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THE RENAISSANCE

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Author Topic: THE RENAISSANCE  (Read 4147 times)
Bianca
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« Reply #90 on: October 17, 2008, 08:20:12 pm »










Traditionally, Regiomontanus's Padua oration has been seen through the lens of its rather obvious humanism. In particular, the Padua oration has come to be understood as the rhetorical embodiment of the fifteenth-and-sixteenth-century revival of ancient (Greek) mathematics. Just as this revival is inextricable from the rise of humanism, so Regiomontanus has come to be seen as an exemplar of humanist mathematics.3 It is the aim of this paper to examine the Padua oration in the context both of contemporary humanist rhetoric and of Regiomontanus's own intellectual background in order to argue that, while the oration is stylistically consistent with humanist norms, the vision of mathematics presented in it is also deeply grounded in the university mathematical curriculum and in Regiomontanus's own reading of mathematical texts.

Regiomontanus was educated primarily at the University of Vienna (he was also briefly at the University of Leipzig), where he enrolled in 1450, completed his baccalaureate in 1452 and became a master in 1457.4 He remained at Vienna until 1461, when the death of his friend and teacher Georg Peurbach prompted him to travel to Italy with his patron, Cardinal Bessarion.5 In Regiomontanus's day Vienna was probably the most important of the German universities, rivaled only by Prague, whose prestige had declined after it was stripped of its theology faculty in the aftermath of the Hussite Wars.6 The curriculum at the University of Vienna was modeled [End Page 42] after that of Paris, and Parisian scholars played a major role both in its founding in 1365 and in its re-establishment (this time with a theology faculty) in 1384.7 
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Bianca
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« Reply #91 on: October 17, 2008, 08:21:31 pm »










Most importantly for the purposes of this paper, the Viennese mathematical curriculum of Regiomontanus's day included all of the traditional authorities taught at Paris in the fourteenth century. For arithmetic and algebra, various "algorisms" (prose or poetry instructions for carrying out arithmetical operations that often also included a small amount of number theory) were the basic texts, supplemented in the fourteenth century by the Quadripartitum numerorum of Jean de Murs. Jordanus de Nemorare's De numeris datis and al-Khwarizmi's Algebra were common sources for those engaged in more advanced studies (i.e., they were not normally the subject of ordinary lectures, but were readily available to interested students, and would perhaps have been the subject of occasional extraordinary lectures). Euclid's Elements, supplemented by the commentaries of Pappus and Campanus, was the central text for geometry, and a number of medieval texts on practical and speculative geometry were in circulation as well. For astronomy, the Sphere of Sacrobosco and the Theorica planetarum were the most commonly used teaching texts, sometimes supplemented by al-Farghani's Elements of Astronomy (the subject of Regiomontanus's Padua lectures). Advanced students could read numerous more specific treatises by Arabic and Latin authors. For optics, the Perspectiva communis of John Peckham was the most common basic text, with texts by Witelo and Roger Bacon (both generally known as Perspectiva) being the most common advanced texts. Alhazen's Perspectiva (also known as De aspectibus) was likewise available, as were the optical writings of Euclid and Ptolemy.8 [End Page 43]

Most of the above texts were the subjects of lengthy commentaries, which often circulated independently. In addition, numerous encyclopedic texts played an important role in university culture and, while not specifically about mathematics, included information on its history or place in the division of knowledge (one of these, Isidore of Seville's Etymologies, will be discussed in more detail below). It should be noted that mathematics at Vienna, as at Paris, was never central to the curriculum; logic and philosophy were more important. Nevertheless, all students who took arts degrees at Vienna were expected to acquire basic competence in mathematics, and more advanced instruction was certainly available for those who desired it. A succession of mathematically inclined masters, most notably including Johannes von Gmunden and Regiomontanus's own teacher Georg Peurbach, taught at the university in the first half of the fifteenth century.   
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Bianca
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« Reply #92 on: October 17, 2008, 08:26:07 pm »








Humanism, too, was present at the University of Vienna in Regiomontanus's day. Ordinary lectures on humanist subjects began to appear in the arts curriculum at Vienna in 1451, shortly after Regiomontanus's arrival, when Phillip Mautther lectured on the Nova rhetorica (that is, the Rhetorica ad Herrenium) of Cicero.9 For the next decade, humanist topics regularly appear on the lists of ordinary lectures. Even before this, there had been a humanist presence in Vienna. The great Italian humanist Aeneas Silvius Piccolomini (later pope Pius II) was a member of the Habsburg court in Vienna during the 1440s, and in 1445 even took part in disputations at the university, speaking on, among other topics, the question of why so few poets existed in the present day.10 Regiomontanus's mentor Peurbach also held humanist sensibilities, lecturing on Virgil and Juvenal and writing classical-inspired poetry.11

Regiomontanus's background at Vienna, and as university-trained mathematician in general, provides the context for his vision of the mathematical arts. He was not among those humanists who denigrated their university educations. There is no reflection of Petrarch's famous sentiment regarding the University of Bologna, "I never did anything prudently . . . but, if I did, this is among the first things which, if not wise, has certainly been fruitful, namely, that I saw Bologna and that I did not remain there," [End Page 44] in Regiomontanus's attitude towards his own education.12 When he mentions Vienna in the oration, it is to speak highly of it. "You, Vienna, created the most worthy doctor of the arts," he exclaims in the oration (he is speaking here not of himself but rather of his beloved mentor Peurbach).13

It is no accident that the Padua oration has been strongly associated with humanism: it is the most immediately striking aspect of the text. The grammar and vocabulary of the oration both display classicizing tendencies, but the clearest sign of humanism is the use of elaborate periodicity that was so central to humanist stylistic ideals.14 For example, Regiomontanus opens his oration by declaring to his audience:
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Bianca
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« Reply #93 on: October 17, 2008, 08:27:42 pm »









Unless your clemency should favor the beginning of the course as I will make it, and would set straight

the thread of trembling oration, Oh most distinguished men, I would prefer to have remained silent than

to ascend this philosophical pulpit too boldly, particularly because I could easily be deterred by both

the novelty of this enterprise, especially in so great a gathering of celebrities, and by my long

abstinence from academic rituals.15
 
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« Reply #94 on: October 17, 2008, 08:28:45 pm »









The intricate relationship between the clauses and the setting up and resolving of suspense ("unless . . . I would rather . . . because") are both hallmarks of what Michael Baxandall has called
"one of [humanists'] most anti-popular interests."16

Regiomontanus was clearly aiming to impress his audience with his oratorical prowess.

Of course, the oration itself is a genre that is closely associated with humanism, and Regiomontanus's oration, praising the mathematical arts [End Page 45] and their practitioners, falls firmly into that most humanist genre of oration, epideictic.17 Indeed, it is a member of a specific sub-category of epideictic rhetoric, the oration in praise of the arts (or an art).

A number of such orations survive, and many of them were given on the same occasion on which Regiomontanus gives his: the inauguration of a course of lectures.18

The conventions that Regiomontanus follows in praising mathematicians are largely those followed by his contemporaries. For example, Lorenzo Valla in his encomium of St. Thomas Aquinas compares Thomas to St. Dominic:   
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« Reply #95 on: October 17, 2008, 08:29:46 pm »









Dominic, then, founded the house of the Preachers, Thomas covered its pavements with marble;

 Dominic raised its walls, Thomas adorned them with beautiful paintings; Dominic was the crowning ideal

of the Brothers, Thomas was their model. Dominic planted, Thomas watered. One shunned and rejected

the distinctions and the bishoprics which were offered to him, the other fled from nobility, wealth,

family, parents, as if they were the Sirens. . . .19
 
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Bianca
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« Reply #96 on: October 17, 2008, 08:31:21 pm »









Valla's rhetoric is particularly broad here because the oration is a parody of traditional epideictic. He begins by offering, in classical style, extravagant praise of Aquinas's sanctity, only to continue by denigrating, in equally classical style, the saint's ability as a theologian.20 Regiomontanus uses similar techniques, though of course without the parodic inversion, in comparing the great classical mathematicians Archimedes and Apollonius: [End Page 46]





Archimedes was a citizen of Syracuse, and Apollonius of Perga [was] customarily called divine

because of the height of his genius; I cannot easily say which should be preferred to the other.

For although Apollonius most subtly composed the conic elements in eight books that have not yet

appeared in Latin, nevertheless the variety of Archimedes's publications seem to have won him the

prize. . . . When Apollonius is translated by someone from Greek into Latin, all of you will come to

admire him.21
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« Reply #97 on: October 17, 2008, 08:32:34 pm »









The back-and-forth of "the one . . . the other" is a typical example of classical rhetorical techniques. Again, Regiomontanus's aim is both to educate his audience (Apollonius, in particular, is a figure who was unlikely to be appreciated by the average university student), and also to impress them with his proficiency.

Regiomontanus also makes use of the kind of visual language that John O'Malley has shown is central to humanist epideictic.22 He speaks of "the most visible splendor of mathematics,"23 or tells his listeners "you have the cause before your eyes. . . ."24 He does not take the use of epideictic as a kind of spoken painting so far as some of the preachers that O'Malley cites, but there is certainly an element of this in his rhetoric.

However, when one begins to delve into the actual content of Regiomontanus's history, it is apparent that he departs somewhat from humanist norms. Humanists were interested in the literature of antiquity not only for the sake of eloquent Latin, but also for the historical information contained therein, and Regiomontanus shows little indication of sharing this interest. This can first be seen in his stories of the origins of the quadrivial arts, which derive from, and have a great deal in common with, the standard medieval accounts given in Isidore of Seville's Etymologies. For example, this is Isidore's version of the origin of geometry:     
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« Reply #98 on: October 17, 2008, 08:40:47 pm »









The science of geometry is said to have been discovered first by the Egyptians, because when

the Nile overflowed and all their lands [End Page 47] were overspread with mud, its origin in the

division of land by lines and measurements gave the name to the art.

And later, being carried further by the keenness of the philosophers, it measured the spaces of

the sea, the heavens, and the air.25
 
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« Reply #99 on: October 17, 2008, 08:41:55 pm »









And, in comparison, this is what Regiomontanus tells his audience:






In those times when the River Nile, violently overflowing, erased the boundaries of the Egyptians'

fields and completely flooded everything, the farmers began to argue; each one desired to define

the fields according to his own estimate—as it is the nature of man to be fixated on that which is

more than enough—each one, whether by speech or by force, wanted more than the others; even

if he had a narrow field before [the flood], he set up [his borders] where he pleased; when this

affair went beyond what was equitable, the matter fell to the prince of that land, who by certain

just calculations and sure measurements restored the borders of each [field].

Thus, [the Egyptians] were turned by a certain general and extraordinary impulse of the human

mind towards making measurements; they began to pose questions to one another, and whatever

they considered well invented in this sort of exercise, however confused it was at that point, they

desired to commit to writing.26
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« Reply #100 on: October 17, 2008, 08:42:52 pm »










Regiomontanus's account is obviously much more elaborate than Isidore's, as befits his style of oration; he also adds a narrative to the process of the [End Page 48] discovery of geometry that is lacking in Isidore's account. However, what it has in common with Isidore, in addition to a general agreement on the facts of the story, is that it lacks the detailed sourcing that humanists tended to give their histories. There are no references to Cicero, Virgil, Strabo or other authorities to give the history the grounding in the world of classical texts that was so important to the humanist sensibility.

Regiomontanus does, however, go beyond Isidore and other readily available texts in compiling his account. The figure of the prince who restores the fields to their proper sizes, in particular, is not found in the standard medieval sources. Rather, it seems to be a modified version of Herodotus's story of the king Sesostris, who, when a portion of a subject's lands were destroyed by flooding, would measure the remainder and reduce his rent accordingly.27 Lorenzo Valla had recently completed his translation of Herodotus, so it is possible that Regiomontanus knew of it through Cardinal Bessarion, their mutual acquaintance.28 At any rate, Regiomontanus never actually mentions Herodotus, which in itself is a departure from humanist norms. In fact, the only classical authorities mentioned in any of Regiomontanus's histories are mathematicians. He mentions Plato and Cicero at the beginning of the oration, but as models of eloquence rather than as sources of historical knowledge.29   
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« Reply #101 on: October 17, 2008, 08:44:22 pm »










In contrast, this is the English humanist Polydore Vergil's version of the story, given in his
On Discovery:






In all the world the most famous river is the Nile. Each year from the summer solstice to the
autumnal equinox, as Herodotus shows in book 2 and Diodorus in book I, it floods all of Egypt
with an immense quantity of water.

In its rise the Egyptians foresee either abundance or scarcity for the crops to come. For when
the Nile rises 12 cubits, Egypt knows famine; with thirteen cubits she goes hungry; fourteen
cubits bring cheer, fifteen security and sixteen exultation.

This is said to be the regular level of the rise, but the [End Page 49] highest was eighteen cubits
in the time of the Emperor Claudius, and the lowest came with the battle of Pharsalus as if by
some prodigy to show horror at the killing of Pompey the Great, according to Pliny in book 5 and
Strabo in book 17.

Thus, since these Nile floods muddled the borders of the fields . . . it was necessary to measure
the land over and over again. For this reason, some have claimed that the Egyptians first disco-
vered geometry, just as the Phoenicians invented arithmetic, the science of numbers, for
commercial purposes, according to Strabo in book 17 of the Geography and Herodotus in book 2.30 
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« Reply #102 on: October 17, 2008, 08:45:27 pm »










This account generally agrees with those of Regiomontanus and Isidore—not surprising, as Isidore's ultimate source was Cassiodorus, who knew his classics. But, it contains a wealth of citations of classical authors. The references to Pliny, Strabo, and Herodotus signify to the reader that this is not just an origin story, but also a part of the classical corpus. Of course, On Discovery is not in the same genre as the Padua oration: it is a learned encyclopedia, meant to be read rather than heard, and will therefore display the author's erudition in a manner different from that of a spoken oration. However, the same phenomena can be seen in fifteenth-century orations on the arts. For example, Andrea Brenta, a Paduan who was professor of rhetoric and ancient languages at Rome, cites Livy and Thucydides in the brief section on mathematics in his oration in praise of the arts.31

Turning to astronomy, Regiomontanus's own most beloved discipline, here is what the Padua oration has to say about the origin of that art: [End Page 50]






Indeed, some claim to hold Abraham of the Hebrews, and Moses, as the father of astronomy.

Others attribute the theft of the divine fire, which carried the light of astronomy to mortal men,

to Prometheus.

They say that Hercules carried the heavens on his shoulders for Atlas, either because he learned

astronomy under Atlas or because he served as an overseer when Atlas was absent from his reign.32   
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« Reply #103 on: October 17, 2008, 08:46:23 pm »









Again, this account generally agrees in its details with that given in the Etymologies.33 Also note that Regiomontanus actually offers two different histories of mathematics, one rooted in pagan, the other in scriptural, antiquity, without giving any indication of which his audience ought to prefer. For sixteenth-century historians of mathematics, the issue of the Greek or Hebrew origins of astronomy would be a subject of heated discussion. Regiomontanus seems not to care; it is enough for him that astronomy is an ancient discipline.

In comparison, this is the history of astronomy given by the fifteenth-century Italian humanist Gregorio da Città di Castello, also known as Tiphernas, in his oration on astrology:   
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« Reply #104 on: October 17, 2008, 08:47:55 pm »









They say that the discovery of astronomy came from the Egyptians.

For because of the perpetual calmness of the air in that region, and because the Egyptian priests,
who desired learning, abounded in leisure and in those things necessary to life, as Aristotle says,
they applied themselves to observing and discovering many things that concern the science of the heavens . . . but after many thousand years, as certain men have recorded, [this knowledge] was carried to Greece, and was studied by many, as can be seen particularly from the poets.

Then followed Pythagoras and Pherecydes, who, being learned in mathematics, added many things
to this discipline.

After this came Erathosthenes, Berosus [Note that Berosus was not Greek, but Babylonian], Hipparchus, and many other men, in whom Greece abounded.

But Plato and [End Page 51] Eudoxus, when they went to Egypt in order to learn, brought many

secrets learned from Egyptian priests back to Greece, both those of other arts, and notably those
of astronomy, as Strabo says.

Thus, little by little, through the passing of time and continual increase, [the knowledge] came
down to the time of Ptolemy, by whom that wonderful, almost divine book of astronomy was
written, so that, in some ways, nothing seems to be able to be added to that science.34 
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