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THE RENAISSANCE

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Bianca
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« Reply #105 on: October 17, 2008, 08:48:59 pm »









Tiphernas grounds his account in references to classical tradition that are lacking in Regiomontanus's account: Aristotle, Strabo, and the "certain men," who are, in fact, Cicero, and Pliny.35

Of course, Tiphernas, though he gives an account of the Egyptian origins of astronomy, focuses primarily on classical figures, and ends with Ptolemy—after him, "nothing seems to be able to be
added to that science."

In contrast, Regiomontanus, after relating the story of astronomy's origins with either Abraham
and Moses or Prometheus, Atlas, and Hercules, goes on to say this:   
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« Reply #106 on: October 17, 2008, 08:50:22 pm »










It can be said without injustice that while Hipparchus of Rhodes was the first parent of this
discipline, Claudius Ptolemy of Alexandria was its author and chief.

For, before Hipparchus, very few men freely observed the motion of the stars, and no one had
yet thought that the fixed stars revolved with anything other than a simple, diurnal motion, which Hipparchus noticed, concluding [End Page 52] that the said stars move with their own very slow
motion towards the east.

Then Ptolemy, taking up the discoveries of the ancients . . . pronounced that motion to be one
degree in one hundred years, as can be seen in the third theorem of chapter 7 [of the Almagest;
the printed text of the oration actually cites the seventh theorem of chapter 3, which is incorrect
and likely the result of a misreading] . . . many other esteemed Greek professors of that art are being passed over in silence . . . moreover, how much value the Arabs placed on that art is shown by the writings of the most worthy al-Bategni, whom Plato of Tivoli translated into Latin. Likewise, a certain Gerard of Cremona translated Geber of Spain, whom Albertus Magnus did not fear to call the corrector of Ptolemy in the Speculum astronomiae. . . .36 
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« Reply #107 on: October 17, 2008, 08:51:37 pm »









Regiomontanus's account of Hellenistic philosophy is much more detailed than Tiphernas's, giving details about the theories of Hipparchus and Ptolemy gleaned from his reading of the Almagest, an Epitome of which he had only recently completed.37 Hipparchus's theories are discussed in chapter 7 of the Epitome (for which Regiomontanus was responsible), although interestingly, he is there referred to as "Abrachis," a garbled version of his name stemming from Gerard of Cremona's translation of the Almagest.38 Nevertheless, it is clear that Regiomontanus derives his history of Hellenistic astronomy from his reading of the Almagest itself.

Further, whereas Tiphernas ends his account with Ptolemy, Regiomontanus continues, mentioning medieval Arabic and Latin sources, and describing [End Page 53] how the Arabic sources were translated into Latin. As with his description of Hellenistic astronomy, his treatment of Arabic and Latin shows a deep familiarity with the texts themselves. He portrays Arabic and medieval Latin sources in a positive light, implying that they can supplement and even supersede (Geber is the "corrector of Ptolemy") the body of knowledge left behind by classical authorities. Humanist hostility towards canonical medieval texts can be overstated, but it is nevertheless the case that there was a strong tendency to compare medieval authors unfavorably to their antique predecessors. Valla's encomium on Aquinas, which assigns the great Dominican theologian to the lowly place of cymbalist in the orchestra of theologians ("the first pair, Basil and Ambrose, [are] playing on the lyre; the second Nazianzen and Jerome, playing on the zither . . . the fifth, the Damascene and Thomas, playing on the cymbals") is a good example, as is Petrarch's marshalling of classical sources in opposition to St. Bernard in his invective Against a Detractor of Italy.39   
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« Reply #108 on: October 17, 2008, 08:53:15 pm »








More important than the mere inclusion of favorable references to medieval sources, however, is the concept of the history of astronomy that is apparent in the passage.

For Regiomontanus, there is a break in the history of astronomy, but it occurs not between classical and medieval authors, but rather between the origins of astronomy and what might be considered the actual discipline of astronomy.

Ptolemy is the "author and chief" of astronomy because it is his texts that survive and form the basis of the astronomical tradition that persists in Regiomontanus's own time. This attitude can be seen most clearly in Regiomontanus's account of the history of arithmetic:   
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« Reply #109 on: October 17, 2008, 08:54:28 pm »







Although, through his skill and numbers, Pythagoras attained immortality among future generations, both because he submitted himself to wandering Egyptian and Arab teachers, who were greatly
skilled in that study, then because he tried to probe all the secrets of nature by the certain
connection of numbers, nevertheless, Euclid made a much more worthy foundation of numbers in
three of his books, the seventh, eighth and ninth, whence Jordanus gathered the ten books of elements of numbers and from this produced his three most beautiful books on given numbers.

Diophantus, however, produced thirteen most subtle books (which no one has [End Page 54] yet translated from Greek into Latin), in which lie the very flower of all arithmetic, namely the art of assessing and accounting, which today is called algebra after its Arabic name.

Indeed, Latin authors treat many fragments of that most beautiful art, but after Giovanni Bianchini,
an excellent man, I find a scarcity of greatly learned men in our own time.

In our time, the Quadripartitum numerorum is certainly thought to be quite distinguished, likewise
the Algorithmus demonstratus and the Arithmetic of Boethius, the introduction [of which] was
taken from the Greek Nichomacus.40 
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« Reply #110 on: October 17, 2008, 08:58:50 pm »









Again, it is Euclid, rather than Pythagoras, who "made a much more worthy foundation" of arithmetic because Euclid's writings actually survive, and thus form a part of the arithmetical tradition that has, in Regiomontanus's view, existed continuously until his own time (Giovanni Bianchini was a correspondent of Regiomontanus's). Boethius, Jordanus, and Jean de Murs (author of the Quadripartitum numerorum) all deserve mention in a history of arithmetic because their writings were widely read. The same pattern is apparent in Regiomontanus's history of optics, where he cites Greek works by Euclid and Archimedes, the Arabic author Alhazen, and Latin texts by Witelo and Roger Bacon, all of whom were important authorities in Regiomontanus's time.41 Diophantus's Arithmetic, which was not widely read (this was the first public mention of the text, and it would not be translated into Latin until the next century) is an interesting case, because it seems to be an example of Regiomontanus trying to establish the same kind of pattern for algebra, which until then had been considered an Arabic art.42 [End Page 55]

Paul Lawrence Rose has noted the great emphasis that Regiomontanus places on the translation and transmission of mathematical knowledge, which contribute to a vision of mathematics as existing in a continuous tradition stretching back to antiquity.43 This is true, but it should be noted that given the distinction that Regiomontanus makes between the origins of the mathematical arts and their true founders—men like Euclid and Ptolemy, whose works still survive and are used—the real continuity in the mathematical disciplines is between the earliest mathematical texts and the mathematics of the fifteenth century.   
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« Reply #111 on: October 17, 2008, 08:59:43 pm »








Regiomontanus's history of mathematics is one that is founded in the practice of university mathematicians. Other than Diophantus (who, again, had only recently been discovered), the figures that Regiomontanus cites in his post-origin accounts of the mathematical disciplines were all known to contemporary mathematicians. Many of them were central to the traditional university mathematical curriculum and therefore to Regiomontanus's own education. He praises works like Euclid's Elements, Jordanus's De numeris datis, Jean de Murs's Quadripartitum numerorum, and Witelo's Perspectiva that, as mentioned above, had been widely used since the thirteenth and fourteenth centuries. He also mentions his contemporaries, men whose works had not yet had time to circulate extremely widely, but whom he saw as having made particularly important contributions to the mathematical arts, and also the great patrons of mathematics, men like Bessarion (his own patron) and Nicholas of Cusa. The history of mathematics, for Regiomontanus, is necessarily bound up with those figures who are central to the education and practice of his fellow mathematicians.

However, while Regiomontanus's vision of mathematics is grounded in university practice, it goes beyond it in scope. The authorities whom he praises most highly, Archimedes, Apollonius, and Ptolemy were known to university mathematicians but very rarely used. Latin translations of the works of Archimedes and of Ptolemy's Almagest were both available, and Archimedes, at least, was occasionally used.44 Apollonius had not been translated, although Witelo's Perspectiva shows at least a passing acquaintance [End Page 56] with his Conics.45 Regiomontanus, following the work of his mentor Peurbach, had recently completed an Epitome of the Almagest, but the scarcity of detailed knowledge of the Almagest even in Peurbach's Vienna can be seen in his Theoricae novae planetarum, written about a decade before he began work on the Epitome, in which Peurbach seems unaware, for example, of Ptolemy's solution for finding stationary points.46 By emphasizing the importance of Archimedes, Apollonius, and Ptolemy in addition to the traditional medieval sources, Regiomontanus advocated an augmentation of the standard practice of mathematics. The contemporaries for whom he had the highest praise, Peurbach and Bianchini, were men whom he knew shared his interests; Peurbach, of course, was his collaborator on the Epitome, and in 1463, Regiomontanus had begun corresponding with Bianchini over a variety of astronomical questions.47   
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« Reply #112 on: October 17, 2008, 09:01:45 pm »









Regiomontanus's comments on the utility of mathematics provide more insight into his understanding of the discipline. He begins conventionally enough, listing, by way of claiming that he neglects to mention them, those practical pursuits to which mathematics is important, including architecture, commerce, and military matters.48 Emphasis on practical utility was generally characteristic of those Italian humanists who considered mathematics in a positive light. Andrea Brenta, for example, focuses on Archimedes's ability to delay the overrunning of Syracuse by Marcus Marcellus's troops (a story told by Valerius Maximus).49 Aeneas Silvius Piccolomini, in his The Education of Boys, also relates that story, along with accounts of generals whose knowledge of eclipses allowed them to soothe their soldiers' terror during those unsettling events.50

Regiomontanus, however, quickly moves on to a more specific kind of utility: understanding the Aristotelian corpus. For example, "I think there is no one who is able to learn the seventh [book] of the Physics without [End Page 57] understanding proportions."51 According to Regiomontanus, important sections of De caelo et mundo, the Meteora, the Physics, and the Metaphysics all require that the reader be fluent with mathematics. The idea that a grounding in mathematics was necessary for the study of philosophy was one propounded by a number of Byzantine educators, particularly John Argyropoulos, who probably taught mathematics in conjunction with the works of Aristotle.52 In a letter to his son, Argyropoulos's pupil Alamanno Rinucinni explained the importance of mathematical studies:
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« Reply #113 on: October 17, 2008, 09:02:46 pm »









When the listener arrives at natural philosophy, he will taste the first elements of astronomy
 
and geometry. . . . However, after understanding the principles of those disciplines, he will

easily understand what is said by Aristotle. And so, he will be considered to have learned

enough of those disciplines that pertain to philosophy if he has studied that brief little work

on the sphere [i.e, Sacrobosco] and the Theorica planetarum, in which the elements of

astronomy are contained, and in geometry, the first book of Euclid.53 
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« Reply #114 on: October 17, 2008, 09:03:57 pm »









Note, however, that Alamanno is here recommending a much lower level of mathematical learning than Regiomontanus advocates; the Sphere, the Theorica planetarum, and the first book of the Elements were among the most basic mathematical texts taught at the university level. Alamanno is interested in providing the necessary background for philosophical studies, not in exploring the glory of Ptolemaic astronomy or Archimedean geometry.

Claims about the utility of mathematics, whether for civic or academic purposes, still place it in an ancillary position. Mathematics is praiseworthy because it is necessary for other pursuits. This, to be sure, is praise, but not of the highest order, and Regiomontanus follows it with a much more [End Page 58] forceful argument for not just the utility, but rather the supremacy of mathematics.   
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« Reply #115 on: October 17, 2008, 09:05:06 pm »









How many different branches have grown from the trunk of that [Aristotelian] sect? Some follow Scotus; others St. Thomas; a few, out of innate promiscuity, follow both. . . . However many
leaders philosophy has, by that much less is our time learned.

Meanwhile, the prince of philosophers is completely abandoned, and he who is better than others
in sophismata usurps his name, so that if Aristotle himself were revived, he would not, I believe,
even understand his followers and disciples.

No one, unless insane, would dare speak these things of our own discipline [i.e., mathematics],
since indeed neither time nor the ways of men are able to detract from them.

The theorems of Euclid are just as certain today as they were a thousand years ago and the discoveries of Archimedes will be no less admired after a thousand centuries [than they are now].54 
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« Reply #116 on: October 17, 2008, 09:06:03 pm »









For Regiomontanus, it is the certainty offered by mathematics that demands it be ranked highly among the disciplines. This was not a unique argument, as certainty was occasionally invoked by Regiomontanus's humanist contemporaries as one of the beneficial aspects of mathematics. For example, Pier Paolo Vergerio, who taught at Padua, mentions that "knowledge of [geometry] is most pleasant and contains within itself a high degree of certainty."55 Vergerio, however, is primarily focused on the pleasure offered by mathematical pursuits, particularly astronomy:





as we gaze upwards, it is pleasant to pick out the constellations of the fixed stars . . .

there is nothing that is not pleasant to understand, [End Page 59] but it is especially

pleasant to concern ourselves with those things which cause sensible effects in the air

and round about the earth.56
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« Reply #117 on: October 17, 2008, 09:07:28 pm »









Vittorino da Feltre, another Paduan, also accorded mathematics a great deal of importance, so there was at least a tradition of respect for mathematics in Paduan humanism that Regiomontanus could build upon.57

However, Regiomontanus's ideas about the certainty of mathematics cut against the grain of many of his humanist contemporaries. Hanna Gray, in her seminal article "Renaissance Humanism: The Pursuit of Eloquence," argues that humanists contrasted the scholastic pursuit of universal truth to their own emphasis on the virtuous life.58 Gray quotes Petrarch's invective On his own Ignorance and that of Many Others: "the object of the will is to be good; that of the intellect is truth. It is better to will the good than to know the truth."59 A more proximate example can be found in Piccolomini, who notes that "although arts of this sort [i.e., geometry and logic] engage in investigation of truth, it is contrary to duty to be drawn away from attending to our affairs by studying them, since all the glory of virtue, as [Cicero] says, consists in action."60 Regiomontanus, on the other hand, opposes the search for truth through philosophy with the even greater certainty offered by mathematics. Mathematics is superior to scholastic philosophy not because it leads its students towards the good but because it offers real certainty, not continually contested opinions. Astrology, in Regiomontanus's view the greatest of the mathematical arts, is praised even more highly. "You [astrology] are without doubt the most faithful messenger of the immortal God, you who provide the rule for interpreting his secrets, by whose grace the omnipotent decided to regulate the heavens, in [End Page 60] which he everywhere placed starry fires, signs of future events."61 Astrology is preeminent among the arts because of the knowledge it offers: insights into the secrets of God himself. 
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« Reply #118 on: October 17, 2008, 09:08:30 pm »









Like his conception of the history of mathematics, Regiomontanus's vision of its utility is linked to mathematical practice. The outward sign of philosophy's lack of certainty is manifest in the state of contemporary scholarship, namely the endless warfare between the followers of various authorities. Mathematics, on the other hand, suffers from no such divisions. The truths offered by mathematics make the idea of rival camps of "Euclideans" and "Archimedians," as philosophy has its Thomists and Scotists, absurd. Both men's theorems have stood the test of time and, Regiomontanus assures his audience, will continue to do so for "a thousand centuries." Likewise, the implication of juxtaposing a revived Aristotle's presumed inability to understand the work of his followers with the continuing certainty of Euclid's theorems is clearly that Euclid, were he to rise from the grave, would have no trouble understanding his disciples: Regiomontanus and his fellow mathematicians, who continue to build on the certain foundations that he established. The practice of mathematicians, like that of philosophers, relies on the use of authorities, but for mathematicians, this reliance is progressive and cumulative. Geber of Spain can be the "corrector of Ptolemy" and Jordanus can use the number theory in Euclid to produce his own De numeris datis without causing rifts among mathematicians. Certainty is the tie that binds ancient, medieval, and contemporary mathematics together.

In the end, Regiomontanus's vision of mathematics is that of a mathematician, rather than that of a historian, an educator, or a philosopher. It is simultaneously humanist and deeply rooted in the traditional university curriculum because a mathematician can (and for Regiomontanus, probably should) be both of those things. Above all, it is rooted in mathematical texts, both curricular and extra-curricular.   
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« Reply #119 on: October 17, 2008, 09:12:28 pm »









Footnotes



An earlier version of this paper was delivered at the 2004 History of Science Society meeting in Austin, Texas. Thanks to Robert Goulding, Lauren Kassell, Nicholas Popper, and Anthony Grafton for their comments at the HSS meeting.

1. Regiomontanus, "Oratio Iohannis de Monteregio, habita in Patavii in praelectione Alfragani," in Opera collectanea, ed. Felix Schmeidler (O. Zeller: Osnabrük, 1972), 43–53. Further citations of the Padua oration refer to this edition.

2. "Memorare possem in primis originem nostrarum artium, et apud quas gentes primum coli coeperint, quo pacto ex linguis peregrinis variis ad Latinos tandem pervenerint, qui in hisce disciplines apud maiores nostros claraverunt, et quibus nostra tempestate mortalibus palma tribuitur." Regiomontanus, Padua oration, 43.

3. See, for example, Paul Lawrence Rose, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo (Geneva: Droz, 1975); Helmuth Grössing, Humanistische Naturwissenschaft: zur Geschichte der Wiener mathematischen Schulen des 15. und 16. Jahrhunderts (Baden-Baden: V. Koerner, 1983); N.M. Swerdlow, "The Recovery of the Exact Science of Antiquity," Rome Reborn: The Vatican Library and Renaissance Culture, ed. Anthony Grafton (Washington, D.C.: Library of Congress, 1993), 125–68; Jens Høyrup, "A New Art in Ancient Clothes: Itineraries Chosen Between Scholasticism and Baroque in Order to Make Algebra Appear Legitimate and their Impact on the Substance of the Discipline," Physis 35 (1998): 11–50.

4. Ernst Zinner, Regiomontanus: his Life and Work, tr. Ezra Brown (New York: North-Holland, 1990), 13–50.

5. Ibid., 51–55.

6. On the medieval history of the University of Vienna, see Rudolf Kink, Geschichte der kaiserlichen Universität zu Wien (Vienna: C. Gerold und Sohn, 1854), vols. 1–2; Joseph Ritter von Aschbach, Die Wiener Universität und ihre Gelehrten (Vienna: Verlag der k.k. Unversität, 1888), vols. 1–2; Alphons Lhotsky, Die Wiener Artistenfakultät, 1365–1497 (Vienna: Hermann Bohlaus, 1965); Paul Uiblein, Mittelalterliches Studium an der Wiener Artistenfakultät (Vienna: WUV-Universitätsverlag, 1995); Uiblien, Die Universität Wien im Mittelalter: Beitrage und Forschungen (Vienna: WUV-Universitätsverlag, 1999).

7. The curricular bonds between German universities and the University of Paris are further discussed in Astrik L. Gabriel, The Paris Studium: Robert of Sorbonne and his Legacy (Notre Dame: United States Subcommission for the History of Universities, 1992), 113–68.

8. On medieval mathematics in general, see Michael S. Mahoney, "Mathematics," in Science in the Middle Ages, ed. David Lindberg (Chicago: University of Chicago Press, 1978), 145–78; A.P. Iushkevich, Geschichte der Mathematik im Mittelalter, trans. Viktor Ziegler (Leipzig: Teubner, 1964); on arithmetic, see, Jordanus de Nemore, De numeris datis, ed. and trans. Barnabas Hughes (Berkeley: University of California Press, 1981); Ghislaine L'Huillier, Le Quadripartitum numerorum de Jean de Murs: Introduction et edition critique (Geneva: Droz, 1990); on geometry: H.L.L. Busard, The Latin Translation of the Arabic Version of Euclid's Elements Commonly Ascribed to Gerard of Cremona (Leiden: Brill, 1984); on astronomy: Lynn Thorndike, The Sphere of Sacrobosco and its Commentators (Chicago: University of Chicago Press, 1949); Francis S. Benjamin, Jr. and G.J. Toomer, Campanus of Novara and Medieval Planetary Theory (Madison: University of Wisconsin Press, 1971); on optics: David C. Lindberg, Theories of Vision from al-Kindi to Kepler (Chicago: University of Chicago Press, 1976).

9. Acta Facultatis Artium. Vol. 3. Universitätsarchiv, Vienna. Codex Ph. 8, 51r.

10. Lhotsky, Die Wiener Artistenfakultät, 139–41.
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