3
The Status of Astronomy as a Science
in Fifteenth-Century Cracow:
Ibn al-Haytham, Peurbach, and Copernicus
Nicholas Copernicus’s first known astronomical writing is Nicolai Copernici de hypothesibus motuum caelestium a se constitutis commentariolus (Nicholas Copernicus’s Small Commentary on the Hypotheses of Celestial Motions Constructed by Him).1 In the Commentariolus, Copernicus expounds his hypotheses, among others, that the Earth has three motions, whereas the Sun and fixed stars have none.2 The Commentariolus, which is the best evidence we have of Copernicus’s conception of astronomy when he first proposed heliocentrism, is a work of the same type as Georg Peurbach’s Theoricae novae planetarum – that is, theoretical rather than practical, narrative rather than demonstrative, and based on the assertion of hypotheses or principles.3
In his very thorough and foundational 1973 article on the Commentariolus, Noel Swerdlow writes:
Now, the principal textbook of planetary theory in Copernicus’s time, the Theoricae Novae Planetarum by Georg Peurbach, contains descriptions, and very elaborate descriptions they are too, of spherical representations of Ptolemy’s planetary models. Peurbach gives exceedingly careful attention to the proper alignment of the eccentric sphere that carries the epicyclic sphere through its proper path, to the inclinations of the axes about which the spheres rotate, and to all the different motions of the spheres and axes required to produce the apparent planetary motions in longitude and latitude, and the precession of the apsidal and nodal lines along with the sphere of the fixed stars. This entire apparatus of spheres and axes, rotations and inclinations, is taken over by Copernicus in the Commentariolus, although his description is not as thorough since the reader’s familiarity with such models is taken for granted. Copernicus usually describes planetary motions in terms of rotations of spheres and inclinations of axes.4
The genre of Peurbach’s Theoricae novae planetarum is made explicit by Albert of Brudzewo, master at Cracow University in the late fifteenth century. In his commentary on Peurbach’s Theoricae novae planetarum, which in its closing statement is also called a commentariolus, Brudzewo describes the various genres of astronomical writing.5 According to the definition of Isidore of Seville, Brudzewo reports, astronomy provides the laws of motion of the heavenly bodies.6 According to Claudius Ptolemy and Haly (ʿAlī ibn Riḍwān), astronomy has two principal parts, the first concerning the motions of the heavenly bodies and the second concerning their effects on Earth. The first principal part is theoretical or speculative and the second is practical.7 Speculative or theoretical astronomy is divided into narrative and demonstrative. The Theoricae novae planetarum is theoretical and narrative, not demonstrative, he continues, and is intended as an introduction. The Theorica planetarum of Campanus of Novara is likewise narrative, Brudzewo goes on, as is the XXXI Differentiis of Alfraganus (al-Farghānī). In contrast, Ptolemy in the Almagest, Geber (Jābir ibn Aflaḥ), and Albategnius (al-Battānī) provide demonstrative presentations.8 Works on tables and instruments are parts of practical astronomy. It is worth noticing here how many works Brudzewo includes in his survey that are translated from Arabic.9
The earliest recognized source of the configurations of celestial orbs (theoricae orbium) in Peurbach’s Theoricae novae planetarum is AbūʿAlī al-Ḥasan ibn al-Haytham’s On the Configuration of the World, a work that was transmitted to Latin-speaking Europe at the latest by the end of the thirteenth century and that could have reached Peurbach by so many alternative routes that it does not make sense to look for a primary route of transmission. The conception of multipart physical orbs carrying around the planets was widely familiar in Europe already by the fourteenth century. It is distinctly different from conceptions found in Ptolemy’s Planetary Hypotheses, which, in any case, was not known in Europe. But the European descendants of Ibn al-Haytham’s configurations also included material that postdated On the Configuration, such as inclusion of the apsides of the Sun among astronomical variables affected by the one degree per century motion of the eighth orb, so there was not a single early transmission from Ibn al-Haytham and then further development in Europe, but there must have been later transmissions from the Islamic science of configuration (ʿilm al-hayʾa). In this chapter, I argue that Ibn al-Haytham’s On the Configuration of the World and Peurbach’s Theoricae novae planetarum are genealogically linked, while not attempting to trace the descent of ideas within Islamic hayʾa or the likely multiple migrations of the ideas to Europe beginning in the late thirteenth century before they appear in Peurbach’s Theoricae novae planetarum. Without attempting further to contribute to the study of transmission, one might immediately suggest the possibility of oral rather than written transmission and consider the many possible intercultural connections over time between those with interests in astronomical or astrological subjects and methods. That the Europeans at some point seem to have stopped making translations from the Arabic to Latin does not mean that other modes of transmission had to end, let alone that astronomical activity in Islamic areas ceased to progress because European interest in it declined.
When the idea of physical orbs spread from Islamic to European areas, the use of Ptolemy’s mathematical methods was not abandoned, for the astronomy of physical orbs was incomplete, compelling astronomers to use Ptolemaic mathematics in order to do their work (e.g., compiling almanacs and ephemerides, predicting eclipses, and so forth). So in his commentary on Peurbach, Brudzewo quotes Richard of Wallingford as explaining that, in order to do their work, astronomers were forced to use imaginary mathematical devices, which they in no way thought really existed in the heavens.10
It followed that Peurbach’s Theoricae novae planetarum presented alternate complementary, neither unified nor mutually exclusive, physical and mathematical approaches for understanding the motions of the planets. The physical approaches involved real three-dimensional orbs, while at the same time there were mathematical approaches represented in theoricae figures as two-dimensional geometrical circles and lines, which were understood not to be real things existing in the external world but to be the products of mathematical construction or imagination. Thus astronomical research could proceed along two simultaneous paths, one mathematical and the other physical.
Islamic astronomers had long followed a program of trying to propose physical bodies that might lie behind the observed motions described mathematically in Ptolemaic astronomy. They had not, however, completed the job of finding a physical configuration consistent with every mathematical regularity. As a result, there was a continuing felt need for new and better physical hypotheses. Astronomers, therefore, could seek to advance independently both mathematical astronomy, on the one hand, and possible physical configurations, on the other.
Existing at the same time as this situation in the discipline of astronomy, there was a wider Averroistic-Aristotelian conception of science according to which there were many autonomous scientific disciplines, none of which needed to be subservient to another, at least at that time, since scientific knowledge was still unfinished. In relation to astronomy in particular, it was possible to accept a situation in which mathematical astronomy, such as contained in the Almagest, was considered a tool with proved success and in which physical astronomy involving real, three-dimensional, uniformly rotating aethereal orbs was also considered worthy of attention, despite the fact that existing mathematical astronomy and physical astronomy were, at that given moment, not fully consistent with one another. No one thought that bare mathematical lines were in the heavens, but astronomers had to use mathematical approaches lacking physical support because they had no other way to do their work.
This picture of the components of the conception of astronomy as a science present in Cracow when Copernicus was a student is what I have arrived at after looking for the last several years at the major sources relevant to the background of the Commentariolus at Cracow. The Commentariolus was composed at a time when Copernicus had not worked out significant parts of the mathematics that would be included in De revolutionibus orbium coelestium.11 At the time of the Commentariolus, Copernicus assumed that there were real, three-dimensional orbs made out of aether. He recognized that the existing physical astronomy was a work in progress. He seems to have believed that he had caught a glimpse of a more satisfactory configuration or system of orbs than the ones that had been devised for an Earth-centred system. So he decided to communicate his ideas to some others, short of publication, to test the waters and see what reception his ideas might receive. For the rest of his life, he continued to pursue his ideas until, finally, Georg Joachim Rheticus persuaded him that it was time for him to publish his large work.
The Commentariolus mirrors the Theoricae novae planetarum in starting with a statement of principles. In Copernicus’s work these principles are stated postulates (petitiones), and in Peurbach’s work they are the theoricae (figures) themselves, together with their descriptions. Each work then elaborates a configuration consistent with these principles. Copernicus’s principles (or postulates) are as follows:
1.There is no one centre of all the celestial orbs or spheres.
2.The centre of the Earth is not the centre of the universe, but only the centre towards which heavy things move and the centre of the lunar sphere.
3.The orbs surround the Sun as though it were in the middle of all of them, and therefore the centre of the universe is near the Sun.
4.… [T]he distance between the Sun and the Earth is imperceptible compared to the great height of the sphere of the fixed stars.
5.Whatever motion appears in the sphere of the fixed stars belongs not to it but to the Earth …
6.Whatever motions appear to us to belong to the Sun are not due to the [motion] of the Sun but [to the motion] of the Earth and our orb with which we revolve around the Sun just as any other planet …
7.The retrograde and direct motion that appears in the planets belongs not to them but to the [motion] of the Earth.12
These petitiones state guiding principles for constructing a configuration of the world that could account for what astronomers had observed in the heavens and described mathematically. In his Letter against Werner (1524) concerning Johannes Werner’s treatise On the Motion of the Eighth Sphere (1522), Copernicus describes how astronomers worked from measured positions to arrive at their theories:
The science of the stars is one of those subjects which we learn in the order opposite to the natural order. For example, in the natural order it is first known that the planets are nearer than the fixed stars to the earth, and then as a consequence that the planets do not twinkle. We, on the contrary, first see that they do not twinkle, and then we know that they are nearer to the earth. In like manner, first we learn that the apparent motions of the planets are unequal, and subsequently we conclude that there are epicycles, eccentrics, or other circles by which the planets are carried unequally. I should therefore like to state that it was necessary for the ancient philosophers, first to mark with the aid of instruments the positions of the planets and the intervals of time, and then with this information as their guide, lest the inquiry into the motion of heaven remain interminable, to work out some definite planetary theory, which they were seen to have found [quam tum visi sunt invenisse] when the theory agreed in some harmonious manner with all the observed and noted positions of the planets.13
Thus in the Commentariolus Copernicus’s petitiones represent hypotheses derived from experience, which are to be accepted as true, even though they could be wrong given that astronomy is a science still in the process of development. The currently existing theories may not be perfect, but they have been adopted for the present “lest the inquiry into the motion of heaven remain interminable.”
In Peurbach’s Theoricae novae planetarum, the parallel to Copernicus’s postulates are Peurbach’s statements about the orbs of the planets and the figures (theoricae) that represent them, starting with the Sun:
The sun has three orbs, separated from one another on all sides and also contiguous to one another. The highest of them is concentric with the world on its convex surface, but is eccentric on its concave surface. The lowest, on the other hand, is concentric on its concave but eccentric on its convex surface. The third, however, situated in the middle of these, is eccentric to the world on both its convex surface and its concave surface … Therefore, the first two [orbs] are eccentric relatively, and they are called the deferent orbs of the apogee of the sun. The apogee of the sun varies according to their motion. The third is eccentric absolutely and is called the deferent orb of the sun. The body of the sun is attached to it and moves indeed according to its motion. These three orbs take two centres. For the convex surface of the highest and the concave of the lowest have the same centre, which is the centre of the world. From that fact the whole sphere [tota sphaera] of the sun, just as the whole sphere of any other planet, is said to be concentric with the world. But the concave surface of the highest orb and the convex of the lowest, together with the surfaces of each side of the middle orb, share another center, which is called the centre of the eccentric.14
Peurbach then goes on to describe and to represent by a figure (i.e., the theorica orbium) the basic configuration for the Sun, which is repeated with variations for the other planets (see figure 6.2). These whole three-dimensional, three-part spherical shells, or orbs, are the identifying DNA, so to speak, of the configuration that Peurbach’s Theoricae novae planetarum shares with Ibn al-Haytham’s On the Configuration of the World, a configuration that by Peurbach’s time had been shared between European authors since at least the late thirteenth century – so, for instance, a passage very similar to this long and detailed paragraph about the three component orbs of the whole orb of the Sun can be found in John of Sicily’s Scriptum super canones Azarchelis de tabulis Toletanis (ca. 1290).15 John of Sicily’s description of the orbs obviously belongs to the family tree linking Ibn al-Haytham and Peurbach even if it is not in the direct line linking the two.
Within each whole thick spherical shell of this configuration, there is an eccentric deferent just thick enough to contain the Sun (or the epicycle of a planet), which touches the convex and concave surfaces of the whole orb at two opposing points. Inside and outside of this eccentric deferent, there are unevenly thick orbs that not only fill out the rest of the volume of the whole orb but also carry around the apsides (i.e., apogee and perigee) of the planet west to east, around an axis parallel to the axis of the ecliptic, at a rate of one degree every hundred years. These outer shells are called the deferents of the apogee or the deferents of the apsides of the eccentric of the planet. They are not simply there filling up space but have a function of moving the apsides west to east at the very slow rate of one degree per century. Between these two outer suborbs, there is a fixed cavity within which the eccentric deferent of the Sun (or of the epicycle and planet) rotates continuously at a uniform velocity into itself, never impinging upon the deferents moving its apsides.16
These orbs are rigid, not fluid. Edward Grant is in error in his Planets, Stars, and Orbs when he argues that orbs were not thought to be rigid until the sixteenth century.17 If Grant had followed the development of configuration astronomy, or ʿilm al-hayʾa, from Ibn al-Haytham through its transmission into Europe, he would not have made this mistake. For instance, in his Al-Tadhkira fīʿilm al-hayʾa, which built upon Ibn al-Haytham’s On the Configuration of the World, Naṣīr al-Dīn al-Ṭūsī wrote, “Nothing having the principle of circular motion can undergo any rectilinear motion at all, and conversely, except by compulsion. Thus the celestial bodies neither tear nor mend, grow nor diminish, expand nor contract: neither does their motion intensify nor weaken. They do not reverse direction, turn, stop, depart from their confines, nor undergo any change of state except for their circular motion, which is uniform at all times.”18 Instead of paying attention to Ibn al-Haytham, Grant credits the idea of three-part orbs to Ptolemy’s Planetary Hypotheses and makes little use of works with the title Theorica planetarum in his research:
The old Theorica planetarum, probably composed in the latter half of the thirteenth century, includes nothing relevant to cosmology, omitting even discussion of the orbs. Apart from a discussion of the order and distances of the planets, there is little that is cosmological in Campanus of Novara’s similarly named Theorica planetarum. The same judgment applies to Georg Peurbach’s Theoricae novae planetarum of 1460–61, which is virtually devoid of cosmological content, although Peurbach does discuss the orbs as if they were real physical bodies. Bernard of Verdun chose to include more cosmology than most astronomers. In his Tractatus super totam astrologiam, he devoted the first eleven brief chapters to cosmological themes, approximately 4 percent of the entire treatise.19
When he comes to the introduction of what should be recognized as Ibn al-Haytham’s configuration, Grant writes,
References to epicycles and eccentrics appear in widely used thirteenth-century works like [Johannes de] Sacrobosco’s On the Sphere [of the World] and in the anonymous Theorica planetarum, although neither author implies or suggests that they might be real, material, solid orbs. If Roger Bacon (ca. 1219–ca. 1292) was not the first to mention material eccentrics and epicycles in the Latin West, he may well have been the first scholastic natural philosopher to have presented a serious evaluation of their cosmological utility. After some hesitation and ambivalence, Bacon rejected physical eccentrics and epicycles and opted for Aristotle’s system of concentric spheres.20 Ironically, it was his description of the system of eccentrics and epicycles that was most widely adopted by medieval natural philosophers and which still found defenders well into the seventeenth century.21
Actually, Roger Bacon did not grasp all of the features of the three-orb model – for instance, that the orbs surrounding the eccentric deferents of the planets move one degree per century with the motion of the eighth sphere. Bacon called the three-orb model the “imaginatio modernorum,” leaving it open to speculation whom he meant by moderns. John of Sicily’s more organized and complete statement of the theory in his Scriptum super canones is a much more likely route for the spread of ʿilm al-hayʾa into Europe.22
After a long list of writers who gave the view described by Bacon “more than a cursory glance,” Grant goes on to describe the three-orb view in detail: “I shall frequently refer to the ‘modern’ theory as the ‘three-orb system,’ but I shall also refer to it as the ‘Aristotelian-Ptolemaic system,’ since Aristotle’s concentric spheres were assigned a significant role within the system of eccentrics … Bacon introduces another interpretation – ‘a certain conception of the moderns,’ as he put it – in which the external surfaces of each planetary orb are concentric but which contain at least three eccentric orbs.”23 Despite Grant’s Latin-language perspective, the authors that Bacon called “the moderns” should be understood to be Islamic astronomers in the ʿilm al-hayʾa tradition or those authors writing in Latin who took up their views.24 Among scholastics who accepted the physicalized three-orb system, Grant lists Albertus Magnus, Duns Scotus, Aegidius Romanus, and Durandus de Sancto Porciano.25 He then describes in more detail how the system was defended by Pierre d’Ailly in his 14 Questions on the Sphere of Sacrobosco and by many others in succeeding centuries.
It is clearly the case in Ibn al-Haytham’s On the Configuration and in the Islamic hayʾa tradition generally that the orbs are rigid bodies and that the included deferents, epicycles, and planets are held tightly in place, except that they can rotate uniformly, never exceeding the place or cavity they are in. Moreover, these orbs spin. The Latin orb is a good translation of the Arabic falak. Not only are these orbs and suborbs geometric spherical shells, but they are also physical: they uniformly spin, and there are movers (i.e., separate or immaterial substances or intelligences) that cause their spinning.
Recently, historians such as Peter Barker and Michela Malpangotto have supported the view that Peurbach and Copernicus accepted the existence of orbs.26 In the meanwhile, however, the connection of Peurbach’s orbs to Ibn al-Haytham’s orbs, known perfectly well in 1976 to Swerdlow, has been neglected, undermining the correct interpretation of the place of orbs in sixteenth-century astronomy.
In the Commentariolus, Copernicus is commenting on the hypotheses that he himself has set out (de hypothesibus motuum caelestium a se constitutis), whereas in commenting on Peurbach, Brudzewo states what he plausibly understands to be the principles of Peurbach’s Theoricae novae planetarum. According to Brudzewo, they are as follows:
1.The heaven is a simple body.
2.Of any simple body there is only one simple motion proper to it naturally.
3.A motion belonging to one body unnaturally, necessarily is proper to another body naturally.
4.One orb is not moved with several motions by the same intelligence, nor is the same orb moved by several intelligences equally proper to it.
5.To this may be added that a lower sphere does not influence the motion of a sphere above it, but rather the opposite, the superior influences the inferior. Not everything in a lower orb that is derived from a higher orb is natural to it, but it may belong to the lower sphere beyond [praeter] nature, because the lower orb has a nature obedient [oboedientialem] in everything, by which it obeys the higher spheres. Therefore, the higher orbs may influence by their motion the lower orbs and carry them around, whereas the opposite does not happen.27
These principles have a relation to Peurbach’s Theoricae novae planetarum similar to the relation of Copernicus’s petitiones to his Commentariolus. They derive ultimately from thinking about observations and how they could be explained by underlying reality. They are physical principles rather than mathematical ones.
Thus an idea about the nature of astronomy as a science that Copernicus was exposed to when he was a student at Cracow was that at least one genre of astronomical writing (works that are theoretical but not demonstrative) can start by stating principles or postulates upon which the following exposition will be based. This format could also be found within the determinations of scholastic questions in which, after stating principal arguments for one conclusion, the author mentions an opposite solution and then starts a determination of the question with definitions, suppositions, and the like.
In fact, the fourth and fifth of the principles that Brudzewo states for Peurbach’s Theoricae novae planetarum are principles found earlier in Albert of Saxony’s Quaestiones subtilissime in libros de caelo et mundo, where, as is typical, he states principles or suppositions before answering questions. In this case, Albert of Saxony’s question is how many celestial spheres or orbs there are.28 After some discussion, he proposes the opinion that there are nine orbs, starting with the suppositions that lead to this conclusion:
For the proof of this position, first it is supposed that the eighth sphere or orb is moved with more than one motion: it has one motion from east to west on the poles of the world [once a day], and another motion from west to east on poles of the Zodiac, one degree in a hundred years. The second supposition is that a single orb is not moved with several motions by the same intelligence, nor is the same orb moved by several intelligences equally proper to it. Aristotle proves this in the Metaphysics, Book XII.29 The third supposition is that a lower sphere does not move with itself a higher sphere, but a higher sphere does move a lower sphere.30
Later, in 14 Questions on the Sphere by Pierre d’Ailly (1350–1420), also in a question about the number of the orbs, a similar set of suppositions appears.31
Thus Albert of Saxony, Pierre d’Ailly, and others before Brudzewo put their theories or parts of their theories into a structure in which there are suppositions, principles, or premises (i.e., hypotheses) on which conclusions are based. These principles are typically physical rather than mathematical, and they are thought to derive from experience. They are not demonstrated but are the foundations of demonstrations.
Knowing that the principles are not proved and that the processes by which they are arrived at are not logically rigorous, practitioners of the discipline of astronomy could in exceptional circumstances think that a reformation of principles was called for. This is what Copernicus says at the beginning of the Commentariolus:
I understand that our predecessors assumed a large number of celestial orbs (multitudinem orbium coelestium) principally in order to account for the apparent motion of the planets through uniform motion … the stronger opinion (potior sententia), in which the majority of experts finally concurred, seemed to be that it is done by means of eccentrics and epicycles. Nevertheless, the theories concerning these matters that have been put forth far and wide by Ptolemy and most others, although they correspond numerically [with the apparent motions], also seemed quite doubtful, for these theories were inadequate until they also envisioned certain equant circles, by which it appeared that the planet never moves with uniform velocity either in its deferent sphere or with respect to its proper center. Therefore, an opinion of this kind seemed neither perfect (i.e., complete – absoluta) enough nor sufficiently in accordance with reason. Therefore, when I noticed these [difficulties], I often pondered whether perhaps a more reasonable way [modus] composed of circles could be found from which every apparent irregularity would follow while everything in itself moved uniformly, just as the principle of perfect motion requires. After I had attacked this exceedingly difficult and nearly insoluble problem, it at last occurred to me how it could be done with fewer and far more suitable things [rebus] than had formerly been put forth if some postulates called axioms are conceded to us which follow in this order.32
And then Copernicus lists the seven postulates (petitiones) quoted earlier in this chapter. That Copernicus, like the authors of theoricae planetarum, starts with physical principles supports the contention that he conceived his research program within the theorica planetarum genre.
Thus I claim that as a student at Cracow where Peurbach’s Theoricae novae planetarum was a model for the status of astronomy as a science, Copernicus would have learned that astronomy was both mathematical and physical and that, although it had many real achievements, it might still be improved by new insight into the hidden physical structures behind the appearances. Whether or not Copernicus was aware of it, the conception of astronomy as a science reflected in Peurbach’s Theoricae novae planetarum derived from Islamic hayʾa astronomy going back to Ibn al-Haytham’s On the Configuration of the World, to which I now turn.
The Theoricae novae planetarum and the Commentariolus not only presuppose the research tradition based on Ptolemy’s Almagest but also presuppose the Islamic tradition of hayʾa astronomy. An ancestor of Peurbach’s Theoricae novae planetarum (and hence of Copernicus’s Commentariolus) is Ibn al-Haytham’s On the Configuration of the World.33 In one Arabic manuscript of Ibn al-Haytham’s text, although not in the Latin or Hebrew translations, there are principles like those in fourteenth-century authors and in Peurbach. These principles may in some way have affected the statements of principles in Brudzewo and other Latin authors.
Historians of astronomy looking for the origins of the concept of thick physical celestial orbs have typically suggested that it derives from the orbs of Ptolemy’s Planetary Hypotheses, but Ptolemy himself, having first described three-dimensional orbs, then decided instead in favour of rings, tambourines, or “sawed-off-pieces,” and it is for the latter that he was known in the Islamic world. Moreover, Ptolemy suggested that these rings could interpenetrate and that the planets were the movers of their orbs rather than vice versa, both rejected by Peurbach.34
Unlike Ptolemy’s Planetary Hypotheses, Ibn al-Haytham’s On the Configuration of the World was translated and assimilated over time, so evidence of its reception in the West is widespread, although it is difficult to distinguish what comes from Ibn al-Haytham and what comes from other parts of the Islamic tradition of physical representations of celestial orbs, or hayʾa. As F. Jamil Ragep defines it, “The Arabic term hayʾa had several distinct significations when used in the medieval astronomical context. The basic meaning is ‘structure’ or ‘configuration’ … An astronomer writing in the hayʾa tradition was charged with transforming mathematical models of celestial motion, usually those of Ptolemy’s Almagest, into physical bodies that could be nested, along with the sublunar levels of the four elements, into a coherent cosmography (hayʾa).”35 As to the influence of Ibn al-Haytham’s work and the hayʾa tradition in Europe, Ragep states, “Given that the Planetary Hypotheses was unknown during the Latin Middle Ages, it is clear that the main source for the European theorica tradition was Islamic hayʾa; Ibn al-Haytham’s On the Configuration of the World was certainly quite influential.”36 Robert Morrison writes, “The best known achievement of Islamic astronomers from the thirteenth century onward was the creation of physical models that could represent available observations. In fact, the genre in which these astronomers wrote, ʿilm al-hayʾa (astronomy, literally ‘science of the configuration’) was a product of Islamic civilization.”37
What about evidence for the connection of Peurbach’s Theoricae novae planetarum to the longstanding history of ʿilm al-hayʾa in Islamic astronomy? In Peurbach’s Theoricae novae planetarum, the word “theoricae” may be understood to refer to the figures in the book representing astronomical hypotheses or parts of theories, many of which include the word theorica in their titles. Some of Peurbach’s theoricae represent physical bodies called orbs, whereas other theoricae represent mathematical lines or motions with no pretense of being bodies. Typically, between the first theorica representing coloured physical orbs of unequal thickness and the third theorica representing circles and the motions on them in two dimensions, there is a second theorica axium et polorum, which by means of lines and circles shows the axes and directions of rotation of the physical orbs. The distinction between the physical and the mathematical may have been made clearer in the printed versions of the Theoricae novae planetarum by their coloured representations of the partial orbs, in contrast to the narrow black lines used for the mathematical theoricae, but the distinction was not new.
In any case, the relation of the physical and the mathematical in this sort of astronomy can be appreciated better by seeing its roots in Ibn al-Haytham’s On the Configuration of the World and subsequent hayʾa astronomy. Peurbach’s theoricae novae have many features linking them to the configurations found in Ibn al-Haytham’s On the Configuration of the World, especially the postulation that the “complementary” orbs surrounding each planet’s eccentric deferent move the apsides of the deferent with the slow motion of the eighth sphere, most commonly one degree per century. Importantly, the three-part orbs are concentric with the world on their outside and inside surfaces, thus avoiding the problem that epicycles might be thought either to stick out and conflict with nearby deferents or to require that there be a vacuum or compressible material between celestial spheres in order to allow room for rotations of spheres with epicycles protruding (see figure 6.2).
In the introduction to the On the Configuration of the World, Ibn alHay-tham states the motivation of his work, which is to find physical configurations that might lie behind mathematical astronomy in the Ptolemaic tradition. The introduction is not included in the only known complete translation of the On the Configuration of the World into Latin,38 but it is present in thirteen of the fifteen manuscripts of the work in Hebrew found in European libraries, so it may well have entered the European context through Jewish intermediaries.39 Ibn al-Haytham’s goal is to associate each of the motions that Ptolemy describes with a real, solid, uniformly rotating orb that does not conflict with any other orbs that may exist. Like others in this genre, Ibn al-Haytham is going to describe the bodies and motions in the heavens without attempting to demonstrate them. The audience for his book is people who desire a rapid way of learning the basic facts of astronomy. Ibn al-Haytham carries his program out most completely for the Sun. As far as I can judge, he completely succeeds in carrying out his program only for the Sun, for which Ptolemy had not resorted to the equant. For the three outer planets and Venus, both Ibn al-Haytham and Peurbach succeed in embedding the planet within the epicycle and embedding the epicycle within the deferent – making a consistent combination of orbs explaining a good part of the motions, as was the case for the Sun, with the addition of the epicycle containing the planet. Nevertheless, Ibn al-Haytham has something corresponding to Ptolemy’s equant for the planets, and Peurbach likewise has equants in the mathematical theorica linearum et motuum. Thus astronomers working in the hayʾa, configuration, or theorica traditions had arrived at a view of astronomy that included both physical configurations and mathematics functioning in a complementary way. Late-fifteenth-century conceptions of scientific disciplines, for theoretical rather than practical reasons, supported this conception of a diversity of approaches within astronomy.
Much of historians’ efforts to understand Copernicus’s position on the status of astronomy as a science has concerned his De revolutionibus orbium coelestium and its reception. A common conception is that medieval astronomers following Ptolemy saw their task as being limited to saving the phenomena by making use of deferents, eccentrics, epicycles, and even equants or other mathematical devices for which they did not claim physical reality.40 Then (Andreas Osiander’s introduction notwithstanding) Copernicus in De revolutionibus asserted that his heliocentric (or heliostatic) planetary system represented the real system of the world. In the years that followed the publication of De revolutionibus in 1543, according to this commonly received view, many astronomers following what is called the “Wittenburg interpretation” made use of Copernicus’s mathematical techniques, while still following the old conception of astronomy as a science, claiming only to save the phenomena, not to assert the reality of a Sun-centred system.41 Only with Johannes Kepler, according to this view, was a unified physical and mathematical astronomy proposed.42
This chapter, however, seeks not to track the conceptions of astronomy as a science that existed after the publication of De revolutionibus in 1543 but to examine the conceptions of astronomy that were common in Cracow University at the time that Copernicus was a student, considering how these conceptions may have shaped Copernicus’s Commentariolus. At that time, as already indicated, Peurbach’s Theoricae novae planetarum, the main astronomical textbook of the day, already attempted to provide a physical basis for as much as possible of Ptolemaic astronomy. Although late-fifteenth-century celestial physics was not the same as the physics of sublunar bodies – it described the uniform rotations of bodies made of aether rather than the rectilinear motions of sublunar elements and compounds – it was a physics with movers as well as moved bodies. Astronomy in the late fifteenth century did include equant circles, which were purely mathematical and not paired with physical bodies, but this could be understood to be the case because the program of supplying a physical basis for Ptolemaic astronomy had yet to be brought to a successful completion.
One reason why many historians continue to assume that the astronomers who received and reacted to Copernicus’s De revolutionibus in the way that they did were typical of the whole later medieval period is that they find similar positions among Aristotelians of the thirteenth century and among those of the later sixteenth century. It was the case, however, that medieval Aristotelianism went through alternating phases of conservativism or traditionalism, following the so-called via antiqua, and progressivism or renovation, following what was called the via moderna. The dominant form of Aristotelianism present during the decades of reception of De revolutionibus reflected the decisions of the Council of Trent and the Catholic Counter-Reformation, which advocated a return to the Christian Aristotelian synthesis found in the work of Thomas Aquinas and of other thirteenth-century thinkers such as Albertus Magnus and Aegidius Romanus. The advocates of this form of Aristotelianism were later called antiqui, as contrasted with fourteenth-century Aristotelians such as William of Ockham, John Buridan, Albert of Saxony, Marsilius of Inghen, and others who were called moderni.43
On the basis of the evidence described here, I argue that the conception of astronomy as a science that Copernicus encountered as a student at Cracow University, the one reflected in the Commentariolus, was closer to the attitudes of the moderni than to those of the antiqui. From the fourteenth century up through Copernicus’s time at Cracow, the conception of celestial orbs as physical rather than only mathematical was widely held. More important, astronomy was conceived as a progressive scientific discipline in which principles were derived a posteriori from experience and hence could be rederived from new or added experience.44 The conception of science in general and of astronomy in particular can be seen not only in works of astronomy but also in commentaries on the works of Aristotle, particularly in commentaries on the Posterior Analytics. Although the adoption of physical orbs as well as purely mathematical methods of calculation may have developed for internal reasons within the discipline of astronomy proper, corresponding attitudes toward the relation of mathematics and physics were supported by what was written in commentaries on Aristotle’s Posterior Analytics.45 Although I think the conception of astronomy in the Theoricae novae planetarum was probably of greater influence on Copernicus’s Commentariolus than were commentaries on Aristotle, the background of Aristotelian philosophy at Cracow also helps to explain why Copernicus might have proposed a new configuration of the world in the Commentariolus.46
Aristotle had understood astronomy to be intrinsically both mathematical and mechanical, similar to what was understood to be the case for terrestrial mechanics in the pseudo-Aristotelian Mechanica, where the properties of a rotating rigid body are used to explain the law of the lever.47 In the Mechanics, the mathematical and the physical are inextricably bound together in the analysis of simple machines. The same could be the case in astronomy, where, rather than planets moving on elliptical orbits through space, one has the planets carried around by rigid orbs or aether shells, constraining their motions to the possibilities for a rigid orb rotating in place with uniform velocity.
If late-medieval astronomy was both mathematical and physical or mechanical, at the same time mathematics was conceived to be a product of the human mind rather than referring to quantitative forms really existing in the external world. In book 6 of the Physics, Aristotle had argued that all kinds of continua – geometric, corporeal, temporal, and so forth – are isomorphic. Thomas Bradwardine, in the fourteenth century, still assumed the same isomorphism in his De Continuo.48 Averroes (Ibn Rushd) had argued, however, that physical continua were not necessarily isomorphic to geometric continua. The latter were assumed to be divisible ad infinitum, but physical continua might be composed of atoms, or minima naturalia, and therefore might have a limit to their divisibility.49 Like Averroes, the late-medieval moderni did not assume that geometrical truths must always be consistent with physical truths or vice versa.50
Aristotle’s conception of demonstrative science, as set out in his Posterior Analytics, posits that demonstrative sciences have structures similar to Euclidean geometry. They have principles (i.e., axioms, postulates, definitions, suppositions, or hypotheses) on the basis of which conclusions are demonstrated. In some cases, a scientific principle may be known in itself (per se nota), such as the principle that the whole is greater than the proper part, but in other cases a principle may be known from experience, such as the principle that fire is hot or the principle that the stars are on a sphere rotating around the Earth once a day. For Aristotle, there are many distinct and autonomous scientific disciplines, but in a few cases, including astronomy, one scientific discipline may be “subalternate” to another. Astronomy is subalternate to geometry, and music or harmonics is subalternate to arithmetic, in the sense that the subalternated science makes use of arguments from the scientific discipline subalternating it. For the moderni as a rule, astronomy was not subalternate to natural philosophy.51
Aristotelian commentators over time developed many different interpretations of the Aristotelian conception of scientific knowledge. A first polarity within medieval views on demonstrative science concerns the emphasis on necessity and certainty in science as opposed to a conception of science as developing over time. According to a taxonomy of senses of “science” frequently quoted from Robert Grosseteste’s commentary on Aristotle’s Posterior Analytics, there are four levels of scientific knowledge. Science, in the fourth and most proper sense, is the apprehension of a necessary truth, of which the necessity is known by its cause, and in this way we know only the conclusions of demonstration. Science, in a broader sense, however, includes apprehension of contingent truths or of principles of demonstration.52
In the thirteenth century and again in the sixteenth, the antiqui paid attention to Aristotle’s view that, in the most proper sense, science is demonstrative, universal, and necessary. In commenting on Aristotle’s Physics, Averroes had, however, modified what Aristotle had said by proposing a distinction between mathematics and empirical science, saying that physics differs from geometry because in geometry the same thing (e.g., an axiom) is better known in itself and to us, but in physics, what is better known in itself or to nature (e.g., the causes) is not better known to us because we observe the effects before knowing their causes. For example, astronomers first know by observation that lunar eclipses occur (this is the effect); then they reason to causes of eclipses, namely that the Moon shines by reflected sunlight and that an eclipse occurs when the Moon moves into the Earth’s shadow (this is the cause). Once the cause of eclipses is known, an astronomer can predict or demonstrate that a lunar eclipse will occur when, by the relative positions of the Sun, Moon, and Earth, it follows that the Moon will be in the Earth’s shadow.53
What does such a history of the discovery of the causes of eclipses mean with regard to the status of the science of eclipses? A commentator following the via antiqua would likely hold that the demonstration that an eclipse will occur based on its causes (i.e., a demonstration propter quid) is certain in Grosseteste’s fourth, most proper sense. He would understand this to mean that the principles or premises of such a demonstration are known for certain to be true; some would even say that they are known per se or a priori. In contrast, a commentator following the via moderna might explain that astronomers do not doubt the principles of such a demonstration because the practitioners of any demonstrative science do not doubt their principles – there are no more certain principles in the science by which the principles at issue could be demonstrated – but, in fact, the principles of eclipse science could be false. For the moderni, most of the principles of the natural sciences were in fact thought to be based on experience, not on a priori or innate knowledge. Even with respect to a mathematical science like geometry, John Buridan, a quintessential modernus, went so far as to say that the principles or postulates of geometry itself are not certain. Thus for Buridan, it was not necessarily true in all sciences that a continuum is divisible in infinitum (as a principle of geometry states) since physicists might discover that the world is made up of indivisible atoms. Nevertheless, geometers do not doubt divisibility in infinitum because, if it were not true, many of their theorems would be false.54
Consistent with this first polarity, a second, related polarity between the views of the antiqui and moderni concerns the certainty of the method by which the principles of an empirical science come to be known. The moderni followed Averroes in believing that, in contrast to mathematics, in empirical sciences, there is a two-fold process in which scientists first work a posteriori from observed effects to causes and then later, once the causes have been established, work a priori from causes to effects. They call a demonstration from effects to causes a demonstration that or of fact (quia). They call a demonstration from cause to effect a demonstration of the reasoned fact (propter quid). But must one always begin from sense, memory, and experience, or might there be arguments entirely a priori? Might there be a third kind of demonstration (potissima, most powerful) that could at the same time demonstrate the fact and the reason for the fact? Avicenna (Ibn Sīnā) is said to have held that there is only one species of demonstration, the propter quid demonstration; Averroes is said to have held that there are three species of demonstration: quia, propter quid, and potissima, the last of which proves both the cause and the existence of the effect.55
Related to these questions is the question of how, in fact, demonstrations that or quia leading to principles are supposed to work. Clearly, principles cannot be demonstrated syllogistically within the given discipline; otherwise, they would not be principles or there would be circular demonstrations. Walter Burley argued for confidence in science derived from experience on the grounds that humans have the natural ability to understand rationally what they observe, just as fish have the ability to swim and birds to fly. If such an ability is denied to humans, Burley said, what justifies their high place in the cosmos?56 By the time of Giacomo Zabarella (1533–89), among the most prominent of the Renaissance Aristotelians, who in this respect belongs among the antiqui, there was great concern about what was called regressus, or a multistage process by which, after causes have been found a posteriori by a demonstration quia, they can somehow be stabilized or proved true before being used as the premises in a propter quid demonstration.
Buridan, for his part, responded to arguments that the process of deriving principles from experience is not certain by saying that anyone who was not willing to accept theories that were true only “for the most part,” rather than always and necessarily, did not deserve to take part in natural philosophy, where this uncertainty is a condition of the work.57 In the second question of book 2 of his Metaphysicen Aristotelis, Buridan argued against the view (defended by Nicholas of Autrecourt) that all principles have to be traced back to the principle of noncontradiction.58 In empirical sciences, Buridan said, many principles are based on sense, memory, and experience, and there may be as many principles as there are conclusions. We do not doubt these principles based on sense experience, but they cannot be demonstrated. Rather, we assent to a principle such as “fire is hot” because we have always observed it to be the case and have never observed an exception. We have developed a habit of assenting to the principle. Nevertheless, we could begin to worry that a principle we have thought was certain is not certain. Here, Buridan gave an example found in more than one place in his work, where old women are led to doubt a first principle. Buridan first asks them whether it is possible for them to sit and not sit at the same time. They say it is impossible. Then he asks whether they believe that God could do it, and they answer that they do not know.59
In the Prior Analytics, Aristotle himself had said that most of the principles of natural science are based on experience: “But in each science the principles which are peculiar are the most numerous. Consequently, it is the business of experience to give the principles which belong to each subject. I mean for example that astronomical experience supplies the principles of astronomical science; for once the phenomena were adequately apprehended, the demonstrations of astronomy were discovered. Similarly, with any other art or science. Consequently, if the attributes of the things are apprehended, our business will then be to exhibit readily the demonstrations.”60 Generally speaking, in discussions of learning from experience in science, we now think of induction, where by induction we have in mind collecting a body of data and then reasoning from the data to the generalization that encompasses these data. In Aristotle and in the Middle Ages, however, the process was described as moving from sense, to memory, and then to “experiment,” where experiment means grasping clearly in the mind what has been sensed. An astronomical example would be observing the positions and motions of the stars, noticing that they all move together without changing their positions relative to one another, and then concluding that the stars are attached to or embedded in a rigid rotating sphere. Thus the process involved insight, not the collection and methodical analysis of sets of data.
Thus, on this second point, the antiqui differed from moderni in their estimation of the certainty of the principles of natural science. The antiqui concentrated on the ideal demonstrative science, whereas the moderni had in mind the scientific disciplines as they currently existed. This focus does not mean, however, that the moderni were “skeptics”; this label – which, to me, has negative connotations – was placed on them by antiqui, whose goal was certain knowledge. They are better understood as critical realists; they aimed for true theories and they had a high opinion of the success of natural philosophy, but they were not dogmatic.61 The current theories, although the best available, might be wrong.
A third polarity between antiqui and moderni has to do with their conceptions of the relations of scientific disciplines to each other. Aristotle generally held that there are many scientific disciplines, each with its own principles, subject matter, and conclusions, and he argued that demonstrations should not mix concepts or terms from different disciplines. Geometry as a demonstrative science is separate from the demonstrative science of physics. It follows that Aristotle would have rejected René Descartes’s later claim that he could deduce all of physics a priori from mathematics, Aristotle’s grounds being that this deduction would involve an illegitimate transgression of disciplinary boundaries (metabasis).62 The antiqui, by contrast, tended to the view that scientific disciplines could and perhaps should be synthesized or at least coordinated under metaphysics into a unified worldview, one that is sometimes called by historians the “Christian-Aristotelian synthesis.”
In how many subject matters is demonstrative scientific knowledge possible? In the Nicomachean Ethics, Aristotle had said that it was foolish to expect scientific demonstrations in ethics, the subject matter being far too complex; at best, only probable arguments could be obtained.63 In other sciences, however, it might be difficult but not impossible in the long run to develop a demonstrative science. In the case of such a complex subject matter as meteorology, Aristotle said that a scientist may have to be satisfied with a hypothesis that saves the phenomena and is at least possible, containing nothing impossible or self-contradictory.64 At the time of Copernicus, eccentrics and epicycles had been shown to be possible if they were embedded within thick concentric orbs. Equants, however, had not been shown to be physically possible. Because of the observed regularity of celestial motions over long periods, from which it had been inferred that celestial bodies are incorruptible and celestial movers infatigable, it was taken to be more likely that astronomy could reach high levels of scientific certainty than meteorology.
Taken together, what did these opinions mean about the present state of astronomy for the given practitioner around 1500? In the Aristotelian scheme, the practitioner of a given science is expected to take the principles of his science as true; they are the ultimate basis of proofs and cannot be proved themselves. When, however, a natural science is in the process of being established on the basis of observation, the most obvious facts of observation – on the basis of which the scientist reasons a posteriori to causes – should not be denied. In this sense, a physicist does not need to dispute with a follower of Parmenides who denies that anything moves because the most obvious empirical truth of physics is that all or some things move.65 These are Aristotelian ideals of demonstrative science, but a given science at a given point in time may not have reached the ideal. Since this is the case, it is always possible that in the future someone will be able to propose better or more satisfactory principles or hypotheses for the given science. In the 1270s and 1280s, at the University of Paris and at Oxford University, there were lists of condemned theses, many of which seemed to assert that God could not contravene the truths of Aristotelian physics – saying, for instance, that God could not cause a vacuum to exist because vacuums are physically impossible. After the condemnations, it became normal to distinguish between what is naturally possible and what would be supernaturally possible if God so chose. Thus it became common to discuss conceptions of the world differing from Aristotle’s.
On the one hand, then, the antiqui and moderni generally agreed with Aristotle that ideally scientific disciplines should demonstrate conclusions on the basis of principles and that our ideas of the world originate in sensation. On the other hand, antiqui and moderni tended to differ in that the antiqui emphasized that science should be universal, necessary, and certain, as well as that the disciplines should be consistent with each other, coordinated under the discipline of metaphysics. In contrast, the moderni took scientific disciplines to be distinct rather than tied together in a single overarching worldview, and they took them to be not absolutely certain but still in progress, true for the most part, but not perfected.
What would happen if a conflict arose between an established theory and observation? In the thirteenth century, Roger Bacon considered one possibility. In responding to Averroes’s rejection of epicycles and eccentrics on the grounds that they conflict with truths of natural philosophy, although the observation that planets are sometimes nearer and sometimes farther seems to require them, Bacon supposed that those like Averroes would choose established theory over observation. So Bacon wrote that the naturalists who say that the heavens must contain only uniform circular motion and who reject epicycles and eccentrics say that it is better to contradict sense than the order of nature because sense is known to fail at a great distance.66 But then Bacon seems to go off on a different tack, saying that if the natural mathematicians succeed in saving the appearances just as well as the pure mathematicians do, it is better to follow the natural mathematicians, for the principles are in natural things and mathematicians accept principles from natural scientists.67 Moreover, Bacon had determined earlier that there is solid evidence from eclipses that the Sun and Moon are sometimes nearer to the Earth and sometimes farther away.68 Everything visible that sometimes appears under a larger angle and sometimes a smaller one, although the visible thing itself is not changed, nor the medium, nor the vision, is sometimes nearer and sometimes farther away. But this is true of planets, notably the outer planets, which always appear larger when they are in opposition to the Sun.69 From this it follows that no theory that does not allow the distances to the planets to vary can be correct. In the end, Bacon concluded in a conciliatory way, saying that it should be known that although the pure method of mathematicians differs from that of the one who knows natural things with regard to saving appearances in the heavens, everyone has the same goal, albeit approaching it by different routes, namely to find the places of the planets and stars with respect to the zodiac. Thus, however much pure mathematicians and mathematical astronomers may diverge along the way, one and the same goal terminates their endeavours.70
Another response to a situation in which theory and observation disagreed within a given discipline or in which there were competing theories concerning the same subject would be to reconsider the principles that had been assumed. In his Optics, Ibn al-Haytham addresses the lack of unanimity in previous physical and mathematical optics, one camp holding that rays come from objects into the eye and the other camp holding that rays go out from the eye.71 He says that when there are conflicting theories, both theories may be false, one may be true and the other false, or the two theories could be shown to be consistent if the inquiry were taken farther.72 To resolve the disagreement between the intromission and extromission theories in optics, Ibn al-Haytham proposes to “recommence the inquiry into the principles and premises, beginning our investigation with an inspection of the things that exist and a survey of the conditions of visible objects.”73 Ibn al-Haytham’s suggestion was known in the Latin Middle Ages.74 On a smaller scale, in giving their answers to or determinations of scholastic questions, the moderni often began by listing several suppositions on which they would base their conclusions. This habit of listing suppositions as so-to-speak lower-level principles corresponded to the practice of authors of theoricae planetarum.75
A fourth, more particular polarity between antiqui, exemplified in the person of Thomas Aquinas, and moderni, exemplified in the person of William of Ockham, has to do with their conceptions of mathematics. Aquinas thought that there are quantitative forms existing in bodies, which may be abstracted from bodies and studied in themselves by mathematicians, although they never really exist separately as Platonic forms are supposed to do. The resulting abstract mathematics is the most certain of any science, but it is not empirical. In contrast, Ockham thought that there are no quantitative forms; everything that exists is either a substance (i.e., a substance consisting of matter and a substantial form) or a qualitative form existing in a substance. Following Ockham, the fourteenth-century moderni treated mathematical quantities as concepts existing in the minds of mathematicians, not in the external world.76 Rather than abstracting quantitative forms from the bodies in which they actually exist, mathematicians, thinking creatively, come up with mathematical concepts. Sometimes the concept might be based on a “phantasm” existing in the mind as a result of seeing a body in the external world. The concept of a geometric line might be based on the vestige or trace of the motion of a body, reduced to a single dimension. Or the mathematician could imagine a “latitude of heat” by analogy to a line, without there being any such qualitative line in the external world.77 At Cracow in the late fifteenth century, John of Głogów and Albert of Brudzewo tacitly assumed what Ruth Glasner calls a “divorce” between mathematics and physics, together with the belief that mathematical entities are conceptual or imaginary.78 In relation to astronomy, it is worth noting that the moderni treated indivisibles, such as points, lines, and surfaces, as mathematical, not physical, entities. Already, this means that Peurbach’s theoricae orbium will be understood very differently from his theoricae linearum et motuum. In the next section, then, I examine what Głogów and Brudzewo had to say about mathematics and physics in relation to the polarities I have sketched between the antiqui and moderni.
Copernicus almost certainly became acquainted with Aristotle’s Posterior Analytics when he was a student at Cracow.79 He might have seen Albert of Saxony’s questions on the Posterior Analytics, which are typical of the via moderna, and/or he might have heard more eclectic lectures by John of Głogów or one of his students. The publication dates of the questions on the Posterior Analytics of Albert of Saxony (1497) and of John of Głogów (1499) are both after Copernicus left Cracow for Bologna, so if Copernicus knew of these works or others like them, it was from manuscripts or from the lectures of his teachers.80
John of Głogów’s commentary is in effect an edition of, or supercommentary on, John Versor’s commentary on Aristotle’s Posterior Analytics, and it takes some of its colouration from Versor. Basing himself on Versor’s questions, Głogów says he will also draw upon the commentaries of Aegidius Romanus, Thomas Aquinas, Albertus Magnus, and Paul of Venice, among others.81 These are names mainly associated with the antiqui.
In answer to the question of whether it is possible to know something de novo (utrum possibile sit aliquid scire de novo?), Głogów says that we can have scientific knowledge and that it can be new rather than, as Plato said, always something that we knew previously but forgot. First of all, this new knowledge comes by way of sense experience. Our senses are actualized by sensible species, and our intellect is actualized by intelligible species poured into it by the agent intellect.82 Beyond new knowledge (intellectus) from experience, Głogów describes how we demonstrate scientific conclusions by reasoning and argumentation. Here, Głogów seems to have in mind a science like geometry in which mathematicians gradually prove new theorems on the basis of the definitions, axioms, and postulates. Not everyone who is presented with the principles of geometry immediately sees the truth of all possible theorems.83
In Aristotle’s opinion as Głogów reports it, everything we come to know originates in sense. Once we have concepts in mind, the intellect reasons with phantasmata.84 Correspondingly, as Albertus Magnus wrote in the thirteenth century, in physical or mathematical reasoning phantasmata are required.85 Would Głogów have thought that humans can, in this way, gain certain scientific knowledge about the world? Or, given the role of human sensation and cogitation, is all putative scientific knowledge fallible? It appears that Głogów, like most fourteenth- and fifteenth-century scholastic Aristotelians, holds that scientific knowledge is fallible.86 This view would put Głogów more in the camp of the moderni.
With regard to the relation of the special sciences to the more general sciences, such as metaphysics or logic, and with regard to the relation of subalternate sciences to those to which they are subalternated (e.g., astronomy to geometry), Głogów notes Aristotle’s argument that scientific demonstrations must be based on proper rather than common principles.87 A geometer would not use the common principle “every whole is greater than its part,” he says, except only insofar as it is contracted or limited to the subject matter of geometry, namely magnitude.88 In particular, demonstrations propter quid, or the most powerful demonstrations, should be from the proper principles of the given science. If special sciences use common principles, it is only as they are contracted or limited to the particular subject matter.89 Głogów does say that special sciences might in some way be corroborated or strengthened by a common science such as metaphysics,90 but he qualifies this effect by repeating Aristotle’s statement that it is difficult to know whether we have scientific knowledge, explaining why this is the case, namely that we often do not know whether we have proper principles for our demonstrations.91
In response to the question of whether sciences such as mathematics are the most certain, Głogów says that they are because in them one is less likely to err, given that they are universal rather than particular and given that the arguments are often convertible. But the first argument he makes is more surprising, namely that mathematical sciences are more certain because they offer more to sense and nothing is in the intellect that was not previously in the sense. Thus, he says, mathematics points to a circle drawn in the sand.92 For such reasons, demonstrative mathematical sciences are the most certain and rarely err.93 Here, Głogów cites Euclid, Aegidius Romanus, Averroes (Ibn Rushd), and Paul of Venice, as well as other books of Aristotle. From Paul of Venice, Głogów quotes an answer to the argument that mathematics assumes something false when it says a line may be extended infinitely. According to Paul of Venice, Głogów says, this assertion is to be understood according to mathematical imagination, not physical reality. For any given line, one can imagine a larger one.94
What Głogów writes in his commentary on Sacrobosco’s On the Sphere of the World is also consistent with the views of the moderni. Commenting on the different types of scientific proof, Głogów says that the daily rotation of the primum mobile can be proved with an argument “from a sign,” namely by the paths of the fixed stars rising in the east and setting in the west. This is a weak form of argument, he says, but of a type that Aristotle frequently uses in his writings. Such an argument from a sign shows that something is the case but not why. The same conclusion can be shown a priori, he says, by referring to the purpose (or final cause) of such a situation in nature. The reason why there is daily rotation is to cause the generation and corruption observed in the sublunar realm. In other words, causing generation and corruption is the purpose for the rotations of the heavenly bodies.95
Later in his commentary on On the Sphere, Głogów repeatedly distinguishes between the parts of astronomical theories that are supposed to represent physical bodies and the parts that are merely mathematical. First, the various circles drawn on the celestial sphere are not real, unless perhaps the zodiac, understood as being twelve degrees wide and containing constellations, is real.96 Sacrobosco had said that every planet except the Sun has three circles, namely the equant, the deferent, and the epicycle. The equant of the Moon, he said, is a circle concentric with the Earth.97 Głogów explains that equants for the three outer planets are imagined circles, which astronomers proposed because the planets did not move with constant velocity either around the centre of the world or around the centre of their own deferents.98 In contrast, the main three-part orbs of the planets are real and thick even if sometimes they are referred to as circles. They are only represented as projected on a plane because otherwise they cannot be painted or shown in a figure.99 Głogów observes that Gerard of Cremona writes about the eccentric and equant circles of the superior planets in the same terms, but Georg Peurbach and Johannes Regiomontanus often criticize him.100 Epicycles should be understood to be embedded within the depth of the deferent.101
So in his commentary on On the Sphere, Głogów distinguishes between what is mathematical (and hence imaginary) in astronomical theories and what is physical. He includes final causes or purposes as belonging to the theory. Although commenting on On the Sphere, which deals mainly with circles, not solid orbs, Głogów imports into his commentary the solid orbs of Peurbach’s Theoricae novae planetarum. Thus, in harmony with the views of the moderni, Głogów distinguishes in On the Sphere between what is to be understood as real or physically existing and what is mathematical and hence dependent on human thought.
Finally, clear evidence concerning the conceptions of astronomy as a science present at Cracow when Copernicus was a student is contained in Albert of Brudzewo’s commentary on Peurbach’s Theoricae novae planetarum.102 If Brudzewo did not lecture on Peurbach’s Theoricae novae planetarum with Copernicus in attendance, he had taught such a course in previous years, and whoever taught Peurbach with Copernicus in attendance had likely heard Brudzewo himself earlier. Before the commentary was printed in 1495, there were likely manuscript versions in circulation in Cracow. Brudzewo has a clear conception of astronomy as a science in part physical and in part mathematical; the commentary makes this view abundantly clear, even if it were not obvious from Peurbach’s work itself.
Near the start of his introduction, Brudzewo raises two questions. First, he asks what moved the wise (sapientes) to posit several celestial orbs. He replies that the movement of the fixed stars together in an unchanging configuration led them to posit that there is a single sphere to which the fixed stars are attached. Then they noticed that there were certain stars (the planets) that changed positions relative to the fixed stars, so they posited separate spheres for each of them.103 Second, he asks how many mobile orbs there are. In reply to the second question, he cites Averroes’s questioning of the existence of any sphere not containing a visible star and then cites the theoricae, which posit three or more orbs for an individual planet. He is not going to discuss every opinion on the number of orbs, he says, but only the more probable theories. He describes three different senses in which the word “orb” is used – taking this from Albert of Saxony or some other previous author – and then lists the suppositions or principles discussed above.104
If one looks in the commentaries of Averroes and Albert of Saxony in connection with Brudzewo’s reference to Averroes or with his silent borrowing from Albert of Saxony, two features of these sources emerge. First of all, Averroes repeatedly analyzes the logical structure of Aristotle’s arguments and describes many and various astronomical opinions. On the basis of Averroes’s report of Aristotle’s views, one easily gets the impression that cosmology and astronomy are works in progress. With regard to the number of orbs, for instance, Averroes writes, “What [Aristotle] says, that the Eighth Orb is near the first orb, we find written thus. The opinion of the ancients is that the Eighth Orb, that is the starry orb, is the first orb. Ptolemy, however, posited a ninth, because he said that he himself found in the fixed stars a slow motion in the order of the signs [i.e., west to east], and the modern Arabs say that this motion is a motion of access and recess, and that it is not perfected, and they say that this motion was spoken of by the ancient Babylonians.”105 The fundamental motions of the celestial orbs are uniform, so if planets appear to move with changing velocities, Averroes writes, it must be because they are moved by several uniform motions, which combine in varying ways to produce the observed effect.106 Here, Averroes says, Aristotle does not have in mind epicycles and eccentrics, but Averroes himself is motivated on this point to remark that there is no way to demonstrate whether a given irregular motion is caused by an eccentric or by an epicycle. Just because, if an epicycle is assumed, the observed positions can be calculated or just because, if an eccentric is assumed, the observed positions can be calculated, does not mean that from the agreement between the observations and the assumptions, one can infer that there is an epicycle or alternatively that there is an eccentric. Some people later argue that you might infer that there is one or the other but not which one.107
In the conclusion of his introduction, Brudzewo provides an alternate explanation of the relations of the three-dimensional orbs of the Theoricae novae planetarum to the two-dimensional eccentrics and epicycles of standard Ptolemaic astronomy.108 The major two-dimensional figures are simply the projections of the three-dimensional orbs or, alternately, what results if the three-dimensional figures are collapsed into two dimensions. More than one projection of the three-dimensional figures is possible. If the celestial sphere is collapsed along its north-south axis, the celestial equator will be the largest circle, and the Tropic of Cancer will be superimposed on the Tropic of Capricorn. According to another projection, however, the various circles are represented as they would appear from a particular point of view; the nearest tropic may appear larger because it is nearer to the viewer. Astronomers use plane figures to represent astronomical theories because the theories are much easier to understand if they are represented visually. But it is the three-dimensional, thick orbs that are physically real, whereas the two-dimensional eccentric circles and epicycles are mathematical fictions, ones that play an important role in aiding the human intellect to understand the theory.109 The figures in the printed editions of Peurbach’s Theoricae novae planetarum no doubt reinforced in the minds of his readers the distinction between physical orbs and fictional lines by colouring the unequally thick real orbs. Here, Brudzewo increases the potential impact of the Theoricae novae planetarum in this respect by emphasizing the artistic or artificial properties of the figures. Brudzewo further clarifies the conditions of the art, or professional work, of the astronomer – why, for instance, astronomers make use of mathematical devices that are not related to real bodies – by contrasting astronomy and natural philosophy. The latter presents a less detailed view and tries to explain why the stars and planets display the positions and motions that have been observed. Whereas the astronomer considers partial orbs, the philosopher considers only total orbs. Whereas astronomy measures motions by the angles passed through by the planets, philosophy measures motions by the distances traversed. Thus, whereas the Moon may move through angles more quickly in the astronomers’ view, being nearer, it traverses shorter distances in the philosophers’ view.110 When Averroes objects to Ptolemaic astronomy, Brudzewo says, he is writing as a natural philosopher, not an astronomer, and he cares only about total orbs.111
After his introduction, Brudzewo begins his comments on individual passages of Peurbach, marking those passages by the opening words. The commentary is too detailed to be summarized here, but there are some notable points worth reporting. For instance, in agreement with Aristotle’s doctrine that the practitioners of a given scientific discipline do not doubt the principles of their science, Brudzewo argues that uniform circular rotation of the celestial bodies is a basic principle of astronomy, not to be disputed by astronomers. Thus he writes, “And if it is a first principle of astronomy that the Sun moves regularly in its eccentric (and therefore there is no further dispute in astronomy with someone denying it), nevertheless such a principle can be demonstrated by a subalternating science, namely mathematically.”112 Contrary to what might be thought at first, Brudzewo here has in mind an a posteriori, or quia, demonstration, based on observed positions of the Sun at different times: “The Sun in equal times describes equal angles upon its center and cuts off equal arcs. Therefore it is moved uniformly [aequaliter]. The consequence holds by the definition of ‘to be moved uniformly,’ which is taken from Book VI of the Physics. But the antecedent is clear in this figure.”113 In a related use of mathematics, astronomy can determine the eccentricity of the Sun’s deferent by determining how much longer it takes the Sun to move from the spring equinox to the fall equinox than to move from the fall equinox to the spring equinox. Thus mathematics provides a method of reasoning quia from observed positions of the stars and planets over time to suppositions or principles of astronomy – as astronomers infer from observation that the stars are on a rotating sphere. This process establishes that something is the case.
A difficulty might arise, however, when more than one theory, such as an eccentric or an epicycle, produces results that fit the appearances equally well. This difficulty had been recognized from the time of Apollonius of Perga (ca. 262–190 BCE). Concerning alternative theories of the motion of the Sun, Ptolemy had said in the Almagest that there is no way to demonstrate that one of the two alternative theories is the correct one, eccentric or epicycle, so he is limited to probable rather than demonstrative arguments. However, it would seem more reasonable to associate it with the eccentric hypothesis since that is simpler and is performed by means of one motion instead of two.114 Later, Naṣīr al-Dīn alṬūsī made this same characterization of Ptolemy’s situation.115 Thus in this conception of Ptolemaic astronomy, many of the principles come from observation, memory, and experience, but there may be no unique set of principles corresponding to experience. The same observations could be explained by an eccentric and by a corresponding concentric deferent together with an epicycle.
Like Ptolemy, Peurbach in the Theoricae novae planetarum chose to use an eccentric to explain the motion of the Sun. Unlike Ptolemy, but following Arabic predecessors in the configuration or hayʾa tradition, Peurbach embedded the Sun’s eccentric within a thick concentric shell (see figure 6.2). For Venus, Mars, Jupiter, and Saturn, Peurbach proposed that within the eccentric there was embedded a solid epicycle containing the planet. Although, mathematically speaking, some motions produced by a concentric deferent and epicycle could equally well be produced by an eccentric, physical factors might favour one configuration over another. If Peurbach never raises the question of any possible interference of small circles near the centre, this is no doubt because they are not real. Where he has such a small circle for Mercury, he provides a second set of unequally thick nested orbs that cause the imagined centre of the deferent to trace the small (mathematical or imaginary) circle to which no body corresponds.116
In addition, there were some aspects of the planets’ motions that had not been explained using uniformly rotating concentrics, eccentrics, and epicycles. Like previous astronomers, Peurbach tracked these other motions mathematically using geometric lines or circles rather than physical orbs (they could also be tracked arithmetically using tables). In his commentary on Peurbach, Brudzewo carefully notes where lines rather than physical orbs are being described. In particular, Brudzewo does not consider the equant to be a real physical thing in the celestial realm because it does not correspond to any aether sphere. So he inserts a note that does not correspond to any particular text in Peurbach:
It should be noted that there is no work [opus] for the equant insofar as the motion in itself of the orbs is concerned. For the equant does nothing with regard to the regular motion of the orb since it is an imaginary circle. But it has a function with regard to the astronomical work and the calculation of tables. Tables are calculated using mathematical principles and conclusions, which conclusions indeed often cannot be accommodated and applied to the motions either in themselves or as they appear to us. Therefore, these mathematicians sometimes take the motions of celestial bodies other than they are in their nature or other than they appear to us, and they consider them in a way that serves their art and their operation, since it is certain that otherwise there is no way that they can arrive at their intended work correctly and precisely. Therefore, they imagine that the motion is uniform, which does not appear uniform in itself, on account of their work, that it be done more correctly. And from this they are convinced and compelled to posit imaginary equant circles, on which they consider the diverse and unequal motions of the orbs to be equal, and they reduce those diverse motions first to uniformity, as to that on the basis of which the irregular motions should be taken. “The regular is the judge of itself and of the irregular” [as Aristotle says in book 1 of De anima].117
Thus Brudzewo repeatedly makes clear the distinction in scientific status between physical orbs and mathematical or imaginary circles.
To repeat: how does the use of equants fit into the proper role of an astronomer? Since it is a goal of practical astronomy (e.g., astrology) to support prognostications concerning the effects of the heavenly bodies on Earth, it should be understood that astronomers have adopted mathematical methods that go beyond the limits of their confidence in theoretical physical astronomy in order to make those predictions. Thus Peurbach and Brudzewo posit equant points in order to make the tables and predictions that are part of their task as practical astronomers. They do not think that equant points – or the circles that may be drawn around them – correspond to real physical bodies. Nevertheless, they posit and use them because they see no other way to carry out their craft. What Brudzewo says here is related to what Copernicus wrote in his Letter against Werner, saying that astronomers have settled on a chosen theory, “lest the inquiry into the motions of heaven remain interminable.”118 Astronomers are confident that their mathematical descriptions are correct in connecting the observational data in a systematic way, even though they have not yet been able to devise a system of real celestial orbs embodying that motion because the results of their mathematical calculations agree with what is observed.
Thus, in Brudzewo’s interpretation of Peurbach, we are left with an astronomy partly grounded in what are supposed to be real, uniformly rotating aether spheres in addition to which astronomers imagine other mathematical devices, which are not based on physical bodies but are necessary to make the predictions called for. The physical orbs provide a pretty good theory, so they are taught, although the theory has not been perfected. As far as the real physical orbs in the heavens are concerned, it should be possible to explain the relation of rotations to their causes, including intelligences or final causes. If the heavenly bodies are the causes of being and life of the terrestrial bodies, how, Brudzewo asks, can one explain the existence of empty spheres, which presumably would have no terrestrial influence? And why do seven orbs have one body each, whereas the eighth orb contains all the fixed stars? These questions and their answers are taken, nearly verbatim, from the paraphrase of Aristotle’s Metaphysics by Albertus Magnus. The questions are answered in terms of the intended effects of the stars and planets on Earth.119
But there is another aspect of the work of the theoretical astronomer, namely to fine-tune the celestial theories in accordance with the observed positions of the stars and planets. If there remain open questions about astronomical theory, new data about celestial positions may help to decide between alternatives. In this connection, Brudzewo has more to say than Peurbach about the importance of accurate observations, especially as aided by instruments, and he notes that the measurements of earlier astronomers may have been inaccurate.120 In the course of his commentary, Brudzewo reports differences of opinion between past astronomers, such as with regard to the apparent sizes of the Sun and Moon.121 To determine how far a star or planet has moved, it is necessary to have a fixed point of reference, he says, and if there is trepidation of the equinoxes, the point of intersection of the celestial equator and the ecliptic may itself be uncertain.122 So another method of measurement is to use the angles between a planet and one or more fixed stars.123
When Ibn al-Haytham’s On the Configuration of the World was translated into Latin, Europeans recognized his embedding of eccentrics and epicycles within concentric total orbs as showing how eccentrics and epicycles could be reconciled with real, nonoverlapping physical orbs. On the Configuration of the World became known in Europe in multiple ways. A Latin translation (or reworking) found in Oxford was “part of the corpus of Arabic scientific writings that were made available in Latin under the patronage of Alfonso X (the Wise) of Castile (1221–1284).”124 Translations of the work into Hebrew were even more numerous.125 By whatever route, the idea that Ptolemy’s mathematics could and should be matched with thick physical orbs rotating uniformly spread widely in Europe beginning in the later thirteenth century. However the ideas spread, works called theoricae planetarum evolved from describing the motions of planets using mathematical lines to describing them, at least in part, using physical orbs.126
Thus from the late thirteenth century, a growing number of Western astronomers working in the theorica planetarum or configuration tradition explained as much as they could using solid orbs or aether shells but then presented other theoricae or figures containing lines and motions that did not correspond to bodies, these other theoricae of lines, motions, axes, and so on being necessary to ground the calculations of positions required in astronomical or astrological practice.127 In his Theorica planetarum, Campanus of Novara begins to move from lines to physical orbs insofar as he describes how to draw three-dimensional orbs extending from the nearest possible approach of the planet to the Earth to the greatest possible distance away.128 In his Scriptum super canones Azarchelis de tabulis Toletanis (ca. 1290), John of Sicily notes that words like “eccentric” are ambiguous, sometimes referring to physical orbs and other times to mathematical lines, such as the path of the centre of a planet. Whereas he takes the three-part orbs descended from On the Configuration of the World to be real, he says that the equant is an imaginary circle (circulus imaginarius).129
From the later thirteenth century through the fourteenth and fifteenth centuries, the embodiment of as much as possible of Ptolemy’s mathematical astronomy within physical orbs was accepted by European astronomers, but for purposes of astronomical prediction, they continued to use astronomical tables based on Ptolemaic mathematics that lacked a physical counterpart. In fourteenth-century Aristotelian commentaries, a typical treatment of epicycles first described Ptolemy’s use of eccentrics and epicycles, then noted Averroes’s rejection of them on the grounds that they would require either vacua or more than one body in the same place, and finally went on to say that this objection can be overcome by embedding epicycles and eccentrics within the depth of concentric shells such that any given orb remains within the same cavity in the body surrounding it, always rotating into itself. This, for instance, is what John of Jandun writes in his commentary on the Metaphysics.130 John Marsilius Inguen (pseudo Marsilius of Inghen) makes a similar remark in his commentary on the Physics to refute the view that if there are epicycles, there must be vacua or fluids in the heavens.131
In this chapter, I have not tried to provide detail of possible routes of transmission between Ibn al-Haytham’s On the Configuration of the World in the late tenth or early eleventh centuries, the hayʾa tradition of Islamic astronomy, and the appearance of similar configurations in Europe at the end of the thirteenth century; there were many routes, of which John of Sicily’s Scriptum super canones is only one. Works on configuration or hayʾa continued to be produced in Islamic areas, and I hope that historians of Arabic astronomy can to a great degree trace the history of the science of configuration forward.132 On the European receiving end, it is clear that European astronomers took it for granted that they had many Islamic predecessors. Albert of Brudzewo cites by name Alfraganus (al-Farghānī), Albategnius (al-Battānī), Averroes (Ibn Rushd), Geber (Jābir ibn Aflaḥ), Haly (ʿAlī ibn Riḍwān), and Thābit ibn Qurra.133
If I have shown that Copernicus wrote his Commentariolus within the conception of astronomy as a science exemplified by Peurbach’s Theoricae novae planetarum and, moreover, that this conception descended from Ibn al-Haytham’s On the Configuration of the World, the transition from Theoricae novae planetarum to Copernicus’s Commentariolus can be more correctly understood. In several papers in recent years, Peter Barker and Michela Malpangotto have described in detail the place of orbs in Peurbach’s Theoricae novae planetarum and in Brudzewo’s commentary on that work. Barker has argued that what was new in Peurbach’s Theoricae novae planetarum may have stimulated Copernicus’s innovations in astronomy, and Malpangotto has given even more credit to Brudzewo, whom she sees as critical of Peurbach’s failure to replace Ptolemy’s equant and mean motion of the apogee of the epicycle with uniformly rotating orbs or circles.134 I argue that both Peurbach and Brudzewo were carrying on a research program initiated by Ibn al-Haytham and carried on by Islamic hayʾa astronomy. All three – Ibn al-Haytham, Peurbach, and Brudzewo – describe complementary figures or theoricae. Some of these figures portray physical orbs, but others represent Ptolemaic mathematical astronomy not matched with bodies because so far no one had succeeded in matching those parts of mathematical astronomy to physical bodies. Before Copernicus, Regiomontanus had tried to advance the program of finding homocentric astronomical configurations, but he had failed to complete this work.135 Much earlier, in the twelfth century, Averroes had hoped to reconcile Ptolemaic mathematics with bodies but came up short. Thus Copernicus worked within a context in which astronomers were hoping for reform of the physical side of astronomy, a context having its roots in Islamic astronomy. Contrary to Malpangotto, I think that Peurbach and Brudzewo both accept the idea that there are some physical orbs uniformly rotating and other, purely mathematical methods that do not correspond to bodies. Brudzewo is not disappointed with Peurbach but is elucidating positions with which Peurbach would have agreed.
Like Regiomontanus, who took an interest in homocentric theories in astronomy as well as in theories of eccentrics and epicycles, Copernicus could see himself as within the current program of astronomy in proposing an alternate and more satisfactory theorica of the planets. Copernicus was not the first to try a new configuration, but by the time he published De revolutionibus orbium coelestium, he was able to make clear in mathematical detail that a configuration of the world with the planets rotating around the Sun could fit appearances as well as the Ptolemaic Earth-centred system.