimensional spheres of the “later” astronomers. A similar contrast reappears when Regiomontanus portrays as a “ªctitious art” an astronomy that does not go beyond the accommodation of computations to the appearances. To con- clude, I use Regiomontanus’s expression to reinstate (pace Goldstein and Barker in this journal, 1998) ªctionalism as an actor’s category in Osiander and sixteenth-century astronomy.

I. Introduction Sooner or later, historians of science who work in ancient, medieval, or early modern astronomy must come to terms with the work of Bernard

I thank Dr. V. Soboliev and the Archive of the Russian Academy of Sciences (St. Petersburg branch), without whose hospitality and permission to use their holdings, this research would not have been possible. I also wish to thank Richard L. Kremer and David Lindberg for their criticisms and comments; the International Research and Exchanges Commission, the National Science Foundation (grant SBR-9729712), the Graduate School of the Uni- versity of Wisconsin-Madison, and especially the UW’s Institute for Research in the Hu- manities for their support.

Perspectives on Science 2002, vol. 10, no. 2 ©2003 by The Massachusetts Institute of Technology

179 180 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

Goldstein. His contributions span not only an enormous chronological range, but also multiple languages, even more cultures, and often the in- teractions between them. His work in almost every case has opened up new vistas on the ªeld. His research has frequently taken a thorough look at neglected ªgures, works, or problems, exploiting his knack for demon- strating that some marginalized topics were in fact central episodes of the history of astronomy. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 His recovery of the missing half of Book I of Ptolemy’s Planetary Hy- potheses is no doubt his most famous ªnd (Goldstein 1967), but many other important examples abound. He has brought out of the shadows several crucial issues and ªgures in the history of astronomy and its cultural pere- grinations. Consider Ibn al-Muthanna, whose tenth-century commentary on the zij of al-Khwarizmi is now available in a Goldstein edition and translation. Preserved in Hebrew, this text offers clues about the fate of Greek astronomy at the interface between India and the Arabic world. Or consider the original homocentric astronomy of the twelfth-century Arabic-writing Andalusian al-Bitruji (º. 1200), the edition and transla- tion of which we also owe to Goldstein’s scholarship. The Latin world im- mediately picked up al-Bitruji’s Principles of Astronomy, which became grist for the cosmological mill down to the time of Copernicus. The list must be cut short, but not without mentioning a ªnal, signally important ex- ample. Consider Levi ben Gerson (d. 1344), the polymath rabbi from Bagnols in Provence, who worked at the interface between Hellenistic, Hebrew, Arabic, and Latin science. Thanks to Goldstein’s analyses, Levi has become the most original astronomer of fourteenth-century Europe (Goldstein 1985, and most recently 2001). To gauge where this scholar- ship has taken us, one need only notice how impoverished the ªeld looks if the insights gained from these ªgures, works, and analyses drop out of our historical consciousness. A second striking feature of Goldstein’s work is its conceptual range. He has dealt with a spectrum of problems in the history of astronomy that stretches from the geometrical foundations of theoretical models, through problems of parameters, observations, instrumentation, and the computa- tion of tables, to issues of cosmology including, in each case, the philo- sophical underpinnings and cross-cultural diffusion. He has proved him- self a rare specimen of scholar—a hybrid between a hedgehog, a fox, and a gadºy, who does not shrink from asking unsettling critical questions about our favorite narratives or sources. What did Aristotle and his con- temporaries really know about planetary motion anyway? Just what was the relation between theory and observation in medieval astronomy? What difference would it make if historians of astronomy took more seri- ously Pliny’s clues about the state of the ªeld in the ªrst century? (for ex- Perspectives on Science 181 ample, Goldstein 1972 and 1997b). The list of probing questions could go on. Finally, Bernie Goldstein is one of the most cooperative scholars in the ªeld. The fact that he has co-authored articles with more than half a dozen other scholars with very diverse interests says much about the liveliness of his curiosity. In light of this short bio-bibliography, the title of my paper may seem Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 tautological. If Goldstein had touched only the strands of the ªeld men- tioned above, almost any work in medieval astronomy contains some Goldsteinian themes. My title, however, aspires to be an informative. As we shall see, speciªc and strong resonances between Goldstein’s work and that of Johannes Regiomontanus (1436–1476) create challenging prob- lems for our interpretation of the latter’s work. As historians, we can coun- tenance simultaneously such diverse, even contradictory, approaches as the mathematical framework of the Almagest, a physical world picture that models its theories in three dimensions, and a homocentric ideal. Regiomontanus, however, was trying hard to bring these disparate tradi- tions together as a working astronomer. These tensions in his outlook emerge most clearly in a long and long-neglected work that he called “The Defense of Theon against George of Trebizond.”1 This text brings to- gether several puzzling aspects of Regiomontanus’s thought and practice and forces us to deal with them at the core of his work, instead of letting them fester on the fringes. After a brief overview of Regiomontanus’s life and the growing interest in his work during the last generation, I sketch the usual image of him as a mathematical astronomer in the mold of Ptolemy’s Almagest. While this characterization deserves a permanent place in all subsequent evaluations of his work, it conveys a surprisingly incomplete picture of his concerns and of the tensions between them. While we know about some of Regiomontanus’s answers and see how they ªt our own historical prob- lems, the “Defense of Theon” can take us much farther in the direction of understanding what problems he had set for himself. Second, I summarize the relatively recent puzzle of Regiomontanus’s abiding interest in homocentric astronomy, that is, in an astronomy that dispenses with such fundamental Ptolemaic tools as eccentrics and epicycles. Third, I intro- duce brieºy Regiomontanus’s “Defense of Theon,” in which his mastery of traditional mathematical astronomy, his interest in making astronomy physical, and his homocentric ideals stand in direct tension. By coinci- dence, each of these themes recalls the Goldsteinian Leitmotive of the Alma- gest, the Planetary Hypotheses, and al-Bitruji’s Principles of Astronomy.

1. Richard Kremer and I are bringing out the text and collaborating on several studies. 182 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

Finally, I lift out of the “Defense of Theon” two interrelated themes that further resonate with Goldstein’s scholarship. The ªrst is Regiomon- tanus’s intriguing interpretation of the history of astronomy, which shows how surprised, and perhaps pleased, he would have been to come across the complete version of Ptolemy’s Planetary Hypotheses. The second theme is Regiomontanus’s reºection on the status of astronomical models, which offers new insights on the much vexed question of realism and ªctionalism Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 (or instrumentalism) in the astronomy of the ªfteenth and sixteenth cen- turies. Goldstein, in collaboration with Peter Barker, has in this journal made a recent contribution to the problem of Osiander’s preface in partic- ular (Barker and Goldstein 1998). I will use the “Defense of Theon” to of- fer a brief challenge to their interpretation.

II. Brief biography Johannes Regiomontanus (1436–1476) ªrst appears in the university re- cords of Leipzig as Johannes molitoris (“John of the miller”). To this desig- nation, he himself sometimes added de Künigsperg, de Monte Regio, or de Regio monte. He was therefore almost certainly born into a miller’s family in the other Königsberg, a small Franconian town (Zinner 1968, pp. 7, 13). His career can be divided into four phases. From 1448 to 1461, he was associated with the universities of Leipzig and (for two and eleven years, respectively). During his Viennese years, he developed a close working and personal relationship with Georg Peuerbach (1423(?)-1461), who was an astronomer, university master, teacher of Latin poetry, and also an astrologer at the nearby royal courts of Bohemia and Hungary and at the imperial court of Frederick III of Habsburg. In 1460, Peuerbach and Regiomontanus met the Greek Cardinal , who was in Vienna on an extended diplomatic mission for the papacy, advocating a crusade to re- take Constantinople and the Eastern Empire from the Ottoman Turks. The year-and-a-half interaction that followed this meeting proved cru- cial for astronomy and for the “Defense of Theon” in particular. Beyond his political and ecclesiastical duties, Bessarion had many interests, in- cluding astronomical ones that dated back to his studies in the Greek mathematical sciences and philosophy as a youth in Byzantium. His ur- gent goal of trying to save the Greek intellectual heritage from destruc- tion by the Ottoman Turks led him to assemble the largest library of Greek manuscripts in Italy and to pay close attention to the diffusion of this heritage (Mohler [1923] 1967, I, pp. 408–415; Shank 1992a, p. 128). In particular, Bessarion was sharply critical of the 1451 Latin translation of Ptolemy’s Almagest and the lengthy commentary on it by George of Trebizond, a teacher of rhetoric who had served as a translator for various popes, especially Nicholas V (Monfasani 1976, esp. ch. 3). As a coun- Perspectives on Science 183 ter-measure to George of Trebizond’s ºawed commentary, Bessarion urged Peuerbach to write an epitome of the Almagest. When Peuerbach’s prema- ture death in 1461 left the Epitome half-ªnished, Regiomontanus not only took over the project, but also accompanied Bessarion back to Italy. During the second phase of his career (his Italian years from 1461 to the mid-1460s), Regiomontanus completed the crucial second half of the Epitome of the Almagest. His contribution to the project did more than Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 merely summarize Ptolemy’s work; it also examined it critically and in in- teraction with the intervening developments in mathematical astronomy. The last two phases of Regiomontanus’s short life divide equally his ªnal decade. From Italy, he moved in the mid-1460s to Hungary, where he was associated with the circle of Archbishop Janós Vitéz (an old ac- quaintance of his and Peuerbach’s) and the court of King Matthias I Corvinus. When these two patrons turned on one another in 1471, Regiomontanus returned to his native . In , he set up an instrument shop and a printing press to diffuse mathematical and as- tronomical works. While away in Rome, he died in 1476, at age forty, just as his printing and astronomical programs were hitting their stride.

III. Regiomontanus as mathematical astronomer As this overview suggests, the usual picture of Regiomontanus as a mathe- matical astronomer in the vein of the Almagest has much to recommend it. Not only does the Epitome of the Almagest follow closely the structure and geometrical approach of Ptolemy’s work, but also Regiomontanus’s tech- nical analysis in this work is of exceptional historical importance. His crit- ical acumen was to provide Copernicus with a crucial conceptual stepping stone to the emergence the heliocentric theory. At the beginning of Book 12 of the Almagest, Ptolemy discusses the second anomaly of the planets (also known as “the anomaly with respect to the Sun”), the most striking manifestation of which is the retrograde mo- tions. Ptolemy here claims that one can use two different but equivalent models for the superior planets (Mars, Jupiter, Saturn). The ªrst model in- volves a deferent carrying an epicycle that rotates about its center with the speed of the mean Sun. The second uses an eccentric whose center is mov- ing about the Earth with the speed of (and in the same direction as) the mean Sun. For reasons that have never been clearly understood, Ptolemy states that this eccentric model works only for the superior planets, imply- ing that only the deferent-epicycle model works for the inferior planets (Venus and Mercury). When Regiomontanus’s work on the Epitome reached Book 12, instead of merely summarizing Ptolemy, he proved the equivalence of the two models for both the superior and the inferior plan- 184 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism ets, without advertising the fact that he thereby had proved Ptolemy wrong.2 Thirty years ago, in an important argument that is still poorly known in the wider discipline, Noel Swerdlow showed how Regiomontanus’s ec- centric models for the second anomaly point to, and transform directly into, two well-known planetary arrangements—the Tychonic and the Co- pernican. Since the deferent-epicycle model of each superior planet has an Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 epicycle that revolves about its center with the period of the mean sun, the eccentric model for the second anomaly transforms into the geometrical arrangement that we now associate with ’s cosmos (i.e., the planets revolve around the mean Sun, which revolves about a stationary Earth). Since the inferior planets have a deferent rotating with the period of the mean Sun, the eccentric model of the second anomaly transforms into the Copernican system (i.e., the inferior planets revolve around the mean Sun). Copernicus certainly used Regiomontanus’s Epitome, available in the 1496 edition, which probably provided him not only with his ªrst de- tailed acquaintance with Ptolemy, but also with the keys to the world pic- ture for which he is famous. Swerdlow found the smoking gun when he ªgured out the meaning of Copernicus’s manuscript “Uppsala notes,” pre- served in his copy of the 1492 edition of the Alphonsine Tables. Here Co- pernicus was calculating “the proportion of the celestial orbs to an eccen- tricity of 25 parts.” He was measuring the relative sizes of the planetary spheres using as his yardstick an “eccentricity” set at 25 units, that is, the arbitrary number he used for the distance between the earth and the mean Sun in his Commentariolus (Swerdlow 1973, pp. 423–512, esp. 427–29,

2. Today, we may freely visualize the equivalence of the epicyclic and eccentric models merely as the epicycle and the deferent trading places. In one case, a large deferent with ra- dius R carries a small epicycle with radius r; in the other (called “eccentric”), a small defer- ent with radius r carries a large epicycle with radius R. When we take this approach, how- ever, it is difªcult for us to see why this is not one single model represented now with one pair of vectors, now with another. It is therefore important to note that neither Ptolemy nor Regiomontanus saw the problem in this way. Both act as if there is an ontological dif- ference between the deferent–epicycle model and the eccentric one—even when the center of the eccentric is in motion. The distinction is obviously the sharpest and the most puz- zling in Ptolemy. Although he was no mean geometer, something in his outlook made him see the models as asymmetrical. But the distinction remains present in Regiomontanus, who proved their geometrical equivalence. Such ontological distinctions are also present in Theon of Smyrna (early 2nd century), who favors the epicyclic hypothesis as being in the more common, the more generally conceded, in greater conformity with the nature of things, and favored by Plato(!), whereas the eccentric is described “by chance/accident”. Theon chides Hipparchus for being insufªciently attuned to the science of nature ( physiologia) and failing to understand this distinction; Theon of Smyrna 1979, pp. 121–22; Dupuis (1892) 1966, pp. 303–5. Perspectives on Science 185

471–78; more accessible summaries of the argument also appear in Swerdlow and Neugebauer 1984, pp. 54–59; and Swerdlow 1996, esp. pp. 197–200). As Swerdlow has argued, the rationale behind this calcula- tion and the key to its unusual vocabulary are both found in Regiomontanus’s alternative eccentric models for the second anomaly. Apart from this background, Copernicus’s expression “eccentricity of 25 parts” has little meaning. Recall that in the eccentric model for the second Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 anomaly, the center of each eccentric shares the motion of the mean Sun, suggesting that it just might be the mean Sun. Measuring the planetary spheres with this 25-unit standard eccentricity yielded an ordered hierar- chy of planetary spheres from Mercury to Saturn (with, of course, the Earth itself at 25). Thus Copernicus’s ªtting the Earth and the planets around the mean Sun yielded their relative distances around the mean sun, and the “wonderful symmetry/commensurability of the universe” touted with such delight in the ªrst book of his De Revolutionibus (Copernicus [1543] 1966, f. 10r). This account of the emergence of Copernican theory is the most compelling and speciªc that we have. It depends crucially on Regiomontanus’s contribution to the Epitome of the Almagest. The analyses in that work prove him to be not only a critical mathematical astronomer who did not take Ptolemy’s assertions for granted, but also one whose al- ternative analyses played a very signiªcant role in the history of as- tronomy.3

IV. Regiomontanus’s Homocentric Interests We do not yet know speciªcally what, apart from his compulsive thor- oughness, motivated Regiomontanus to explore the eccentric models of the second anomaly.4 But it is now certain that Regiomontanus had stron- ger interests in the physical side of astronomy than anyone has suspected, and these led him to collect and explore a variety of approaches to plane- 3. Jerzy Dobrzycki has recently suggested that most of Copernicus’s postulates in the Commentariolus are deliberate counterparts to those of the Epitome of the Almagest; Dobrzycki 2001, pp. 223–225. 4. It is, however, suggestive that Theon of Smyrna’s Mathematical Knowledge Useful for Reading Plato (early second century) broaches at some length the differences and equiva- lences between epicyclic and eccentric models of the planets. After proving the equivalence of eccentric and epicycle for the Sun, he asserts that the same demonstration applies to the other planets. Surprisingly, he favors the epicyclic as being in greater accordance with na- ture, while the eccentric is an accident—effectively the trajectory of the planet moving on the epicyclic sphere (Dupuis [1892] 1966, pp. 251–311, esp. 279, 305). Regiomontanus was probably aware of the work, for the only surviving manuscripts of it were in Bessarion’s library. The astronomical portion of the work is now Venice, Bibl. Marciana, gr. 303, appearing in the 1468 and the 1474 inventories of Bessarion’s library (numbers 257 and 573, respectively); Labowski 1979, pp. 167, 222, 467; Huxley 1970–1980, p. 326. 186 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism tary modeling. In particular, Regiomontanus was decidedly biased in fa- vor of a homocentric astronomy. This approach was, after Aristotle, tradi- tionally physical rather than mathematical. Antedating Apollonius of Perga, it made no use of epicycles and deferents. After Hipparchus, and es- pecially after the Almagest, however, homocentric astronomy had to con- front these geometrical devices. For some thinkers (Alexander of Aphrodisias, Geminus, Simplicius), the homocentric and epicyclic ap- Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 proaches were simply assigned to different ways of thinking—the physical and the mathematical. For most advocates of a concentric astronomy, how- ever, the Almagest’s predictive success did not excuse its violation or ne- glect of physical principles. Accordingly, they urged the elimination of epicycles and eccentrics on the grounds that these were imaginary circles, not physical entities. Instead, homocentric astronomy sought a physical account of celestial motions compatible with the rotation of physical spheres concentric to the Earth. In the Latin Middle Ages, individuals steeped in the Aristotelian natural philosophical tradition were its primary defenders. Like Averroes (Ibn Rushd) and Alpetragius (al-Bitruji), its chief advocates in twelfth-century Muslim Spain (Sabra 1978 and 1984; Langermann 1990), they were grappling primarily with the problem of the structure of the universe and did not seek to predict planetary positions from it.5 Al- though they could not help being impressed by the predictive success of the problematic epicycles and eccentrics, they tried to work out the geo- metrical details of a homocentric astronomy, the most sustained effort be- ing that of al-Bitruji (º. 1190 in Spain). Al-Bitruji presented his work as having one goal, which he had reached: to show that all celestial motions followed from the daily motion. He candidly conceded at the end of his treatise that he had not succeeded in doing by homocentric means every- thing that the Almagest could do, nor could he, even if he were to spend his life at it (Goldstein 1971, I, p. 154). Al-Bitruji’s work had considerable diffusion in Latin, thanks to Mi- chael Scot’s 1217 translation, which presented the Principles of Astronomy as remedying the incompleteness of Aristotle’s views “on the constitution of

5. It would, however, be a mistake to assume that all natural philosophers favored homocentrism or criticized the Ptolemaic devices. Thus the fourteenth-century natural philosopher Albert of Saxony dismissed out of hand Averroes’s notion that “the opinion of the concerning epicycles and eccentrics is impossible.” He pointed out that Averroes mentions in the commentary on the Metaphysics, book 12, that “in his youth he had hoped to attain an astronomy other than that of Ptolemy, who posits epicycles and eccentrics, but in old age he despaired [of it] and therefore his position and proof here [in Averroes’s commentary on the De caelo] must be judged useless.” Albert of Saxony (1492) 1986, ff. [E4ra], [E4va]. Perspectives on Science 187 the world.” The heyday of Latin homocentrism occurred in the ªrst half of the thirteenth century. Criticisms of it as impossible became more numer- ous in the second half of the century, although Roger Bacon remained a prominent advocate until his death in 1292 (Avi-Yonah 1985, pp. 126–127). Even after its heyday, al-Bitruji’s work continued to feed an undercurrent of dissatisfaction with, and skepticism about, epicycles and eccentrics in the Latin world between the later thirteenth and sixteenth Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 centuries, even among those who used them. On the face of it, these physical concerns seem very remote from the geometrical formality of Regiomontanus’s Epitome of the Almagest or his computations of tables. Yet, already as a student and master in Vienna during the 1450s, he was collecting and copying polemical works that at- tacked and defended homocentrism. Although he was once thought an op- ponent of homocentrism, it now seems certain that Regiomontanus did not compose the one critique of homocentrism long attributed to him (Shank 1992b). On the contrary, in 1460 he wrote to János Vitéz, at the time the bishop of Várad, a letter that brieºy describes homocentric mod- els for the Sun and Moon. In it, he promises “a new work in four treatises” that will refute “[Ptolemy’s] old theory of eccentrics and epicycles” and lead to new astronomical tables. In short, this project was to create a sin- gle theory that would be both physically satisfying (i.e., based on concen- tric spheres) and mathematically potent (i.e., capable of generating predic- tions), a feat that no homocentric schema had ever accomplished. Swerdlow’s recent edition of the “Letter to Vitéz” and his analysis of its models show that Regiomontanus’s proposed homocentric models for the Sun and Moon formally look like a modiªcation of al-Bitruji’s models, in- deed the very modiªcation that Goldstein himself had proposed to make al-Bitruji workable (Swerdlow 1999, esp. p. 4).6 When the best mathematical astronomer and expositor of Ptolemy in ªfteenth-century Europe also seeks to do away with epicycles and eccen- trics, the historian clearly faces a puzzle. Even as he was improving upon Ptolemy’s use of these devices in the Epitome, Regiomontanus was looking for alternatives that would dispense with them altogether. It would be convenient for us if the predicament were limited in time. Since, as far as we now know, Regiomontanus’s most elaborate discussion of homocentrism occurs in the “Letter to Vitéz” (1460), which slightly ante- dates the Epitome (1462), it is tempting to treat the letter as a temporary

6. Whatever similarities we may see between these approaches, Regiomontanus consid- ered himself neither a disciple, nor a modiªer of al-Bitruji. On the contrary, he cast asper- sions upon the latter’s views, from his unsatisfactory homocentric models (see note 12 be- low) to his placement of Venus above the Sun (“Defensio Theonis,” 155v-156r). 188 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

“phase” of cognitive dissonance. Yet brief references sympathetic to homocentrism occur in Regiomontanus’s correspondence with Giovanni Bianchini (ca. 1463–64). From the absence of spherical solutions for the equations of the Sun and Moon in Regiomontanus’s “Tabula primi mobilis,” Swerdlow suggested that Regiomontanus had eventually given up on his homocentric project (Swerdlow 1999, p. 5). The “Defense of Theon against George of Trebizond,” however, suggests that Regiomon- Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 tanus remained committed to a homocentric goal during his Nuremberg years (Shank 1998, p. 162). Indeed, the difªculties with his project not- withstanding, it seems likely that he went to his grave hoping for a homocentric solution to the inconsistencies he saw between Ptolemy’s suc- cessful predictions of planetary positions and his neglect—as Regiomon- tanus perceived it—of physical considerations.

V. The “Defense of Theon against George of Trebizond” Before exploring some of the reasons behind this predicament, a brief in- troduction to the work is in order. Although the “Defense of Theon” now survives only in the autograph manuscript, it was not a secret diary or a set of private notes. It was one long battle in a public twenty-year intellectual war between Cardinal Bessarion and George of Trebizond.7 Regio- montanus had started the work in the early 1460s while a member of the Bessarion household; he extended it in Hungary, at the court of King . Internal evidence shows that he was still working on the “Defense” in the early 1470s in Nuremberg. Nevertheless, the work was already circulating in an incomplete state among trusted friends: the trace of one copy, now lost, appears in the 1474 inventory of Bessarion’s li- brary (“Corruptio Theonis per Trapezuntium et tres quinterniones Ioannis contra eum, in papiro, non ligatus;” see Labowski 1979, p. 238). More ambitious diffusion plans were afoot: Even after Bessarion and Trebizond had died, Regiomontanus intended to print the “Defense,” for he publi- cized it ca. 1474 on the trade list of works forthcoming from his press (Schmeidler 1972, p. 533). His premature death cut all of these plans short. For most of the time since the early nineteenth century, the manu-

7. The signiªcance of the controversy with George of Trebizond in the life of the Bessarion circle may explain Regiomontanus’s move to Hungary. After 1467, George of Trebizond tried to ingratiate himself with the Hungarian court. He dedicated his transla- tion of the Almagest to King Matthias Corvinus and his translation of St. Basil’s Adversus Eunomium to Regiomontanus’s old correspondent, János Vitéz, who was now archishop of and Corvinus’s chancellor (see Monfasani 1976, pp. 194ff). It would scarcely be surprising if Bessarion had dispatched Regiomontanus to Hungary to undercut Trebizond’s bid for inºuence, which evidently proved unsuccessful. Perspectives on Science 189 script, along with two other Regiomontanus autographs, has been in St. Petersburg, Russia. All three are currently housed in that city’s branch ar- chive of the Russian Academy of Sciences.8 The “Defense of Theon against George of Trebizond” is a work of con- siderable importance, for it culminates what is arguably the longest and most acrimonious controversy in all of Latin astronomy before the late six- teenth century. Its target, George’s commentary on the Almagest, is both Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 his most extensive treatise, and the longest quattrocento mathematical work by a humanist (Monfasani 1984, pp. 673–74).9 Both works follow the thirteen-book structure of the Almagest, and interact vigorously with it and the subsequent commentary tradition. Together they comprise roughly 1200 pages of unstudied astronomical polemics in the two de- cades before Copernicus’s birth. In sharp contrast to the Epitome’s formality, the “Defense of Theon” is an angry work that brings out many of Regiomontanus’s working assump- tions. It has its share of geometrical proofs and arguments, but it mixes these with invectives, insults, and asides aimed at George of Trebizond’s equally massive and equally polemical “Commentary on the Almagest.” The criticisms range from howlers and errors in geometrical proofs to substantive differences in both outlook and approach to as- tronomy. While the precise place of Theon as a lightning rod in this affair re- mains to be elucidated, this much is certain: in his own commentary on the Almagest, Trebizond had attacked Theon, whose commentary Bessarion admired. There are, in addition, hints that Regiomontanus thought George was plagiarizing Theon when he was not attacking him.10 In most other places, Theon does not surface: Regiomontanus simply thought George’s interpretations of Ptolemy and criticisms of later astron- omers wide of the mark or plainly incompetent. The “Defense” itself touches on issues ranging from methodology through mathematical modeling and computation to physical astronomy and snippets of natural philosophy. A preliminary transcription of the text 8. The other two autographs contain De triangulis omnimodis (which Petreius eventually printed in Nuremberg in 1533) and the “Commentary on the Geography” (which Willibald Pirckheimer printed at Strassburg in 1525). 9. The “Defensio” is a large work as well, surpassed only by Regiomontanus’s 896-page Ephemerides (Nuremberg, 1474). 10. The twentieth-century editor of Theon of Alexandria, A. Rome, did not know about the “Defense of Theon,” but he had independent reasons for agreeing with Regiomontanus: he promised to discuss George’s plagiarism in the last volume of his own edition of Theon’s commentary, which remained incomplete at Rome’s death (Rome 1931, p. vi). 190 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism is nearing completion. The present remarks are perforce still based on fragmentary soundings rather than on a sure grasp of the whole. Neverthe- less, the early results are exciting and puzzling, not least in the matter of Regiomontanus’s homocentrism.

A. Homocentrism in the “Defense of Theon” Near the beginning of the work, Regiomontanus promises the following: Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 “if time permits...[to] demonstrate how we may save these anomalies in the motions of the Moon without an eccentric or an epicycle, not in the manner of Alpetragius, which is completely un- satisfactory, but in some new and most appropriate way. Nor does it proceed from free will that I should choose or ªnd/invent a manner other than that of the throng of astronomers; in truth, a convincing reason why there is no lunar eccentric and epicycle compels it. For if there is [such a thing], it follows that the angle that the Moon subtends in the center of vision at one moment is almost twice that seen at another, other things being equal, namely the disposition of the air and the relation of the Moon to the horizon and meridian. From this it follows that the area of the Moon, if it appears as one in a speciªc location, appears as nearly four in another, which no one ever sees.”11 Regiomontanus’s observational rationale for eliminating eccentrics and epicycles in lunar theory, which he had brought up on several occasions, was not original.12 At the , where he had been

11. “Quod si posthac per otium licebit demonstratum dabo quonampacto huiusmodi diversitates in motibus lune absque ecentrico et epiciclo salvare possimus, non per modum Alpetragii, qui nequaquam satisfacit, sed per viam quandam novam et convenientissimam. Neque id ex arbitrio libero procedit quod elegerim sive invenerim alium modum quam vulgus astronomorum habeat, verum ad hoc impellit ratio convincens non esse ecentricum et epiciclum lune. Nam si sic, sequitur angulum quem subtendit luna in centro visus pro certo instanti fere duplum esse ad eum sub quo videtur luna in certo alio instanti, rebus ceteris eodem modo se habentibus, videlicet dispositione aeris, habitudine lune ad orizontem et meridianum; hinc quoque sequitur aream lune si in certo situ apparuerit ut unum, in alio situ apparere fere ut quatuor, quod nemo unquam deprehendit.” Regiomontanus, “Defensio Theonis,” St. Petersburg, Archive of the Academy of Sciences, IV-1–935, 7r (cited hereafter as “Defensio Theonis”). 12. Regiomontanus also mentions this argument, together with others about impossi- ble variations in the areas of Mars (52:1) and Venus (45:1), in his last letter to Giovanni Bianchini; Maximilian Curtze (1902) 1990, p. 265; passages translated in Swerdlow 1990, p. 173. Although Regiomontanus implies that eccentrics and epicycles are at the root of these problems, he does not there draw the conclusion that a homocentric system would be preferable. He had mentioned the desirability of a homocentric arrangement, however, in Perspectives on Science 191 trained, the argument went back at least to Henry of Langenstein’s “De reprobatione ecentricorum et epicyclorum.” Langenstein had written his criticism of epicycles and eccentrics in Paris in 1363 and evidently brought it to Vienna when he moved there permanently twenty years later (Kren 1968 and 1969). It was here that Regiomontanus encountered Langenstein’s text, which he copied into his notebook. Immediately before Langenstein, the argument appears in Levi ben Gerson, who also used it, Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 as Goldstein has shown, to provide a rationale for his own alternative to Ptolemy’s lunar theory (Goldstein 1985, p. 105; and 1997a).13 More striking than Regiomontanus’s argument is the role of intentionality and necessity in it, notably his contrast between free will and compulsion in departing from traditional astronomy.14 Contrary to ap- pearances, he argues, his pursuit of an alternative to eccentrics and epicycles is not exceptionalist. Indeed it is not even properly a pursuit, for it is dictated not by choice, but by necessity—the logical necessity of mo- dus tollens. It is important to notice that the use of modus tollens here makes sense only on the assumption of a physical necessity. Regiomontanus does not even consider the possibility that the lunar epicycle might be a geometrical device without physical signiªcance. His argument goes through only on the assumption that the epicycle was intended to play a physical role and, on those grounds, it cannot possibly exist. In this instance, Regiomontanus was eliminating some theoretical op- tions from consideration, focusing on what was necessarily not the case, a complement to the other aspect of his criticism: revealing what was not necessarily the case. As we have seen earlier, he had been extending the range of his options by rooting out pseudo-necessities elsewhere. While Ptolemy had declared the alternative eccentric model for the second anomaly impossible for the inferior planets (“a viable hypothesis only for the three [outer] planets;” Ptolemy 1984, p. 555), Regiomontanus had proved that it was possible. Similarly, in Book 5 of the “Defense,” against George of Trebizond’s claim that the traditional order of Venus and Mer- his earlier (also undated) third letter to Bianchini (the second is dated 21 November 1463); Curtze (1902) 1990, p. 218; see also Shank 1998, pp. 159–160. In the “Defense,” the Moon argument surfaces again in Book 12 (218r ff), as does the problem of reconciling the rotation of the lunar epicycle with the visibility of only one side of the Moon (219r); the problem of seeing no variation (or insufªcient variation) in the sizes of Mars, Venus, and Mercury also enters the discussion (219r-223r); on this issue, see Goldstein 1996. 13. The lunar argument also occurs in Levi’s Latin Astronomy, a translation of excerpts from his Wars of the Lord; see Mancha 1992, esp. p. 28. 14. Rheticus makes a similar argument involving necessity; Rheticus (1540) 1965, f. H2v; Rosen 1971, p. 187; and Wilson 1975, p. 17. 192 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism cury was necessary, Regiomontanus argued that Venus was not necessarily above Mercury.15 With his ongoing interest in the elimination of epicycles and eccen- trics, Regiomontanus had a vested interest in showing that existing plane- tary arrangements were not anchored in necessity. Making astronomical arguments vulnerable to physical ones was one way of advancing his case. Indeed, the controversy with George of Trebizond displays explicitly Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 Regiomontanus’s conviction not only that these two areas have something to say to each other, but also that the astronomer should stimulate that conversation. This topic has a rich and growing historiography, to which I can only nod here, from Duhem’s To Save the Phenomena ([1908] 1990) through Robert Westman’s eponymous article (Westman 1980) to Goldstein and Barker’s recent discussion of realism and instrumentalism in the sixteenth century (Barker and Goldstein 1998).16 Since Richard Kremer and I ad- dress this issue in greater detail in a forthcoming article, sufªce it to quote here Regiomontanus’s pointed remark on the subject in the “Defense,” in the margins of his discussion of Book 12: “...itistherole/duty [ofªcium] of the astronomer to discuss in a worthwhile manner not so much the plane circles of the planets as their corporeal orbs, namely, what their shapes, their order, and their magnitude are, about which poles they rotate, and other ‘acci- dents’ of this sort.”17 Accidents are properties of substances, which is just what corporeal orbs are. With its emphasis on corporeality, this position directly contradicts the late-antique interpretation of Aristotle’s distinction between natural phi- losophy, on the one hand, and and astronomy, on the other.

15. Not content to undermine the perceived necessity of the received order, Regio- montanus argues that one can think up reasons to place Venus next to the Moon (the two feminine planets together); in a marginalium, he alludes to an order that would place the ªve planets with retrograde motions together. On 156r, he explores the possibility that Ve- nus might be immediately above the Moon and restates the possibility of a solar concen- tric. 16. See also Krafft, 1973; and Jardine, 1984. 17. “Presertim cum aliud restet commune placitum, astronomi ofªcium esse non tam de circulis planetarum superªcialibus quam de orbibus corporeis operepretium disserere, qua scilicet ªgura sint, quo ordine, qua magnitudine, quibus polis rotentur et cetera huiusmodi accidentia” (Regiomontanus, “Defensio Theonis,” 211v). The expression “astronomi ofªcium” also occurs in Kepler’s Defense of Tycho; see Jardine 1983, p. 123. Perspectives on Science 193

As expressed pointedly in Simplicius’s Commentary on book 2 of Aristotle’s Physics (reºecting here the views of Geminus):

“It is the business of physical inquiry to consider the substance of the heaven and the stars, their coming into being and their destruction, nay it is in a position even to prove the facts about their size, shape and arrangement; astronomy on the other hand does not attempt to Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 speak of anything of this kind...”(Grant 1994, p. 36).18

Simplicius’s commentary on Aristotle’s Physics was not translated into Latin in the Middle Ages, but Regiomontanus in principle had access to the Greek version of it in Bessarion’s library (now in Vienna, ÖNB, MS phil. graec. 164; see Labowski 1979, p. 491). Grant has claimed that “this important distinction” between natural philosophy and astronomy “was largely preserved during the [Latin] Middle Ages and ” with the “fusion of the two disciplines” taking place with Kepler (Grant 1994, pp. 36–39; Jardine 1984, pp. 247–48). As the appeal of al-Bitruji in the thirteenth century implies, the distinction in practice was probably murk- ier than the methodological statements suggests. In any event, against a view like that of Simplicius, Regiomontanus considered it the task of astronomers to deal with precisely those issues that Simplicius and his followers had restricted to the domain of the natu- ral philosopher. Signiªcantly, Regiomontanus was assigning the astronomer these du- ties speciªcally because he thought that Ptolemy had fallen short of the ideal. On the same page, Regiomontanus wrote in the top margin:

“Two things primarily are to be preserved in celestial motions, namely the primordial and intrinsic equality, and the phenomenon of inequality; the ªrst is brought about by orbs, not by thin circles; the second, by circles describable on planes, by the power of dem- onstration. The ªrst pertains to the nature of celestial bodies, which can tolerate no motion other than the uniform; the second pertains to human observers, to whom these motions seem unequal and dis- orderly. The ªrst, if I may say so, Ptolemy completely neglected; the second, he pursued to the utmost, expressing the quantity of motions with numbers by means of foregoing demonstrations, their quality having been ignored, even though in book 3, concerning

18. The best recent commentary on this passage and its context is Eastwood 1992, esp. pp. 234–38; see also Hugonnard-Roche, 1992 pp. 56–57. 194 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

the Sun, he adopts this almost as a principle, that the celestial mo- tions are equal and regular, etc.”19 In this marginalium, Regiomontanus sets as the agenda not one, but two very different things to be saved or preserved: the “primordial and intrinsic equality” and the appearance or phenomenon of inequality/ anomaly. He treats the ªrst not as a phenomenon, but as a physical princi- ple; indeed, he speciªcally contrasts it with what we can see.20 The “pri- Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 mordial and intrinsic equality” is linked to the nature of the celestial bod- ies, and thus laden with natural philosophical assumptions. Uniformly moving orbs were somehow the solution to that problem. Equally striking is Regiomontanus’s concession that mere circles were in fact adequate to saving the phenomena of inequality. The Ptolemy of the Almagest, whom he greatly admired, had done an excellent job in this one limited area. As Regiomontanus saw the matter, however, Ptolemy had failed to acquit himself of the ªrst task—saving the “primordial and intrinsic equal- ity”—even though he effectively treated it as a principle in book 3 of the Almagest. It was the conjunction of these two items that made for a complete as- tronomy—and it was also this conjunction that created Regiomontanus’s problems. The ideal case of uniformly moving orbs was a homocentric sys- tem, which is probably what Regiomontanus had in mind when he re- ferred to motions appropriate to celestial bodies; the best case of a system capable of “saving the anomalies” was the Almagest with its circles. Be- tween these cases, recent astronomers had been making some headway, as Regiomontanus noted in a historical retrospective.

C. Regiomontanus’s interpretation of the history of astronomy The contrast between orbs and circles also appears in Book 5 of the “De- fense of Theon,” in an intriguing interpretation of the history of astron- 19. “Duo sunt in celestibus motibus precipue servanda: equalitas videlicet primordialis et intrinseca et inequalitatis apparentia: primum per orbes, non per tenues circulos, absolvitur; secundum per circulos in planitie quapiam descriptibiles propter demon- strationis facultatem. Primum ad naturam corporum celestium attinet que nullum nisi equalem motum sustinere possunt; secundum ad spectatores homines refertur, quibus motus illi videntur inequales atque inordinati. Primum Ptolemeus (pace bona dixerim) penitus neglexit; secundum autem summopere prosecutus est, quantitatem quidem motuum per numeros demonstrationibus previis exprimens, qualitate eorum preterita, quamvis et in tertio volumine acturus de sole id quasi principium sumat motus celestes esse equales ordinatosque etc” (Regiomontanus, “Defensio Theonis,” 211v). 20. Note the similarity with Levi ben Gerson’s formulation: “The second principle is that the motion of the celestial bodies is necessarily uniform in itself and that the apparent variation is related to the way we see it, not that the motion in itself varies” (Goldstein 1985, p. 114). Perspectives on Science 195 omy. The context of the passage is Regiomontanus’s response to George of Trebizond’s misunderstanding of the way spherical epicycles and eccen- trics function. In setting George straight, Regiomontanus emphasizes the now-familiar distinction between two approaches to astronomy, now set in a temporal framework. The ªrst is two-dimensional, involving demonstra- tions in plane geometry that lead to arithmetic calculations and generate predictions. The second is three-dimensional: it describes corporeal Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 spheres and worries about such physical constraints as their substance, and such problems as vacua, collisions, and the interpenetrations of bodies. In responding to George of Trebizond, Regiomontanus groups together all the astronomers who, in implicit contrast to the homocentric approach, “embraced eccentrics:” “The ancients from Hipparchus not only to Ptolemy, but also to the Arab mathematicians, whichever ones embraced an eccentric as- tronomy, made mention of circles, simply and in a rather crude manner, if I may say so. They were concerned only that the anoma- lies of the celestial motions be saved by some suitable means; more precisely, that these same anomalies might be predicted by some numbers. The sign of this is that the princes of this art, and above all Ptolemy, were in the habit of moving immediately from linear demonstrations to computation, while saying absolutely nothing about the poles or axes of the motions.”21 Particularly noteworthy in this characterization of the ancients’ proce- dure is the direct move from two-dimensional demonstrations to calcula- tions. This approach cut out—wrongly, as he saw it—any discussion of such three-dimensional or physical parameters as poles and axes of motion, that is, the properties of the corporeal spheres that cause the motion. Attention to poles and axes has a long history of association with physi- cal considerations (Lehti 1986, esp. p. 137). It is perhaps not a coincidence that poles, axes, and circles are the speciªc foci of al-Bitruji’s ªrst com- plaints against Ptolemy in the opening paragraphs of his work. There he criticizes the discrepancy between “many heavens” and the existence of

21. “Hec ille pro bile sua in recentiores videtur iactare astronomos. Veteres enim ab Hipparcho non solum ad Ptolemeum, verumetiam ad Arabos mathematicos, quicumque astronomiam eccentricam amplexi sunt, de circulis simpliciter et subrudi, ut ita dicam, modo mentionem fecere: ad nihil aliud spectantes nisi ut diversitates motuum celestium quodam medio convenienti servari possent; imo potius ut ille ipse diversitates per numeros quosdam pronuntiari possent, cuius rei indicium est quod principes huius artis et in primis Ptolemeus demonstrationibus linearibus e vestigio ad calculum transire solitus est, nil prorsus vel de polis vel de axibus motuum disserendo” (Regiomontanus, “Defensio Theonis,” 65v). 196 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism only two poles, and notes that Ptolemy’s discussion was geared to circles rather than spheres (al-Bitruji 1952, pp. 72–73; al-Bitruji 1971 I, 11, pp. 55, 72).22 Any doubt that Regiomontanus was worried about the ancients’ failure to address the corporeal dimension of the celestial movers disappears as he turns to the improvements in astronomy since the ancients:

“The later [astronomers], penetrating most cleverly and intimately Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 into the nature of the celestial bodies, thought that the carriers of the stars were not merely mathematical circles lacking a third di- mension, but globular bodies that transport the stars by some mul- tifarious and wonderful conveyance. Thus whereas the authority of Ptolemy had allotted the Sun a single eccentric, they understood it to be not a skinny circle, but an orb of equal thickness in every di- rection surrounding the entire elementary region, together with the part of celestial space that lies below it [the Sun]. They thought that the Sun was ªxed in this orb, enclosed by two spherical con- centric surfaces, so that the Sun itself was not wandering, as it were, through the ethereal region, but revolved about its center [the center of the orb] with some given speed under the constrained motion of the aforementioned orb that contained it. Now since this orb recedes from the middle of the universe, its parts necessarily will move away unequally from the center of the universe, and therefore this orb in its motion will run into some celestial body placed near it, and will do violence to it by cleavage or compres- sion; it will also leave a vacuum behind itself. Therefore they thought they would ªll in, on either side, the space that was deªcient in concentricity by means of two other orbs placed near the deferent of the Sun, orbs of uneven thickness, so that the entire congeries of the three aforementioned orbs would have a common center with the world. So much for a cursory account of the Sun.”23

22. The language of ªction is stronger in Goldstein’s translation than in the Latin, but the contrast between imagination and reality is perfectly clear. 23. “Posteriores vero solertius et intimius ad naturam corporum celestium penetrantes, non solum circulos mathematicos tertia dimensione carentes stellarum delatores esse censuerunt, sed corpora quedam globica quorum varia et mira quadam delatione astra ferrentur. Itaque cum auctoritate Ptolemei Sol ecentricum solum sortitus esset, eum intellexerunt non tenuem circulum sed orbem equali undique crassitudine qui totam elementarem regionem una cum parte celestis spatii quod sub eo est ambiret; in quo orbe duobus sphericis superªciebus concentricis clauso solarem globum ªgi arbitrabuntur ut Sol ipse non quasi errabundus per ethereum vagaretur regionem, sed ductu memorati orbis se continentis equali quadam celeritate rotaretur penes centrum suum. Cumque orbis ille Perspectives on Science 197

Regiomontanus’s contrast between the two schools of astronomers could not be clearer. Whereas the “ancients” got their predictions from two-dimensional demonstrations, without references to poles or axes, the later astronomers concerned themselves with the “nature of the celestial bodies,” that is, three-dimensional orbs with physical properties, includ- ing the ability to carry planets. Particularly striking in the passage is Regiomontanus’s emphasis on the constraints that regulate the Sun’s mo- Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 tion, in contrast to its hypothetical “wandering through the ethereal re- gion.” His implicit criticism here is that the ancients’ two-dimensional approach left the motion of the Sun in causal limbo. The language of def- erent “carriers” [delatores] implicitly builds into the circles a causal expec- tation. So strong are Regiomontanus’s natural philosophical instincts here that he mistakes the failure to specify a physical cause of the Sun’s motion for the speciªcation of a failed cause.24 In contrast, the three-dimensional approach that relies on orbs gives an account of the constraints that seem to regulate the Sun’s motion. Al- though here he tells us far less than we would like to know, Regiomontanus clearly is concerned about movers, and he evidently ªnds untenable the notion of planets having voluntary motion. Regiomontanus further emphasizes the physical component of the more recent astronomers’ approach by reference to both the physical enti- ties that the orb encloses (the elementary region) and the problem of the vacuum. Regiomontanus thus examines the natural philosophical conse- quences of Ptolemy’s single eccentric by transposing it into three dimen- sions. He then ªnds that, qua physical body, it implies unacceptable phys- ical consequences, namely the compression and expansion of surrounding bodies (if the surroundings give way) or collisions and a vacuum (if they do not). Additional bodies, namely the partial orbs of uneven thickness, are required to avoid these problems. medio mundi situ recedat, necesse erit partes eius inequaliter centro mundi removeri atque idcirco orbem illum in motu suo occurrere cuidam celesti corpori circa se posito et eidem quasi vim inferre per scissionem aut condensationem: vacuum etiam post se relinquere; ideo spatium quod utrimque concentricitate defecit, suppleri putaverunt duobus aliis orbibus circa Solis delatorem positis, crassitudine quidem impari sic tamen vi- delicet ut congeries trium orbium memoratorum cum mundo centrum haberent com- mune. Hac quidem cursim de Sole narrata sunt” (Regiomontanus, “Defensio Theonis,” 65v-66r). 24. It is important to stress that Regiomontanus did not subordinate astronomy to nat- ural philosophy. On the contrary, he considered astronomy to be the most noble of the sci- ences; Regiomontanus, “Oratio in praelectione Alfragani,” in Schmeidler 1972, p. 46; Swerdlow 1993, pp. 151–152. 198 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

Having explained the simple case of the Sun, Regiomontanus turns to the more complicated case of Jupiter: “In Jupiter, which I will use as an example since it does not have a single anomaly, it was necessary to add an epicycle; for they thought no differently about its eccentric than they did in the case of the Sun, except that they put an epicycle in place of the Sun—not a circular one like that assumed in demonstrations, but a Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 spherical one [orbitum], thinking that there was in the eccentric some spherical hollow [caverna] in which an epicyclic globe of the size required for Jupiter could be placed. But they believed that the Jovial globe was immersed in the epicycle so that the planet ro- tated not on its own and haphazardly, but by the rolling of the epicyclic globe in the spherical hollow. At the same time, the epicycle, together with the planet, would be led around according to the assumed motion of the eccentric rushing along in the hollow.”25 As Regiomontanus understood the history of astronomy, this early “ec- centric astronomy” was, in its Greek and Arabic forms, two-dimensional. Like its archetype (Ptolemy’s Almagest), this tradition proceeded from demonstrations on plane ªgures to numerical predictions of planetary po- sitions. In more recent times, eccentric astronomy had taken a different turn: three-dimensionality had become crucial, and some kind of corpo- real celestial machinery involving partial spherical shells, epicyclic spheres, and hollow “caverns” determined the motions of the planets. Inci- dentally, Regiomontanus depicted the three-dimensional development not as a compromise with Aristotle, but rather as the proper development of “eccentric astronomy.” This “later” approach to eccentric astronomy was, of course, consider- ably more ancient than Regiomontanus supposed. Like the rest of the Latin tradition, he obviously had no inkling of Ptolemy’s Planetary Hy- potheses. He surely would have been pleased, however, to learn that Ptol- emy had given a three-dimensional shape to the models of the Almagest,

25. “In Iove autem, ut exemplo utar, cum non sit simplex diversitas, oportuit adiungi epicyclum; nam de eccentrico suo non aliter quam apud Solem opinati sunt nisi quod loco Solis posuerunt epicyclum, non circularem qualis in demonstrationibus supponitur, sed orbitum, putantes in ecentrico esse quandam cavernam sphericam in qua globus epicyclicus quantus Iovi debetur constitui posset. Sed et ipso epicyclo immersum esse crediderunt globum iovialem; ut sic stella non per se et temere, sed volutione globi epicyclici in caverna spherica rotaretur. Et simul epicyclus una cum stella ad motum ecentrici se in caverna positum rapientis circumduceretur” (Regiomontanus, “Defensio Theonis,” 66r). Perspectives on Science 199 even if its eccentrics and epicycles caused it to fall short of Regiomontanus’s full-blown ideal of uniformly rotating (presumably homocentric) spheres. It is not clear what he would have made of Ptol- emy’s proposed abbreviation of the planetary spheres into rings or “sawed-out pieces.” Regiomontanus’s account of the Sun implies that he would have objected to Ptolemy’s suggestion that the planets drove their spheres, not the other way around.26 Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 Unfortunately, Regiomontanus does not say whom he includes among the “later” astronomers. His schema recalls a tradition that appears ªrst in Latin in William the Englishman (William of Marseille) ca. 1230 and sur- faces for evaluation in Roger Bacon’s Opus Tertium soon after mid-century (Duhem 1913–59, III, pp. 287–290; Duhem 1909, pp. 125–31; Grant 1994, pp. 278–81). The verbal pictures sketched here nicely describe the classic illustrations of the theoricae found in Regiomontanus’s ªrst printed edition of Georg Peuerbach’s Theoricae novae planetarum (Nuremberg, c. 1472). We can therefore be fairly sure that he counted his teacher among the recent astronomers. As Peuerbach had precious little to say about the physical underpinnings of his views, the descriptions of the “Defense of Theon” modestly ªll Peuerbach’s silence about his models. In Regiomontanus’s historical schema, the treatment of the Arabic tra- dition is very cursory, simply lumped together with that of the an- cients—a problem that deserves a detailed study in its own right. A few remarks must sufªce for now. Like Roger Bacon, Regiomontanus evi- dently did not know about the call for an astronomy focused on spheres in Ibn al-Haytham’s On the conªguration of the world. (Despite its translation into Latin, the work seems to have had limited circulation; Millás-Vallicrosa 1942, pp. 285–312). He did know about al-Farghani, on whom he had lectured at Padua (Swerdlow 1993), and also about al-Battani, whom he quoted with respect in the Epitome as well as in the “Defense of Theon.”27 Regiomontanus’s reading of al-Farghani may illus- trate his approach to the Arabic tradition, even if it is not necessarily typi- cal. Al-Farghani did have an interest in planetary sizes and distances, and chapter 12 of his Differentie, which deals with the issue, also touched upon questions of conªguration (“On the description of the ªgures of the circles

26. Ptolemy 1907, pp. 114–18; see the summaries and analyses in Langermann 1990, pp. 13–25; and Murschel 1995, esp. p. 39. During his Viennese years, Regiomontanus copied a passing allusion to a “ring cosmology,” although he did not, as far as I know, com- ment on it; see Shank 1992b, p. 19; and Shank forthcoming. 27.“...nondeerunt mihi testes gravissimi, verum ante omnes Albategnius arabus vir summa modestia et doctrina excellenti cuius abaco astronomico omnes ferme in hunc usque diem schole sunt referte” (“Defense of Theon,” 75v; also 102v, 157v, 159v, 201r, 216r, 220r–21r, 223v, 224r, 260v). 200 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism of the planets and on their order, and on their distances from the Earth”). This chapter, in ’s translation, intermingles the termi- nology of circles and spheres, while the rest of al-Farghani’s exposition is couched largely in terms of circles (Carmody 1943, pp. 22–24; Lerner 1996–97, I, pp. 89–91).28 It would therefore not be surprising if Regiomontanus had read al-Farghani as ªrmly committed to the Alma- gest’s tradition of two-dimensional astronomy. Ibn al-Haytham had read Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 him in just this way: his critique in On the conªguration tells on al-Farghani as much as on the Almagest (Langermann 1990, pp. 25–29).29

E. The Almagest as “fictitious art” Whatever the justiªcation for Regiomontanus’s understanding of the his- tory of astronomy, it is clear that he perceived most of the received tradi- tion, from the ancients through the Arabs, as a disembodied astronomy that leapt from two-dimensional demonstrations based on circles to pre- dictions of the planets’ positions. To cut out consideration of the three- dimensional structure of the celestial spheres—the causes of their motions—was effectively to trade in ªctions. In a memorandum-like mar- ginal note on the page preceding his contrast of globes with tenuous cir- cles, Regiomontanus expressed his vision in this way: “To attain an astronomy such that it not only accommodates com- putation to the appearances, but also truly imparts a complete knowledge of the ªgures of the celestial bodies with the law of their motions; [to do] otherwise is to pass along a ªctitious art [ªctitiam artem]. To safeguard in a worthwhile manner the equality of the ce- lestial motions.”30 This list of maxims concisely encapsulates the themes with which we are familiar—the inadequacy of an astronomy that merely saves the phe- nomena without grappling with the nature and structure of the heavens, and the need to preserve the uniformity of celestial motions. The most striking part of this remark, as we know from its context, is 28. It is not clear whether the early manuscript tradition uses the kinds of diagrams found in the later printed editions and included in the modern edition of the John of Se- ville translation; see Campani 1910, pp. 109–115. 29. Note that al-Farghani and Ibn al-Haytham (at that stage of his career) were both working without the Planetary Hypotheses (Langermann 1990, pp. 11–25). 30. “Ad astronomiam attinere ut non modo calculum apparentibus accommodet, sed et ªguras corporum celestium veraciter(?) cum lege motuum edoceat; alias enim ªctitiam tradere [[astronomiam]] artem Equalitatem motuum celestium opere pretium tutandam esse” (Regiomontanus, “Defensio Theonis,” 210v; the double-bracketed word is crossed out in the manuscript). On Regiomontanus’s use of law, see the suggestive comments of Ruby 1986, esp. pp. 352–57. Perspectives on Science 201

Regiomontanus’s association of “ªctitiam artem” with the tradition of the Almagest. To illustrate its signiªcance, I turn to one of the more recent reapprais- als of the question, classic since Duhem, of “realism and instrumentalism in sixteenth-century astronomy,” namely a recent article by Bernard Goldstein and Peter Barker in this journal (Barker and Goldstein 1998). Like Duhem, Goldstein and Barker revisit this venerable historiographical Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 controversy within the context of an Aristotelian causal framework. At the heart of their concerns are the relative domains of natural philosophy and astronomy, and also—as an attempt to understand these domains—the historical propriety and range of the term “instrumentalism,” which is sometimes used synonymously with “ªctionalism.” Goldstein and Barker take instrumentalism to be an overarching posi- tion of principle that applies “uniformly to all scientiªc disciplines” (Barker and Goldstein 1998, p. 235). In their view, genuine instrumental- ists (as a matter of principle) do not pick and choose the branch of science to which their views apply. Accordingly they ªnd no genuine instrumen- talists before the nineteenth century. The alleged instrumentalists of the sixteenth-century, that is, those who profess agnosticism about astronomi- cal models, are not agnostic in physics, Goldstein and Barker argue, and therefore not genuine instrumentalists. Second, they argue that the so-called instrumentalists of the sixteenth century are not instrumentalists in principle, but merely in practice. It is not that one can in principle never know the true structure of the heavens; it is rather that one can never in fact have such knowledge in this life. While there is much to commend in the clarity of this analysis, it is far from clear that the people who discuss the status of orbs are worried, like Goldstein and Barker, about the distinction between astronomical knowl- edge in principle and in practice. Almost all of the actors under discussion believed that events in the world had natural causes, which were traceable, either mediately or immediately, to a divine agent who also had knowl- edge of them. At stake was neither the existence of causes nor their ulti- mate source, both of which were overwhelmingly, if not unanimously, conceded.31 Rather the issue was the ability of human beings to discover and know such things in this life, without the beneªt of visiting the heav- ens in situ, like Dante and Beatrice. Goldstein and Barker revisit the locus classicus for such a discussion in the famous “Note to the reader, concerning the hypotheses of this work”

31. Nicholas of Autrecourt is of course a notorious critic of causality (see Weinberg 1948, ch. 2). 202 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism that Osiander inserted into Copernicus’s De revolutionibus (1543) while see- ing it through the press. Against the standard view of this note, Goldstein and Barker claim: “Osiander’s Preface was not a statement of ªctionalism or instrumentalism, but a restatement of the impossibility of moving be- yond quia demonstrations in astronomy” (that is, of going from demon- strations of the fact to demonstrations that identify the cause of the fact). To paraphrase their conclusion, Osiander and his contemporaries were not Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 ªctionalists, but “perpetually frustrated realists” (Barker and Goldstein 1998, esp. 250–251, 253). The analytical utility of this phrase is very lim- ited, for it applies to people who had no doubt that there were fundamen- tal truths about the universe known to God, but to which physical theory in this life could not give them access. The designation “perpetually frus- trated realist” thus also applies to the likes of Urban VIII and Duhem himself, whom it is unhelpful to label as realist, not even the perpetually frustrated sort. Aristotle’s argument against eternally unactualized poten- tialities works against perpetually frustrated realists as well. There may well be sound reasons for avoiding the term “instrument- alism” to analyze sixteenth-century views. As Regiomontanus’s position in the “Defense of Theon” shows clearly, however, “ªctitious art” was a ªfteenth-century category applicable, according to him, to astronomy in the tradition of Ptolemy’s Almagest. More importantly, Regiomontanus’s language resonates strongly with the ªctitious language of Osiander’s note “To the reader...,”speciªcally with the notion that the astronomer may concoct “whatever hypotheses” he needs, “hypotheses that need not be true or even probable/likely.”32 Osiander used the absence of conclusive ev- idence for one model over another (the deferent-epicycle and the eccentric hypotheses for the Sun) and well-known inconsistencies between the mod- els and the acknowledged order of the universe (e.g., the physically impos- sible size of Venus’s epicycle) to undercut inferences from the models to that order. Regiomontanus, of course, was worried precisely about hypoth- eses of this sort, from two-dimensional schemata in general to the large epicycle of the Moon in particular, which was necessarily false. Even though they disagreed at a fundamental level, Osiander and Regiomon- tanus would have understood each other’s language perfectly. Both men read the Almagest as belonging to the ªctitious tradition, but whereas Osiander treated the latter as normative, Regiomontanus considered it defective. If Regiomontanus had known about Osiander’s position, he

32.“...seuhypotheses, cum veras assequi nulla ratione possit, qualescunque excogitare et conªngere ...”Inthelast sentence of his “Note to the Reader,” Osiander uses the word conªcta, a term with an undetermined etymology. If derived from conªngere, it retains the ambiguity of “fabrications;” if from conªctare, it loses much ambiguity by being associated with various forms of pretense, including lies and deceptions. Perspectives on Science 203 would have called it advocacy of a ªctitious art, not perpetually frustrated realism. As is well known, in the De Revolutionibus itself Copernicus took a posi- tion very different from Osiander’s, one in which an astronomer could in- deed draw conclusions about the arrangement and structure of the uni- verse. Regiomontanus, the author from whom Copernicus learned his Ptolemy, could not have agreed more: he was already ªghting this very Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 battle in the third quarter of the ªfteenth century.

VI. Conclusion The “Defense of Theon” has taken us far from the Regiomontanus that we once knew. Thoroughly impressed by the successes of Ptolemaic astron- omy, he nevertheless found its models ªctitious. While preferring a three-dimensional alternative along the lines of Peuerbach to the ancients’ two-dimensional approach to “eccentric astronomy,” he nevertheless hoped for a homocentric astronomy different from al-Bitruji’s that would eliminate epicycles and eccentrics (including, presumably, the three- dimensional ones of his teacher) and preserve the uniform motions appro- priate to celestial bodies. As this handful of passages suggests, the “Defense of Theon” holds much promise for a better understanding not only of Regiomontanus himself, but also of the contexts of late-ªfteenth-century astronomy and the .

References Albert of Saxony. (1492)1986. Questiones subtilissime in libros Aristotelis de celo et mundo. Hildesheim, Georg Olms. Avi-Yonah, Reuven. 1985. “Ptolemy vs. al-Bitruji: A Study of Scientiªc Decision-Making in the Middle Ages.” Archives internationales d’histoire des sciences 35 (#114–115):124–47. Barker, Peter and Bernard Goldstein. 1998. “Realism and Instru- mentalism in Sixteenth-Century Astronomy: A Reappraisal.” Perspec- tives on Science 6:232–258. al-Bitruji 1952. De motibus celorum. Edited by Francis J. Carmody. Berke- ley and Los Angeles: University of California Press. ———. 1971. On the Principles of Astronomy. 2 vols. Edited by Bernard Goldstein. New Haven, Yale University Press. Campani, Romeo. ed. 1910. Alfragano: Il Libro dell’ aggregazione delle stelle...Città di Castello: S. Lapi. Carmody, J. ed. 1943. Al Farghani: Differentie. Berkeley: privately printed. Copernicus, Nicolaus. (1543)1966. De revolutionibus orbium celestium. Brussels: Culture et Civilisation. 204 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

Curtze, Maximilian. (1902)1968. “Der Briefwechsel Regiomontan’s mit Giovanni Bianchini, Jacob von Speier und Christian Roder.” Pp. 187–336 in Urkunden zur Geschichte der Mathematik im Mittelalter und der Renaissance. Edited by Curtze. (repr. New York: Johnson Re- prints). Duhem, Pierre. (1908)1990. ⌺⍀⌮⌭⌱⌵ TA ⌽〈⌱⌵⌷⌴⌭⌵〈: Essai sur la notion de théorie physique de Platon à Galilée. Paris, Vrin. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 ———. 1909. Un Fragment inédit de l’Opus tertium de Roger Bacon, précédé d’une étude sur ce fragment. Quaracchi: Collegio S. Bonaventura. ———. 1913–59. Le Système du monde: Histoire des doctrines cosmologiques de Platon à Copernic, 10 vol. Paris: Hermann. Dupuis, J. (1892)1966. Théon de Smyrne, Philosophe platonicien: Exposition des connaissances mathématiques utiles pour la lecture de Platon traduites pour la première fois du grec en français. Brussels: Culture et Civilisation. Dobrzycki, Jerzy. 2001. “Notes on Copernicus’s Early .” Journal for the History of Astronomy 32:223–225. Eastwood, Bruce S. 1992. “Heraclides and Heliocentrism: Texts, Dia- grams, and Interpretations.” Journal for the History of Astronomy 23:233–260. Goldstein, Bernard R. 1967. “The Arabic Version of Ptolemy’s Planetary Hypotheses.” Transactions of the American Philosophical Society, N.S. 57, Part 4. ———. 1971. Al-Bitruji: On the Principles of Astronomy. 2 vol. New Ha- ven, Yale University Press. ———. 1972. “Theory and Observation in Medieval Astronomy.” Isis 63:39–47 (reprinted in Shank, ed., The Scientiªc Enterprise in Antiquity and the Middle Ages: Readings from Isis. Chicago, University of Chicago Press, 2002). ———. 1985. The Astronomy of Levi ben Gerson (1288–1344). Berlin: Springer Verlag. ———. 1996. “The Pre-Telescopic Treatment of the Phases and Appar- ent Sizes of Venus.” Journal for the History of Astronomy 27:1–12. ———. 1997a. “The Physical Astronomy of Levi ben Gerson.” Perspectives on Science 5:1–30. ———. 1997b. “Saving the Phenomena: The Background to Ptolemy’s Planetary Theory.” Journal for the History of Astronomy 28:1–12. ———. 2001. “Before the Sun in the Church.” Journal for the History of Astronomy 32:73–77. Grant, Edward. 1994. Planets, Stars, and Orbs: The Medieval Cosmos, 1200–1687. Cambridge: Cambridge University Press. Hugonnard-Roche, Henri. 1992. “Problèmes méthodologiques dans l’astronomie au début du XIVe siècle.” Pp. 55–70 in Studies on Perspectives on Science 205

Gersonides, a Fourteenth-Century Jewish Philosopher-Scientist. Edited by Gad Freudenthal. Leiden: Brill. Huxley, G. L. 1970–1980. “Theon of Smyrna.” Dictionary of Scientiªc Biog- raphy, 13:325–26. Jardine, Nicholas. 1984. The Birth of History and Philosophy of Science: Kep- ler’s A Defence of Tycho against Ursus with Essays on its Signiªcance and Provenance. Cambridge: Cambridge University Press. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 Krafft, Fritz. 1973. “Physikalische Realität oder mathematische Hypothese? Andreas Osiander und die physikalische Erneuerung der antiken Astronomie durch .” Philosophia Naturalis 14:243–75. Kren, Claudia. 1968. “Homocentric Astronomy in the Latin West: The De reprobatione ecentricorum et epiciclorum of Henry of Hesse.” Isis 59:269–281. ———. 1969. “A Medieval Objection to ‘Ptolemy.’” British Journal for the History of Science 4:378–393. Labowski, Lotte. 1979. Bessarion’s Library and the Biblioteca Marciana: Six Early Inventories. Rome: Edizioni di Storia e Letteratura. Langermann, Y. Tzvi. 1990. Ibn al-Haytham’s On the Conªguration of the World. New York: Garland. Lehti, Raimo. 1986. “Realism and Fictionalism in the Almagest.” Annales Academiae Scientiarum Finnicae, Ser. A (VI, Physica) 431:116–67. Lerner, Michel-Pierre. 1996–97. Le Monde des sphères. 2 vol. Paris: Les Belles Lettres. Mancha, José Luis. 1992. “The Latin Translation of Levi ben Gerson’s As- tronomy.” Pp. 21–46 in Studies on Gersonides, a Fourteenth-Century Jewish Philosopher-Scientist. Edited by Gad Freudenthal. Leiden: Brill. Millás-Vallicrosa, José María. 1942. Las Traducciones orientales en los manuscritos de la Biblioteca Catedrál de Toledo. Madrid. Mohler, Ludwig. (1923)1967. Kardinal Bessarion als Theologe, Humanist, und Staatsmann, 3 vol. Aalen: Scientia Verlag. Monfasani, John. 1976. George of Trebizond: A Biography and A Study of his Rhetoric and Logic. Leiden: Brill. ———. 1984. Collectanea Trapezuntia: Texts, Documents, and Bibliographies of George of Trebizond. Medieval and Renaissance Texts and Studies, vol. 25. Binghamton, N.Y.: Medieval and Renaissance Texts and Studies. Murschel, Andrea. 1995. “The Structure and Function of Ptolemy’s Physi- cal Hypotheses of Planetary Motion.” Journal for the History of Astronomy 26:33–61. Ptolemy. 1907. Opera Minora. Edited by J. Heiberg. Leipzig: Teubner. 206 Regiomontanus on Ptolemy, Physical Orbs, and Astronomical Fictionalism

———. 1984. Almagest. Translated and annotated by G. J. Toomer. New York: Springer Verlag, Rheticus, Joachim. (1540)1965. De libris revolutionum Copernici narratio prima. Facsimile reprint. Osnabrück: O. Zeller Verlag. Rome, A. ed. 1931. Commentaires de Pappus et de Théon d’Alexandrie sur l’Almageste, vol. 1: Pappus d’Alexandrie: Commentaire sur les livres 5 et 6 de l’Almageste. Rome: Biblioteca Apostolica Vaticana. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 Rosen, Edward. 1971. Three Copernican Treatises. New York: Dover. Ruby, Jane. 1986. “The Origins of Scientiªc Law.” Journal of the History of Ideas 47:341–59. Sabra, A. I. 1978. “An 11th-Century Refutation of Ptolemy’s Planetary Theory.” Pp. 117–131 in Science and History: Studies in Honor of Edward Rosen. Studia Copernicana, vol. 16. Wroclaw: Ossolineum. ———. 1984. “The Andalusian Revolt against Ptolemaic Astronomy: Averroes and al-Bitruji.” Pp. 133–153 in Transformation and Tradition in the Sciences: Essays in Honor of I. Bernard Cohen. Edited by Everett Mendelsohn. Cambridge: Cambridge University Press. Schmeidler, Felix. 1972. Joanni Regiomontani opera collectanea. Osnabrück: Zeller Verlag. Shank, Michael H. 1992a. “The Classical Scientiªc Tradition in Vienna.” Pp. 115–136 in Tradition, Transmission, Transformation. Edited by Jamil Ragep and Sally Ragep with Steven Livesey. Leiden: Brill. ———. 1992b. “The ‘Notes on al-Bitruji’ Attributed to Regiomontanus: Second Thoughts.” Journal for the History of Astronomy 23:15–30. ———. 1998. “Regiomontanus and Homocentric Astronomy.” Journal for the History of Astronomy 29:157–166. ———. Forthcoming. “Rings in a Fluid Heaven: The Equatorium-Driven Physical Astronomy of Guido de Marchia (º. 1292–1310).” Centaurus Swerdlow, Noel. 1973. “The Derivation and First Draft of Copernicus’s Planetary Theory: A Translation of the Commentariolus with Commen- tary.” Proceedings of the American Philosophical Society 117:423–512. ———. 1990. “Regiomontanus on the Critical Problems of Astronomy.” Pp. 165–195 in Nature, Experiment and the Sciences. Edited by Trevor Levere and William Shea. Boston Studies in the Philosophy of Science, vol. 120. Boston/Dordrecht: Kluwer. ———. 1993. “Science and Humanism in the Renaissance: Regiomontanus’s Oration on the Dignity and Utility of the Mathemat- ical Sciences.” Pp. 131–68 in World Changes: Thomas Kuhn and the Nature of Science. Edited by Paul Horwich. Cambridge, Mass.: MIT Press. Perspectives on Science 207

———. 1996. “Astronomy in the Renaissance.” Pp. 187–230 in Astron- omy Before the Telescope. Edited by Christopher Walker. London: BCA; New York: St. Martin’s Press. ———. 1999. “Regiomontanus’s Concentric-Sphere Models for the Sun and Moon.” Journal for the History of Astronomy 30:1–23. Swerdlow, Noel M. and Otto Neugebauer. 1984. Mathematical Astronomy in Copernicus’s De Revolutionibus. Berlin: Springer Verlag. Downloaded from http://direct.mit.edu/posc/article-pdf/10/2/179/1789149/106361402321147522.pdf by guest on 01 October 2021 Theon of Smyrna. 1979. Mathematics Useful for Understanding Plato. Trans- lated by Robert and Deborah Lawlor. San Diego: Wizards Bookshelf. Weinberg, Julius R. 1948. Nicholas of Autrecourt: A Study in Fourteenth Cen- tury Thought. Princeton: Princeton University Press. Westman, Robert S. 1980. “The Astronomer’s Rôle in the Sixteenth Cen- tury: A Preliminary Study.” History of Science 18:105–147. Wilson, Curtis. 1975. “Rheticus, Ravetz, and the ‘Necessity’ of Coperni- cus’s Innovation.” Pp. 17–39 in The Copernican Achievement. Edited by Robert S. Westman. Berkeley: University of California Press. Zinner, Ernst. 1968. Leben und Wirken des Joh. Müller von Königsberg genannt Regiomontanus. 2nd ed. Osnabrück: O. Zeller Verlag.