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REGIOMONTANUS

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« Reply #45 on: October 16, 2008, 09:58:33 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 #46 on: October 16, 2008, 09:59:43 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 #47 on: October 16, 2008, 10:00:56 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 #48 on: October 16, 2008, 10:02:00 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 #49 on: October 16, 2008, 10:03:00 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 #50 on: October 16, 2008, 10:04:06 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 #51 on: October 16, 2008, 10:08:34 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 #52 on: October 16, 2008, 10:09:54 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 #53 on: October 16, 2008, 10:14:50 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 #54 on: October 16, 2008, 10:16:13 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 #55 on: October 16, 2008, 10:17:43 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 #56 on: October 16, 2008, 10:18:57 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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« Reply #57 on: October 16, 2008, 10:19:55 pm »










11. Zinner, Regiomontanus, 18. One of Peurbach's poems on nature can be found in Grössing, Humanistische Naturwissenschaft, 210–13.

12. "Me autem nichil unquam provide fecisse . . . aut, si quicquam, hoc in primis, non audeo dicere sapienter, sed feliciter factum est: et quod Bononiam vidi et quod non inhesi." Petrarch, Rerum familiarum libri I-VIII, trans. Aldo S. Bernardo (Albany: State University of N.Y. Press, 1975), 223; Idem, Rerum familiarum IV-VII, ed. Ugo Dotti (Paris: Société d'édition Les Belles Lettres, 2002), 109.

13. "Te Vienna Doctorem Artium creavit dignissimum." Regiomontanus, Padua oration, 48.

14. On humanist periodicity, see Michael Baxandall, Giotto and the Orators: Humanist Observers of Painting in Italy and the Discovery of Pictoral Composition, 1350–1450 (Oxford: Clarendon, 1971), 20–27.

15. "Nisi primitias lectionum facturo mihi mansuetudo vestra aspiraret, orationisque tremebundae filum dirigeret, praestantissimi viri, silvisse prorsus mallem quam pulpitum hoc philosophicum audentius conscendisse, praesertim cum novitate coepti facinoris, tanto tamquam celebri clarissimorum hominum conventu, diuturna demum a scholasticis exercitiis abstinentia facile deterreri potuerim." Regiomontanus, Padua oration, 43.

16. Baxandall, Giotto and the Orators, 21.

17. On humanist epideictic, see John W. O'Malley, Praise and Blame in Renaissance Rome: Rhetoric, Doctrine, and Reform in the Sacred Orators of the Papal Court, c. 1450–1521 (Durham: Duke University Press, 1979), 39–41.

18. A number of such orations can be found in Karl Müllner, Reden und Briefe Italienischer Humansiten (Munich: W. Fink, 1970).

19. "Dominicus igitur domum praedicatorum condidit, Thomas eius pavimenta marmore vestivit, Dominicus parietes struxit, Thomas picturis eos egregiis adornavit, Dominicus fratrum columen extitit, Thomas specimen, Dominicus plantavit, Thomas irrigavit, ille dignationes atque episcopatus ultro delatos refugit atque aversatus est, hic nobilitatem, opes, propinquos, parentes tamquam sirenes effugit. . . ." Lorenzo Valla, "In Praise of Saint Thomas Aquinas," trans. M. Esther Hanley, in Renaissance Philosophy, ed. Leonard A. Kennedy (The Hague: Mouton, 1973), 13–27, 21; J. Valen, "Lorenzo Valla über Thomas von Aquino," Vierteljahrsschrift für Kultur und Litteratur der Renaissance 1 (1886): 384–96, 393.

20. Hannah H. Gray, "Valla's Encomium of St. Thomas Aquinas and the Humanist Conception of Christian Antiquity," in Essays in History and Literature: Presented by the Fellows of the Newberry Library to Stanley Pargellis, ed. Heinz Bluhm (Chicago: The Newberry Library, 1965), 37–51.
 
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« Reply #58 on: October 16, 2008, 10:21:00 pm »










21. "Archimedes Siracusanus civis et Apollonius Pergaeus ob ingenii altitudinem divinus vocari solitus, quorum uter alteri praeferendus sit, non facile dixero. Nam etsi Apollonius elementa conica in octo libris, quos nondum vidit latinitas, subtilissime conscripserit, Archimedi tamen Siculo varietas rerum editarum principatum contulisse videtur . . . Apollonius quoque ubi in Latinum ex Graeco versus fuerit, nemini vestrum non admirandus veniet." Regiomontanus, Padua oration, 45.

22. O'Malley, Praise and Blame, 63–65.

23. "spectatissimus mathematicarum splendor . . ." Regiomontanus, Padua oration, 43.

24. "Ante oculos igitur causam habetis." Regiomontanus, Padua oration, 53.

25. "Geometriae disciplina primum ab Aegyptis reperta dicitur, quod, inundante Nilo et omnium possessionibus limo obductis, initium terrae dividendae per lineas et mensuras nomen arti dedit. Quae deinde longius acumine sapientium profecta et maris et caeli et aeris spatia metiuntur." Isidore of Seville, "On the Quadrivium or Four Mathematical Sciences," trans. Ernest Brehaut, in A Source Book in Medieval Science, ed. Edward Grant (Cambridge, Mass.: Harvard University Press, 1974), 3–16, 8; Idem, Etimologie o Origini, 1.10, ed. Angelo Valastro Canale (Turin: Unione Tipografico-Editrice Torinese, 2002), 290.

26. "Dum Nili flumen solito vehementius exundans limites agrorum Aegyptiorum vastaret, et pene universos delevisset, contendere coeperunt agricolae, sua quisque pro opinione (ut est ingenium hominum ad rem plus satis attentum) agros suos definire cupiebat, quique sive sermone, sive viribus plus caeteris valuit, tametsi antea angustum haberet agrum pro libito terminus instituit, quae res cum praeter aequum et bonum verteretur, ad principem eius patriae devoluta est, qui iusta quadam ratione mensurisque certis suos cuique reparuit limites. Sic generali et inusitato quodam impulsu mortalium animi ad mensuras tractandas converse coepere alter alteri questiones anteponere, et quicquid in huiuscemodi exercitiis bene inventum putabant, quamvis indigestum, ut ita loquar, adhuc foret, literis mandare studuerunt." Regiomontanus, Padua oration, 44.

27. Herodotus, The Histories, 2.109, trans. George Rawlinson (New York: Knopf, 1997), 177.

28. On Valla's translation of Herodotus, see E.B. Fryde, "Some Fifteenth Century Latin Translation of Ancient Greek Historians," in Humanism and Renaissance Historiography (London: The Hambledon Press, 1983), 83–114.

29. "Cum Platonis in dicendo suavitas, tum Ciceronis nostra lingua dissertissima non assint, qui etsi reviviscerent haud quaquam pro dignitate idipsum assequerentur." Regiomontanus, Padua oration, 43. Plato, though he was of course Greek, was seen by fifteenth-century humanists as exemplifying excellent style. See, for example, Baxandall, Giotto and the Orators, 22–24.

30. "Nilus totius orbis fluviorum celeberrimus a solstitio aestivo usque ad autumnale aequinoctium, ut Herodotus libro 2 et Diodorus in 1 testatur, immensa aquarum mole quotannis totam Aegyptum exundat, ex cuius incremento Aegyptii vim aut penuriam futurarum frugum praevident. Aegyptus enim cum Nilus in 12 cubitos excrescit famem sentit; in 13 etiamnum esurit; 14 cubiti hilaritatem afferunt, 15 securitatem, 16 delitias, quod iustum fertur esse incrementum. Maximum autem aetate Claudii principis fuit cubitorum 18, sicut minimum Pharsalico bello veluti caedem magni Pompeii prodigio quodam aversante, autores Plinius libro 5 et Strabo 17. Cum huiusmodi itaque Nili inundationes limites agrorum confunderent, nunc minuendo, alias immutando, nonnunquam delendo signa quaedam quibus proprium ab alieno discerneretur, iterum atque iterum metiri eam terram oportebat, propter quod Strabo 17 Geographiae et Herodotus 2 aiunt nonnullos prodidisse geometriam ab Aegyptiis primo inventam esse, quemadmodum arithmeticam, id est, numeralem scientiam a Phoenicibus propter mercaturas." Polydore Vergil, On Discovery, ed. and trans. Brian P. Copenhaver. (Cambridge, Mass.: Harvard University Press, 2002), 146–49. 
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« Reply #59 on: October 16, 2008, 10:22:16 pm »










31. Andrea Brenta, In disciplinas et bonas artes oratio, in Reden und Briefe, 71–85.

32. "Abraham enim Hebraeorum patrem Astronomiam tenuisse clamant atque Mosen. Prometheo ignis divini furtum imputant, quod Astronomiae lumen mortalibus tradiderit. Herculem pro Atlante coelum humeris suis sustinuisse aiunt, sive quod sub Atlante astronomiam didicerit, sive quod in regno eius absentis praefectus aliquamdiu fuerit." Regiomontanus, Padua oration, 46.

33. Isidore, "On the Quadrivium," 11. The main difference between Regiomontanus's account and that of Isidore is Regiomontanus's mention of Prometheus.

34. "Astronomiam ab Aegyptiis inventam tradunt. Nam cum in ea regione perpetua aeris serenitas contingeret et Aegyptii sacerdotes, qui disciplinis studebant, otio et rebus ad vitam necessariis abundarent, ut Aristoteles, multa et observasse et invenisse perhibentur quae ad caeli rationem pertinebant. . . . Post multa vero annorum milia, ut quidam prodiderunt, in Graeciam delata est et a multis cognita, ut maxime ex poetis cognosci potest. Deinde secuti sunt Pythagoras et Pherecydes, qui mathematicarum rerum studiosi permultum huic disciplinae addiderunt. Hos secutus est Eratosthenes, Berosus, et Hipparchus et innumerabiles alii, quibus Graecia plurimum abundavit. Plato vero et Eudoxus, cum discendi gratia in Aegyptum profecti essent, multa cum rerum aliarum tum vero eius scientiae secreta, ut Strabo refert, ab Aegyptiis sacerdotibus in Graeciam reportarunt. Ita paulatim ac per aetates aucta et ad cumulum perducta usque ad Ptolemaei tempora pervenit, a quo mirandum illud ac prope divinum astrologiae opus editum est, ut nihil quodam modo ad hanc scientiam addi posse videatur." Gregorio da Città di Castello, "Gregorii Tiphernii, viri clarissimi atque Graecarum litterarum eruditissimi, de astrologia oratio," in Reden und Briefe Italienischer Humansiten, 177–78.

35. Müllner, Reden und Briefe, 177 n. 2; Aristotle, Metaphysics, 1.24; Aristotle, Posterior Analytics, 2.19; Strabo, Geography, 14.1.14; Cicero, De finibus, 5.87; Pliny, Historia naturalis, 30.1.

36. "Hipparchus tamen Rhodius huius disciplinae primus parens, Claudius autem Ptolemeus Alexandrinus auctor atque princeps non iniuria praedicabitur. Nam ante Hipparchum pauci admodum astrorum motus expedite contemplati sunt, nemo autem prorsus stellas fixas alio quam diurno motu circumferri putaverat, cui rei Hipparchus oculos adiecit crebriores, conclusitque memoratas stellas motu quodam proprio ac tardissimo Orientem versus mutari. Deinde Ptolemaeus inventa priscorum resumens . . . motum huiusmodi in centum annis per unum gradum pronunciavit quemadmodum in septima dictione capituli 3. intueri licet. . . . Multos denique huius artis Graecos professores silentio praetereundos censeo . . . Arabes praeterea quantum in hoc genere atrium valuerunt, testes ostendunt dignissimi Albategnius quem Latinum fecit Plato quidam Tiburtinus. Item Geber Hispalensis Gerardo quodam Cremonensi traductus, quem Albertus Magnus in speculo Astronomiae correctorem Ptolemaei vocare non formidat." Regiomontanus, Padua oration, 46–47; Ptolemy, Almagest, 7.3, trans. G.J. Toomer, in Ptolemy's Almagest (London: Duckworth, 1984), 329–38.

37. Georg Peurbach and Regiomontanus, Epytoma in almagestum Ptolomei, in Opera collectanea, 59–274; Zinner, Regiomontanus, 51–55.

38. Peurbach and Regiomontanus, Epitome, 7.3, 172–173; Zinner, Regiomontanus, 54.

39. Valla, "In Praise of Saint Thomas Aquinas," 26; Petrarch, Against a Detractor of Italy, ed. and trans. David Marsh, in Francesco Petrarca: Invectives (Cambridge, Mass.: Harvard University Press, 2003), 364–475, 411–25.

40. "Nam etsi Pythagoras numerorum peritia apud posteros immortalitatem reliquerit, tum quod peregrinis praeceptoribus Aegyptiis atque Arabibus, qui plurimum in eo studio valuerunt, se submiserit, tum quod numerorum certa compagine omnia naturae secreta scrutari tentaverit, longe tamen digniora Euclides fecit numerorum fundamenta in tribus libris suis, septimo, octavo et nono, unde et Iordanus decem numerorum elementa decerpsit, hinc tres libros de datis numerorum pulcherrimos edidit. Diofanti autem tredecim libros subtilissimos nemo usque hac ex Graecis Latinos fecit, in quibus flos ipse totius Arithmeticae latet, ars videlicet rei et census, quam hodie vocant Algebram Arabico nomine. Huius equidem artis pulcherrimae multa fragmenta passim Latini contrectant, paucissimos autem egregie doctos offendo nostra tempestate post Ioannem de Blanchinis virum optimum. Habetur demum apud nostros quadripartitum numerorum, opus insigne admodum, item Algorithmus demonstratus et Arithmetica Bohecii, introductio ex Graeco Nicomacho sumpta." Regiomontanus, Padua oration, 46. 
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