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Regiomontanus, Johannes


(b. Konigsber, Franconia, Germany, 6 June 1436; d. Rome, Italy, ca. 8 July 1476)

astronomy, mathematics.

Nothing is known of Regiomontanus before he enrolled in the University of Vienna on 14 April 1450 as “Johannes Molitoris de Künigsperg.”1 Since the name of his birthplace means “King’s Mountain,” he sometimes Latinized his name as “Joannes de Regio monte,” from which the standard designation Regiomontanus was later derived. He was awarded the bachelor’s degree on 16 January 1452 at the age of fifteen; but because of the regulations of the university, he could not receive the master’s degree until he was twenty-one. On 11 November 1457 he was appointed to the faculty, thereby becoming a colleague of Peuerbach, with whom he had studied astronomy. The two men became fast friends and worked closely together as observers of the heavens.

The course of their lives was deeply affected by the arrival in Vienna on 5 May 1460 of Cardinal Bessarion (1403–1472), the papal legate to the Holy Roman Empire.2 Bessarion’s native tongue was Greek (he was bom in Trebizond), and as part of his ardent campaign to bring ancient Greek authors to the attention of intellectuals in the Latin West, he persuaded Peuerbach to undertake a “briefer and more comprehensible” condensation, in Latin, of the Mathematical Syntaxis of Ptolemy, whose Greek style was formidable and whose ideas were far from simple. In those days Greek was not taught at the University of Vienna,3 and Peuerbach did not know it. He had, however, made his own copy of Gerard of Cremona’s Latin translation of Ptolemy’s Syntaxis. Using this twelfth-century version, Peuerbach reached the end of book VI just before he died on 8 April 1461. On his deathbed he pledged Regiomontanus to complete the project.

Complying with Peuerbach’s last wish, Regiomontanus accompanied Bessarion on his return trip to Rome, where they arrived on 20 November 1461.4 When Regiomontanus finished the Epitome, as he entitled the translation by Peuerbach and himself, he dedicated it to Bessarion. In the parchment manuscript, which still survives, he did not address Bessarion as titular Patriarch of Constantinople, an honor bestowed on him on 28 April 1463,5 a decade after the capital of the Byzantine Empire had been captured by the Turks. Thus, sometime before that date the Peuerbach-Regiomontanus Epitome was ready to go to press; but it was not actually printed until 31 August 1496, twenty years after the death of Regiomontanus.

At the end of the fifteenth century, Ptolemy’s achievement remained at the pinnacle of astronomical thought; and by providing easier access to Ptolemy’s complex masterpiece, the Peuerbach-Regiomontanus Epitome contributed to current scientific research rather than to improved understanding of the past. Moreover, the Epitome was no mere compressed translation of the Syntaxis, to which it added later observations, revised computations, and critical reflections—one of which revealed that Ptolemy’s lunar theory required the apparent diameter of the moon to vary in length much more than it really does. This passage (book V, proposition 22) in the Epitome, which was printed in Venice, attracted the attention of Copernicus, then a student at the University of Bologna. Struck by this error in Ptolemy’s astronomical system, which had prevailed for over 1,300 years, Copernicus went on to lay the foundations of modern astronomy and thus overthrow the Ptolemaic system.

Ptolemy was not only the foremost astronomer of antiquity but also its leading geographer; and Jacopo Angeli’s widely used Latin translation (1406–1410)6 of Ptolemy’s Geography was condemned by Regiomontanus because the translator “had an inadequate knowledge of the Greek language and of mathematics.”7 Many of the obscure passages in Angell’s translation could not be explained by Peuerbach, who, as noted above, had not learned Greek. Hence Regiomontanus determined to master the language of Ptolemy. He acquired a remarkable fluency in Greek from his close association with Bessarion, and armed with a thorough comprehension of Ptolemy’s language, he announced his intention to print an attack on Angeli’s translation. But he died before completing this work. Nevertheless, “Johannes Regiomontanus’ Notes on the Errors Committed by Jacopo Angeli in His Translation” formed the appendix (sig. Plr-Q8r) to a new version of Ptolemy’s Geography (Strasbourg, 1525) by a scholar who had access to Regiomontanus’ literary remains.

In a letter written not long after 11 February 1464 to the Italian mathematician Giovanni Bianchini, Regiomontanus reported that he had found an incomplete manuscript of Diophantus and, if he had the whole work, he would undertake to translate it into Latin—“since for this purpose the Greek I have learned in the home of my most revered master would be adequate.”8 Regiomontanus never translated Diophantus nor did he ever find a complete manuscript; nor did anyone else. Nevertheless, the recovery of Diophantus in modern times began with Regiomontanus’ discovery of the incomplete manuscript.

When Bessarion was designated papal legate to the Venetian Republic, Regiomontanus left Rome with him on 5 July 1463.9 In the spring of 146410 at the University of Padua, then under Venetian control, Regiomontanus lectured on the ninth-century Muslim scientist al-Farghani. Although the main body of these lectures has not survived, “Johannes Regiomontanus’ Introductory Discourse on All the Mathematical Disciplines, Delivered at Padua When He was Publicly Expounding al-Farghani” was later published in Continentur in hoc libro Rudimenta astnmomica Alfragani …, whose first item was John of Seville’s twelfth-century Latin translation of al-Farghani’s Elements of Astronomy (Nuremberg, 1537).

Also included in this volume was Plato of Tivoli’s twelfth-century Latin version, “together with geometrical proofs and additions by Johannes Regiomontanus,” of al-Battani’s The Motions of the Stars. One such addition (to al-Battani’s chapter 11, although the printed edition misplaced it in the middle of chapter 12) may have been the germ from which Regiomontanus subsequently developed the earliest statement of the cosine law for spherical triangles. Although he employed the versed sine (1 —cos) rather than the cosine itself and used the law only once, he was the first to formulate this fundamental proposition of spherical trigonometry. He enunciated it as theorem 2 in book V of his treatise On All Classes of Triangles (De triangulis omnimodis ).

The urgent need for a compact and systematic treatment of the rules governing the ratios of the sides and angles in both plane and spherical triangles had become apparent to Peuerbach and Regiomontanus while they were working on the Epitome. At the close of the dedication of that work Regiomontanus stated that he would write a treatise on trigonometry. The manuscript of the last four books contains many blank spaces, which, despite Regiomontanus’ intentions, were never completed. Part of the volume had been written before he left Rome on 5 July 1463. At the end of that year or at the beginning of 1464 he told a correspondent: “I do not have with me the books which 1 have written about triangles, but they will soon be brought from Rome.”11 It may have been in Rome that Regiomontanus propounded, in theorem 1 of book II, the proportionality of the sides of a plane triangle to the sides of the opposite angles (or, in modern notation a/sin A = b/sin B = c/sin C, the sine law ). The corresponding proposition for spherical triangles appears in book IV, theorem 17. Theorem 23 in book II solves, for the first time in the Latin West, a trigonometric problem by means of algebra (here called the ars ret et census). Regiomontanus’ monumental work on Triangles, the first publication of which was delayed until 12 August 1533, attracted many important readers and thereby exerted an enormous influence on the later development of trigonometry because it was the first printed systematization of that subject as a branch of mathematics independent of astronomy.

Regiomontanus dedicated his Triangles to Bessarion. whom Pius II, in 1463, had named titular Patriarch of Constantinople. When the pope died, Bessarion returned to Rome in August 1464 to take part in the election of a successor. Regiomontanus accompanied him and while in Rome composed a dialogue between a Viennese named Johannes (evidently himself) and an unnamed scholar from Cracow. The subject of their conversation was a thirteenth-century planetary theory that was still very popular. Some of its defects were discussed in the dialogue, which was printed by Regiomontanus when he later acquired his own press. Although he published the dialogue without a title, it was often reprinted under some such heading as Johannes Regiomontanus’ Attack on the Absurdities in the Planetary Theory of Gerard of Cremona (Gerard’s pupils did not list this Theorica planetarum in the catalog of their teacher’s productions).12

After an observation on 19 June 1465,13 presumably in Viterbo, a favorite resort of Bessarion’s, Regiomontanus’ activities during the next two years are not known. In 1467, however, he was firmly established in Hungary, where the post of astronomer royal was held by Martin Bylica of Olkusz (1433–1493), who was also present in Rome during the papal election and in all likelihood is the unnamed interlocutor in Regiomontanus’ dialogue on planetary theory.

In 1467, with Bylica’s assistance, Regiomontanus computed his Tables of Directions, which consisted of the longitudes of the celestial bodies in relation to the apparent daily rotation of the heavens. These Tables, computed for observers as far north of the equator as 60°, were first published in 1490 and very frequently thereafter.14 Regiomontanus wrote accompanying problems and in problem 10 he indicated the desirability of abandoning the sexagesimal character of the table of sides by putting sin 90° = 100,000 (105) instead of 60,000 (6 x 104, the base he had used in Triangles (book IV, theorem 25). In that work he had not employed the tangent function; but in Tables of Directions he included a table of tangents (although he did not use this term) for angles up to 90°, the interval being 1° and tan 45° = 100,000, thereby providing the model for our modern tables.

In 1468 in Buda, then the capital of the kingdom of Hungary, Regiomontanus computed a table of sides with sin 90° = 10,000,000 (107). But before he realized the advantage of the decimal base, he had prepared a sexagesimal sine table, to which he had referred in the dedication of his Triangles and which he had used in computing his Tables of Directions, with sin 90° = 6,000,000 (6 x 106), the interval being 1’ and the seconds being found by an auxiliary table of proportional parts. Both of Regiomontanus’ major sine tables, the sexagesimal and the decimal, were first published at Nuremberg in 1541, together with his essay on the Construction of Sine Tables.

While still in Italy, Regiomontanus began to compute his Table of the First Movable [Sphere], or of the apparent daily rotation of the heavens. He completed this work, together with an explanation of its use, in Hungary and dedicated it to his friend King Matthias I Corvinus. He also expounded the geometrical basis of this Table, These three related works constituted an item in the list of his own writings that Regiomontanus intended to print on his own press, an intention he could not carry out. Of these three works, the first two were published in Vienna in 1514, and the third in Neuburg in 1557. Regiomontanus wrote each of these works for the purpose of facilitating astronomical computations. But whatever use was made of them ended with the advent of logarithms.

In 1471 Regiomontanus left Hungary. “Quite recently 1 have made [observations] in the city of Nuremberg … for I have chosen it as my permanent home,” he informed a correspondent on 4 July 1471, “not only on account of the availability of instruments, particularly the astronomical instruments on which the entire science of the heavens is based, but also on account of the very great ease of all sorts of communication with learned men living everywhere, since this place is regarded as the center of Europe because of the journeys of the merchants.”15 On 29 November 1471 the City Council of Nuremberg granted Regiomontanus residence in the city until Christmas of the following year. He installed a printing press in his own house in order to publish scientific writings, a class of books in which the existing establishments were reluctant to invest their capital, partly because the necessary diagrams required special craftsmen and additional expense.

Regiomontanus was the first publisher of astronomical and mathematical literature, and he sought to advance the work of scientists by providing them with texts free of scribal and typographical errors, unlike the publications then in circulation. His emphasis on correct texts was aided by his introduction into Nuremberg printing of the Latin alphabet and, for writings in the German language, rounded and simplified letters that approached the Latin alphabet in legibility.

Regiomontanus’ first publication, a mark of his deep affection for his former teacher, colleague, and collaborator, was Peuerbach’s New Theory of the Planets. This work was the first item in the catalog which Regiomontanus sent out in the form of a broadside, listing his publications, issued or projected, written by himself or others. The second item in the list of his own publications was the Ephemerides, which he issued in 1474 and which was the first such work to be printed. It gave the positions of the heavenly bodies for every day from 1475 to 1506. Of all the books written and published by Regiomontanus, this is perhaps the most interesting from the standpoint of general history: Columbus took a copy on his fourth voyage and used its prediction of the lunar eclipse of 29 February 1504 to frighten the hostile Indians in Jamaica into submission.16

The geographer Martin Behaim “boasted that he was a pupil of Regiomontanus”17 in Nuremberg. More credit is given to the statement that Regiomontanus attracted Bernhard Walther as a pupil. Walther, who was born in Memmingen, in 1467 became a citizen of Nuremberg, where he helped Regiomontanus with his observations and continued them after his teacher left for Rome in the summer of 1475. Regiomontanus’ last observation in Nuremberg is dated 28 July 1475 and Walther’s observations begin five days later.18

According to a Nuremberg chronicler, Regiomontanus went to Rome in response to a papal invitation to emend the notoriously incorrect ecclesiastical calendar. If this report is true, nothing positive resulted from his trip, for he died in less than a year.

In all probability Regiomontanus fell victim to the plague that spread through Rome after the Tiber overflowed its banks in January 1476. But a more sensational rumor concerning the cause of his death surfaced in a laudatory poem that served as the title page of a posthumous edition of his Latin Calendar (Venice, 1482). The rumor gained currency by being repeated in 1549 in Reinhold’s commemorative eulogy of Regiomontanus and again in 1654 19 in Gassendi’s biography of the astronomer. In his catalog Regiomontanus had announced his intention to publish an extensive polemic against George of Trehizond, whose “commentary on the Syntaxis he will show with the utmost clarity to be worthless and his translation of Ptolemy’s work not to be free of faults.” Although Regiomontanus never actually published his attack, which still remains in manuscript in Leningrad, George’s sons poisoned him, according to the rumor. Yet Bessarion died unmolested on 18 November 1472, three years after his own devastating attack on George of Trebizond as a Calumniator of Plato was published in Rome (1469).

“The motion of the stars must vary a tiny bit on account of the motion of the earth.” This portentous statement in the handwriting of Regiomontanus was excerpted from one of his letters by Georg Hartmann, the discoverer of the vertical dip of the magnetic needle and an early supporter of the Copernican cosmology. Hartmann regarded the excerpt as a treasure, undoubtedly because to his mind it provided clear proof that Regiomontanus, the greatest astronomer of the fifteenth century, had accepted the concept of the moving earth and realized one of its numerous implications; Regiomontanus was therefore a Copernican before Copernicus.

The letter from which Hartmann took this excerpt has not survived, nor has the excerpt itself. But it was copied by a professor onto the margin of his unpublished lecture in 1613 on Copernicus’ planetary theory, with the explanation that Hartmann “recognized Regiomontanus’ handwriting because he was also familiar with his features.“Yet Hartmann was not even born until 1489, thirteen years after the death of Regiomontanus.

Nevertheless, it has been suggested that the letter in question may have been sent by Regiomontanus to Novara, who, in an unpublished essay on the duration of pregnancy, called Regiomontanus his teacher. Novara in turn became the teacher of Copernicus. Thus it can be inferred that the concept of the revolutionary geokinetic doctrine was first conceived by Regiomontanus and communicated to Novara, who then passed it to Copernicus. Nevertheless, in the voluminous published and unpublished writings of Regiomontanus, no other reference to the earth in motion has ever been found.


1.Die Matrikel der University Wien,I (Graz-Cologne, 1954), 275. The Johannes Molitoris who entered the University of Leipzig on 15 October 1447 has been identified with Regiomontanus by Zinner, Leben und Wirken des … Regiomontanus. 13. The Leipzig rector, however, did not associate this namesake with any particular place and Molitoris, as a Latinized form of the surname Muller, was extremely common.

2. Ludwig Mohler, Kardinal Bessarionals Theologe, Humanist und Staatsmuan Qucllcn und Forschungen aus dem Gebiete der Gesehichtc, no. 20 (Paderborn, 1923), 298.

3. Joseph Aschbach, Geschichte der Wiener Universittit im ersten Jahrhunderte ihres Bestehens (Farnborough, 1967; repr. of Vienna, 1865), 539.

4. Mohler; op. cit., 303.

5. Conrad Eubei, Hicranhia catholica medii aevi, II (Padua. I960; repr. of 2nd ed., Münster. 1913–1923). 150. Bessarion’s elevation is dated in April 1463, but the exact day is marked as unknown. However, Bessarion’s predecessor, Isidore or Kiev, died on 27 April 1463 (Eubei. II. 36, n. 199).

6. Robert Weiss, “Jacopo Angeli da Scarperia,” in Medioevo e rinascimento. Studi in onore di Bruno Nardi Pubbiicazioni dcHMsiitulo di filosofia deirUniversitii di Roma (Florence, 1955), 824.

7. Regiomontanus’ catalog of the books to be printed on his press; reproduced by Zinner, “Die wissenschaftltchen Bcstrebungen Regiomontans,” in Beitrdge zur Inkunabet-kunde, 2 (1938), 92.

8. Silvio Magrini, “Joannes de Blanchinis Ferrariensis e il suo carteggio scientifico col (1463–64),” in Atti e memorie della deputazione ferrarese di storia patria 22, fasc. 3, no. 2, (1915–1917), lvii.

9. Mohlcr, op.cit., 312,

10. The total eclipse of the moon on 21 April 1464 was observed by Regiomontanus in Padua; sec Scripta clarissimi itiathe-matiei M. Ioannis Regiomontani (Nuremberg. 1544), fol. 41v-42r; or Willebrord Snell, Coeli et siderum … observa-tiones Hassiacae (Leiden, 1618), Ioannis de Montercgio … observationes, fol. 20v.

11. Maximilian Curtze, “Der Briefwechsel Regiomontan’s mil Giovanni Bianchini, Jacob von Speier und Christian Roder,” in Abhandhmgen zur Geschichte der Mathematik, 12 (1902), 214.

12. Olaf Pedersen, “The Theorica Planetarum Literature of the Middle Ages,” in Ithaca, Proceedings of the Tenth International Congress of History of Science (Paris, 1964), 617.

13.Scripta … Regiomontani fol. 42r; Snell, fol. 21v.

14. The manuscript of Regiomontanus’ Tables of Directions that Bylica presented to Cracow University is still preserved there; see Wladyslaw Wislocki, Katulog rekopisow biblioteki jagiellonskiej (Cracow, 1877–1881), 188; and Jerzy Zathey et al., Historia biblioteki jagiellonskiej (Cracow, 1966), 154, n. 64.

15. Curtze, op. c/7., 327. The lunar eclipse on 2 June 1471 was observed by Regiomontanus in Nuremberg—Scripta, fol. 42v; Snell, fol. 22r.

16. Samuel Eliot Morison, Admiral of the Ocean Sea, II (Boston, 1942), 400 403.

17. Jāilo de Barros, Asia, I, decade I, bk. 4, ch. 2 (Lisbon, 1945), 135. If BehainVs claim was correct, he was at most 16 years old when Regiomontanus left Nuremberg; see Richard Hennig, Terrae incognitae, 2nd ed., IV, (Leiden, 1944–1956), 434.

18.Scripta, fol. 27v; Snell, fol. lv; Donald Beaver, “Bernard Walther: Innovator in Astronomical Observation,” in Journal for the History of Astronomy, 1 (1970), 39–43.

19. Gassendi, Tychonis Brahei … vita … accessit … Re-giomontani … vita (Paris, 1654), app., 92; and Opera omnia, V (Stuttgart-Bad Cannstatt, 1964; repr. of Lyons, 1658 ed), 532.


Regiomontanus’ works were reprinted in Joannis Rt’giomontari opera collectanea (Osnabriick, 1972) with a biography by the ed. Felix Schmeidler. An older work is Ernst Zinner, Leben und Wirken des Johannes Midler von Kdnigsberg genannt Regiomontanus, 2nd ed., rev. and enl. (Osnabriick, 1968). A recent trans, of Regiomontanus’ De triangiclis omnimodis is Barnabas Hughes, Regiomontanus on Triangles (Madison, Wis., 1967).

On Regiomontanus and his work, see the anonymous “Regiomontanus’s Astrolabe at the National Maritime Museum,” in Nature. 183 (1959), 508–509; and Edward Rosen, “Regiomontanus’s Breviarium,” in Medievalia et Humanistica, 15 (1963), 95–96.

Edward Rosen

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Regiomontanus, Johannes


(b. Königsberg, Franconia, Germany, 6 June 1436; d. Rome, Italy, c. 8 July 1476), astronomy, mathematics. For the original article on Regiomontanus see DSB, vol. 11.

Studies of Regiomontanus since 1975 have enriched scholars’ understanding of his antecedents, the contemporary context of his work in astronomy and mathematics, and his significance for the Copernican revolution.

Intellectual Debts The astronomical manuscripts Regiomontanus owned or copied during his Viennese years (1450–1461) show a greater engagement with the Viennese astronomical tradition and its library holdings than has been heretofore documented. From Paris, Henry of Langenstein (d. 1397) had brought to Vienna his critique of the epicycles and eccentrics of Ptolemaic astronomy (De reprobatione ecentricorum et epiciclorum, 1364 see Kren, 1968 and 1969). Regiomontanus not only copied it in the 1450s (Vienna, ÖNB cod. 5203), but also drew upon it. Among other points, Langenstein’s treatise criticized Ptolemy’s models for the Moon, inferior planets, and Mars for implying large (but unobserved) variations in their areas. This type of argument motivated Regiomontanus’s efforts to reform astronomical modeling using concentric (homocentric) spheres and surfaced in such later writings as the Epitome of the Almagest, book 5, prop. 22, the correspondence with Bianchini (Swerdlow, 1990, esp. pp. 173–174), and the Defense of Theon against George of Trebizond. In Vienna, Regiomontanus also encountered other astronomical alternatives to the Almagest, notably al-Bitruji’s homocentric De motibus celorum, and criticisms of al-Bitruji in the fourteenth-century Tractatus planitorbii of Guido de Marchia, who also proposed as the carriers of the planets not spheres, but eccentric rings moving in a fluid heaven (Shank, 1992, 1998, and 2003).

In a letter of 1460 to Bishop Janós Vitéz of Hungary, Regiomontanus sketched for the Sun and Moon homocentric models that were silent modifications of a flawed model by al-Bitruji, of whose work Regiomontanus was otherwise critical. He also hoped to eliminate the eccentrics and epicycles of all the planets, producing a “fully concentric” astronomy that would yield improved tables (Swerdlow, 1999). Neither the inherent problems with this program nor his mastery of the nonhomocentric Almagest prevented him from nurturing this hope into the 1470s.

Controversy with George of Trebizond Regiomontanus’s association with Cardinal Basilius Bessarion (1460–c. 1465) drew him into the latter’s controversies with George of Trebizond, which shaped the decade he spent in Italy and Hungary. Bessarion had criticized George of Trebizond’s translation of and commentary on the Almagest soon after they appeared in 1451, initiating a twenty-year feud that extended most famously to their evaluations of Plato. Regiomontanus played the leading role in the astronomical part of these controversies, which framed both his travels and his writings, including two of his most important theoretical works, the Epitome of the Almagest and the Defense of Theon against George of Trebizond.

The Epitome of the Almagest was not a translation (pace DSB XI 349a), but an exposition and updated analysis of Ptolemy’s work. Bessarion had urged Georg Peuerbach to write it as an alternative to Trebizond’s inadequate commentary. When Peuerbach’s death left the Epitome half complete, Regiomontanus finished and revised it circa 1462. Recent work has shown that Bessarion was more than a patron and dedicatee, for he worked through portions of the work with some care (Rigo, 1991). In contrast to the Epitome, Regiomontanus’s Defensio Theonis contra Georgium Trapezuntium (1460s– 1470s) was a highly polemical work that went well beyond addressing Bessarion’s anger at Trebizond’s attacks on Theon of Alexandria’s Almagest commentary, which the cardinal favored highly. This 573-page autograph (St. Petersburg, Archive of the Russian Academy of Sciences) contains a book-by-book attack on George’s commentary on the Almagest, criticizing errors that range from computation and logic to fundamental assumptions and interpretations of Ptolemy. Regiomontanus’s death interrupted his plan of printing both the Epitome and the Defensio on his own press (only the Epitome appeared in print, in 1496). Although these four works—George of Trebizond’s translation of the Almagest and commentary on it, and Regiomontanus’s Epitome and Defensio—constitute hundreds of folios of controversy about the interpretation of Ptolemy in the generation before Nicolaus Copernicus, they have scarcely been studied and outline a new research frontier that will enrich substantially scholars’ understanding of fifteenth-century Latin astronomy.

Commenting on Ptolemy Of the astronomers to whom the Epitome introduced the Almagest, Copernicus is the most famous. The first two propositions from book 12 of the Epitome have special significance for the emergence of Copernicus’s reorganization of the planets around the mean Sun. Regiomontanus’s silent correction of Ptolemy deserves special mention. Ptolemy held that two alternative models could account for the superior planets’ second anomaly (that with respect to the Sun, involving their retrograde motion): an epicyclic model (a traditional small epicycle carried by a large deferent) and an eccentric model (in effect, an inversion of the previous model, with a large epicycle carried by a small deferent). In Almagest, book 12, chapter 1, Ptolemy denied that the eccentric model worked for the second anomaly of the inferior planets (Toomer, 1984, p. 555), and George of Trebizond repeated this claim in his commentary. In Epitome, book 12, proposition 2, however, Regiomontanus proved without comment that an eccentric model could also account for the second anomaly of Mercury and Venus. The corresponding passage in the Defensio attacks George of Trebizond for denying this equivalence, which Regiomontanus asserts to be Ptolemy’s own obvious intention on the grounds that he had already discussed similar equivalences in books 3 and 4 of the Almagest (Shank, 2007). Recent work shows that Regiomontanus was not the first to offer such a proof. The head of the Samarqand observatory, Ali Qushji, had done so before the mid-fifteenth century in explicit criticism of Ptolemy. The strikingly similar orientation of the diagrams of this proof in the Qushji manuscript and in Regiomontanus’s Epitome tantalizingly raises questions of transmission, even as the antithetical interpretations of Ptolemy point to different, and perhaps independent, motivations and contexts (Ragep, 2005; Shank, 2007).

Whatever its origins, the Epitome’s proof in book 12, proposition 2 carries considerable significance for the subsequent history of astronomy, because the eccentric model of the second anomaly of the inferior planets transforms directly into a configuration with Venus and Mercury moving around the mean Sun. The language of Copernicus’s manuscript notes shows that he was working with the alternative eccentric model, in which he treated the distance between Earth and the mean Sun as the eccentricity. When Copernicus used this metric to calculate the size of each planet’s sphere, their spacing gave him a “necessary” sequence of the planets around the mean Sun (Swerdlow, 1973). The new arrangement thus eliminated the longstanding uncertainties in the order of the Sun, Venus, and Mercury and yielded the “commensurability” extolled in De revolutionibus, I, 10. There is, however, no good reason to suppose that Regiomontanus was a proto-Copernican. Contrary to DSB vol. 11, 351b–352a, the claim that “The motion of the stars must vary a tiny bit on account of the motion of the earth,” which Georg Hartmann in the sixteenth century ascribed to Regiomontanus, probably refers not to Earth’s annual motion, but to the small shifts in the central Earth’s center of gravity that some fourteenth-century natural philosophers inferred from the motions of heavy bodies on Earth’s surface (Duhem, 1913–1959).

The Defensio Whereas the Defensio clarifies some of the motivations behind Regiomontanus’s mathematical proofs in the Epitome, it also sheds light on Regiomontanus’s physical assumptions and the polemical context of his work. Begun in the 1460s, the Defensio was at one point to be offered to King Matthias Corvinus of Hungary, for whom Regiomontanus drafted a dedication and to whose court he also moved (c. 1467–1471). The timing of both the dedication and the move follows heightened antagonism between Bessarion and George of Trebizond. The cardinal had prevailed on Pope Paul II to imprison George after discovering that the latter had sought to dedicate his works on the Almagest to Mehmed II, the conqueror of Constantinople, in 1466. Upon his release, George attempted to dedicate these works to King Matthias Corvinus in 1467–1468 (Monfasani, 1976, pp. 286–287). Both Regiomontanus’s move to Hungary and his Defensio were no doubt meant to undercut George of Trebizond’s quest for patronage from Matthias. Although Regiomontanus spent several years in Hungary (c. 1467–1471), books 12 and 13 of the Defensio were not finished until after he had settled in Nürnberg, where he listed it circa 1474 among the works slated for publication on his own press.

A preliminary examination of the polemical Defensio has already offered new insights into Regiomontanus’s natural philosophical concerns, physical assumptions, and understandings of the astronomer’s role. Regiomontanus was clearly in theoretical turmoil in the early 1460s, as is evident in the fact that his Ptolemaic expositions in the Epitome (1462) are bracketed by his homocentric (non-Ptolemaic) astronomical yearnings in both the Letter to Vitéz (1460) and his correspondence with Giovanni Bianchini during 1464 (Swerdlow, 1990). The Defensio confirms that this was not a temporary phase, but a career-long predicament. The same juxtapositions appear in passages from the Defensio that date from the 1470s. Alongside its praise of Ptolemy, the Defensio also contains criticisms of the Almagest on physical grounds. Regiomontanus assails Ptolemy and his followers for using circles to save the phenomena and to produce numerical results.

But planets, he argues, cannot be moved by mere circles, that is, by two-dimensional models devoid of physical properties. Elsewhere, he rails against such an astronomy as a “fictitious art.” Underlying these criticisms was a vision of astronomy as an enterprise that integrated physical considerations (including size, movers, media, etc.) with the task of producing excellent numerical predictions (Shank, 2002).

Sources of Mathematical Results Recent scholarship on Regiomontanus’s mathematics has qualified earlier claims about his originality while tightening his links to the Arabic and late-medieval Latin traditions, the foundation on which his access to Bessarion’s rich library later built. His work on the computation of the first five perfect numbers, formerly considered original, was in fact copied from a thirteenth-century manuscript in Bessarion’s library (Venice, Biblioteca Marciana, f.a. 332). Likewise, while he was working on solving the general cubic equation algebraically, his solution of a special case of the cubic equation goes back to the fourteenth century (Master Dardi of Pisa and others). The reassessment of his mathematics anchors Regiomontanus firmly in his late-medieval intellectual context, whereas his unusual aim of understanding fully and capturing the original meaning of Euclid and Archimedes set him apart from it, and led him to collate Greek and Latin manuscripts by these authors. His efforts reveal a deep understanding of the Elements and of the Archimedean corpus, both of which he intended to print in corrected versions. He read very carefully Campanus of Novara’s recension of Euclid’s Elements, catching problems in the Latin and in Campanus’s attempts to make sense of them. His philological and codicological sensibilities also shaped his mathematics, as they did his astronomy. Thus he disapproved of Campanus’s inclusion of the “parallel postulate” among the postulates, believing—probably on the evidence from a Greek manuscript—that it belonged among the axioms (Folkerts, 1996, pp. 94–96, 104–105, 108). His several hundred corrections of and comments on the translations of Archimedes by Jacobus Cremonensis also drew on his study of several Greek manuscripts (Clagett, 1978, pp. 357–365).

The last phase of Regiomontanus’s forty-year life focused heavily on his Nürnberg press (Stromer, 1980). He planned to devote it overwhelmingly to significant and philologically correct mathematical, astronomical, and astrological works, including some of his own, and it probably motivated his large personal library (see Kremer, 2004). Although his output was modest, he pioneered the printing of complex astronomical tables and geometrical diagrams imbedded in the text in a single pull of the press. Unpublished research on his output shows Regiomontanus’s active involvement in the day-to-day operation. A survey of the twenty-four extant copies of his Disputationes contra deliramenta cremonensia (c. 1475) reveals that he stopped the press for aesthetic resettings of pages (the hypercorrect Latin orthography can be only his) and for corrections of typographical errors. He even made systematic pen corrections of errors missed during the press run (as he also did in the Ephemerides). This attention to detail illustrates his hopes for the production of texts that would be correct philologically as well as typographically. In terms of diffusion, the most important reader of Regiomontanus’s press was Erhard Ratdolt, whose presses in Venice and Augsburg reissued many of Regiomontanus’s works and adopted much of his unrealized printing program.


Clagett, Marshall. Archimedes in the Middle Ages. Vol. 3. Philadelphia: American Philosophical Society, 1978, pp. 357–365.

Duhem, Pierre. Le Système du monde: histoire des doctrines cosmologiques de Platon à Copernic. Vol. 9. Paris: Hermann, 1913–1959, pp. 237–323.

Folkerts, Menso. “Regiomontanus als Mathematiker.” Centaurus 21 (1977): 214–245.

———. “Regiomontanus’s Role in the Transmission and Transformation of Greek Mathematics.” In Tradition, Transmission, Transformation: Proceedings of Two Conferences on Pre-modern Science held at the University of Oklahoma, edited by F. Jamil Ragep, Sally Ragep, and Steven Livesey, 89–113. Leiden, Netherlands: Brill, 1996.

———.“Regiomontanus’s Role in the Transmission of Mathematical Problems.” In From China to Paris: 2000 Years Transmission of Mathematical Ideas, edited by Yvonne Dold-Samplonius, et al., 411–428. Stuttgart, Germany: Franz Steiner Verlag, 2002.

Gerl, Armin. Astronomisches Rechnen kurz vor Copernicus: Der Briefwechsel Regiomontanus-Bianchini. Boethius, vol. 21. Stuttgart, Germany: Franz-Steiner Verlag, 1989.

Grössing, Helmut. Humanistische Naturwissenschaft: Zur Geschichte der Wiener mathematischen Schulen des 15. und 16. Jahrhunderts. Saecula Spiritalia, vol. 8. Baden-Baden, Germany: Verlag Valentin Koerner, 1983.

Hamann, Günther, ed. Regiomontanus-Studien. Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1980.

King, David A., and Gerard L’E. Turner. “The Astrolabe Presented by Regiomontanus to Cardinal Bessarion in 1462.” Nuncius 9 (1994): 165–206.

Kremer, Richard L. “Text to Trophy: Shifting Representations of Regiomontanus’s Library.” In Lost Libraries: The Destruction of Great Book Collections since Antiquity, edited by James Raven, 75–90. London and New York: Palgrave Macmillan, 2004.

Kren, Claudia. “Homocentric Astronomy in the Latin West: The De reprobatione ecentricorum et epiciclorum of Henry of Hesse.” Isis 59 (1968): 269–281.

———. “A Medieval Objection to ‘Ptolemy.’” British Journal of the History of Science 4 (1969): 378–393.

———. “Planetary Latitudes, the Theorica Gerardi, and Regiomontanus.” Isis 68 (1977): 194–205.

Monfasani, John. George of Trebizond: A Biography and a Study of His Rhetoric and Logic. Leiden, Netherlands: Brill, 1976.

Ragep, F. Jamil. “Ali Qushji and Regiomontanus: Eccentric Transformations and Copernican Revolutions.” Journal for the History of Astronomy 36 (2005): 359–371.

Rigo, Antonio. “Bessarione, Giovanni Regiomontano e i loro studi su Tolomeo a Venezia e Roma (1462–1464).” Studi veneziani, n.s. 21 (1991): 83–95.

Rose, Paul Lawrence. The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo. Travaux d’Humanisme et Renaissance, vol. 145. Geneva: Librairie Droz, 1975.

Schmeidler, Felix, ed. Johannes Regiomontani opera collectanea. Osnabrück, Germany: O. Zeller Verlag, 1972. Facsimiles of key works, including the Epitome, Disputationes, De triangulis omnimodis, etc.

Shank, Michael H., “The ‘Notes on al-Bitruji’ Attributed to Regiomontanus: Second Thoughts.” Journal for the History of Astronomy 23 (1992): 15–30.

———. “Regiomontanus and Homocentric Astronomy.” Journal for the History of Astronomy 29 (1998): 157–166.

———. “Rings in a Fluid Heaven: The Equatorium-Driven Physical Astronomy of Guido de Marchia (fl. 1309).” Centaurus 45 (2003): 175–203.

———. “Regiomontanus as a Physical Astronomer: Samplings from the Defense of Theon against George of Trebizond.” Journal for the History of Astronomy 38 (2007) (forthcoming).

Stromer, Wolfgang von. “Hec opera fient in oppido Nuremberga Germanie ductu Ioannis de Monteregio: Regiomontan und Nuremberg, 1471–1475.” In Regiomontanus-Studien, edited by Günther Hamann, 271–272. Vienna: Verlag der Österreichischen Akademie der Wissenschaften, 1980.

Swerdlow, Noel. “The Derivation and First Draft of Copernicus’s Planetary Theory: A Translation of the Commentariolus with Commentary.” Proceedings of the American Philosophical Society 117 (1973): 423–512.

———. “Regiomontanus on the Critical Problems of Astronomy.” In Nature, Experiment and the Sciences, edited by Trevor H. Levere and William R. Shea, 165–195. Boston Studies in the Philosophy of Science, vol. 120. Boston and Dordrecht, Netherlands: Kluwer, 1990.

———. “Science and Humanism in the Renaissance: Regiomontanus’s Oration on the Dignity and Utility of the Mathematical Sciences.” In World Changes: Thomas Kuhn and the Nature of Science, edited by Paul Horwich. Cambridge, MA: MIT Press, 1993.

———. “Regiomontanus’s Concentric-Sphere Models for the Sun and Moon.” Journal for the History of Astronomy 30 (1999): 1–23.

Toomer, G. J. Ptolemy’s Almagest. New York: Springer, 1984.

Wingen-Trennhaus, Angelika. “Regiomontanus als Frühdrucker in Nürnberg.” Mitteilungen des Vereins für Geschichte der Stadt Nürnberg 78 (1991): 17–87.

Zinner, Ernst. Regiomontanus: His Life and Work. Studies in History and Philosophy of Mathematics, vol. 1. Translated by Ezra Brown. Amsterdam: North-Holland 1990.

Michael H. Shank

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The German astronomer and mathematician Regiomontanus (1436-1476) constructed the first European observatory and established trigonometry as a separate area of study in mathematics.

Regiomontanus, called after the Latinized form of his birthplace, Königsberg, in the duchy of Coburg, was born Johann Müller on June 6, 1436, the son of a miller. At the age of 12 he began the study of classical languages and mathematics at the University of Leipzig. In 1452 he moved to Vienna and became the favorite pupil of Georg Peurbach, astronomer and mathematician, who interested Regiomontanus in securing a truly reliable version of Ptolemy's Almagest.

A year after Peurbach's death in 1461, Regiomontanus went to Italy and established close contacts with Cardinal Bessarion, the leading Greek scholar of the time. Regiomontanus made quick progress in Greek and studied various Greek mathematical and astronomical texts in addition to Ptolemy's Almagest. The study of this latter work enabled him to complete Peurbach's Epitome in Cl. Ptolemaei magnam compositionem, but it saw print only in 1496.

The most important work of Regiomontanus, completed in 1464 but printed in 1533, was the first fullfledged monograph on trigonometry, De triangulis omnimodis libri quinque (Five Books on All Kinds of Triangles). The first two books dealt with plane trigonometry, while the rest were largely devoted to spherical trigonometry. Although Regiomontanus relied heavily on Arabic and Greek sources, such as al-Battani, Nasir al-Din al-Tusi, Menealos, Theodosius, and Ptolemy, his work was the starting point of a new development leading to modern trigonometry.

In 1468 Regiomontanus went to the court of King Matthias Corvinus of Hungary at Buda to serve as librarian of one of the richest collections of codices in existence in Europe. There he completed his Tabulae directionum et projectionum, the first European study of Diophantes' Algebra.

In 1471 Regiomontanus went to Nuremberg at the invitation of Bernhard Walther, a rich citizen who provided him with the means to set up the first observatory in Europe. It was equipped with instruments of Regiomontanus's own making, which he described in Scripta de torqueto, astrolabio armillari, first printed in 1544. His most important observations concerned the great comet of 1472 (probably Halley's comet). Walther also set up a printing press and published Regiomontanus's calendars and pamphlets. Regiomontanus published Peurbach's planetary theory, Theoricae novae planetarum, and his own ephemerides for 1474-1506, which contained a method of calculating longitudes at sea on the basis of the motion of the moon. The book was used by the leading navigators of the times.

At the summons of Pope Sixtus IV, Regiomontanus, a newly appointed titular bishop of Ratisbon, journeyed to Italy in the fall of 1475 to undertake the reform of the calendar. He died on July 6, 1476, probably the victim of an epidemic.

Further Reading

There is a chapter on Regiomontanus in Lynn Thorndike, Science and Thought in the Fifteenth Century (1929). Also useful are J. L. E. Dreyer, A History of Astronomy from Thales to Kepler (1905; rev. ed. 1953); Lynn Thorndike, A History of Magic and Experimental Science vols. 5 and 6 (1941); and A. C. Crombie, Augustine to Galileo: The History of Science A.D. 400-1650 (1953). □

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Regiomontanus (rē´jēōmŏn´tā´nəs) [Lat.,=belonging to the royal mountain, i.e., to Königsberg], 1436–76, German astronomer and mathematician, b. Königsberg. His original name was Johannes Müller. In 1461 he went to Rome with Cardinal Bessarion and learned Greek in order to translate Greek writings. In 1468 he was called to the court of the king of Hungary to make a collection of Greek manuscripts, and three years later he settled at Nuremberg, where, with his pupil and patron, Bernhard Walther, he established an observatory and a printing press. Among other works they published the Ephemerides for the years 1474–1506, calculated by Regiomontanus, and Georg von Purbach's Theoricae planetarum novae. Summoned by Pope Sixtus IV, Regiomontanus went to Rome in 1475 to assist in reforming the calendar and was made bishop of Regensburg. He died in Rome. He made improved instruments, both mathematical and astronomical, introduced algebra into Germany, and did much to further trigonometry.

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