John of Gmunden

(b. Gmunden am Traunsee, Austria, ca. 1380- 1384; d. Vienna, Austria, 23 February 1442)

astronomy, mathematics, theology.

John of Gmunden’s origins were long the subject of disagreement. Gmunden am Traunsee, Gmü nd were all thought to be possible birthplaces; and Nyder (Nider), Schindel, Wissbier, and Krafft possible family names. Recent research in the records of the Faculty of Arts of the University of Vienna, however, appears to have settled the question. The Vienna matriculation register records the entrance, on 13 October 1400, of an Austrian named “Johannes Sartoris de Gmundin,” that is, of the son of a tailor from Gmunden.¹ He was surely the “Johannes de Gmunden” who was admitted to the baccalaureate examination on 13 October 1402.² If he was the astronomer John of Gmunden, who was accepted as master into the Faculty of Arts on 21 March 1406, along with eight other candidates, then he spent all his student years at Vienna.³ His birthplace could only have been Gmunden am Traunsee, since Gmü nd and Schwä bisch Gmü nd were then known only as “Gamundia”, the locality on the Traunsee, which even in Latin sources, is called “Gmunden” (Gemunden until 1350).⁴ Schwä bisch Gmü nd must be eliminated from consideration because our John of Gmunden was an examiner of Austrian students, which means he had to be of Austrian birth. The family names Nyder and Schindel can be excluded, but that of Krafft is better established. In his wrings and the records of his deanship of the Arts faculty, carefully written in his own hand, he calls himself Johannes de Gmunden exclusively, thus clearly he never used a family name.⁵

John of Gmunden’s career can be divided into four periods. In the first (1406- 1416) his early lectures —besides one given in 1406 on “Theorice” —were devoted to nonmathematical subjects: “Physica” (1408), “Metheora” (1409, 1411), “Tractatus Petri Hyspani” (1410), and “Vetus ars” (1413).⁶ On 25 August 1409 he became magister stipendiatus and received an appointment at the Collegium Ducale.⁷ He gave his first mathematics lecture in 1412. He was also interested in theology, the study of which he completed in 1415 as “Baccalaureus biblicus formatus in theologia.” Two lectures in this field concerned the Exodus (1415) and the theology of Peter Lombard (1416).

In 1416- 1425 John of Gmunden lectured exclusively on mathematics and astronomy, which led to the first professorship in these fields at the University of Vienna; the position became permanent under Maximilian I. When John became ill in 1418, he lost his salary, since only someone actively teaching (magister stipendiatus legens) could be paid; but, at the request of the faculty, the duke removed this hardship. John obtained permission to hold lectures in his own house—a rarely granted privilege.⁸ During his years at the university he held many honorary offices. He was dean twice (1413 and 1423) and examiner of Saxon (1407), Hungarian (1411), and Austrian (1413) students.⁹ In 1410 he was named publicus notarius.¹⁰ In 1414 he was receiver (bursar) of the faculty treasury and member of the dormitory committee ¹¹ for the bylaws for the burse. In 1416 he was “Conciliarius of the Austrian nation,” and from 1423 to 1425 he was entrusted with supervising the university’s new building program.¹²

The third period (1425- 1431) began when John of Gmunden retired from the Collegium Ducale and, on 14 May 1425, became canon of the chapter of St. Stephen.¹³ Previously he had been ordained priest (1417) and delivered sermons.¹⁴ He was also vicechancellor of the university, which had long been closely associated with St. Stephen’s Gymnasium.¹⁵ Henceforth John devoted himself to writings on astronomy, astronomical tables, and works on astronomical instruments. He also lectured on the astrolabe.¹⁶

In the last period (1431- 1442) John became plebanus in Laa an der Thaya, an ecclesiastical post that yielded an income of 140 guldens.¹⁷ In 1435 he wrote his will, in which he bequeathed his books (particularly those he himself had written) and instruments to the library of the Faculty of Arts. He also gave precise instructions for their use.¹⁸ We note the absence from his list of those books on which his own works were based, but these undoubtedly were in the Faculty of Arts library. John died on 23 February 1442 and was probably buried in St. Stephen’s cathedral; no monument indicates where he was laid to rest. Moreover, we possess no likeness of him except for an imaginative representation that shows him wearing a full- bottomed wig.¹⁹

John of Gmunden’s work reflects the goal of the instruction given in the Scholastic universities: to teach science from existing book, not to advance it. He was above all a teacher and an author. His mathematics lectures were entitled “Algorismus de minutii”s (1412, 1416, 1417), “Perspectiva” (1414), “Algorismus de integris” (1419); and “Elementa Euclidis” (1421).²⁰ In this series one does not find a lecture on latitudines formarum, which was already part of the curriculum and a topic on which John’s teacher, Nicolaus of Dinkelsbühl, had lectured in 1391. John’s main concerns, however, lay in astronomy, and thus even in his mathematical writings he treated only questions of use to astronomers. Tannstetter cites three mathematical treatises by him: an arithmetic book with sexagesimal fractions, a collection of tables of proportions, and a treatise on the sine.²¹ Only the first of these was printed, appearing in a compendium containing a series of writings that provided the basis for mathematics lectures.²² In this area John introduced no innovations.²³ For example, in extracting the square root of a sexagesimal number, he first transformed the latter into seconds, quarters, and so forth (therefore into minutes with even index), added an even number of zeros (cifras in order to achieve greater accuracy, then extracted the root, divided through the medietates cifrarum, and expressed the result sexagesimally (thus . Even the summation of two zodiac signs of thirty degrees into a signum physicum of sixty degrees had appeared earlier.²⁴ The treatise on the sine was recently published.

John’s other mathematical works are contained in manuscipt volumes that he himself dated (Codex Vindobonensis 5151, 5268).²⁵ From his writings on angles, arcs, and chords it is clear that Arabic sine geometry was known in Vienna. Yet it is doubtful whether—as has been asserted²⁶—John was also acquainted with the formula (corresponding to the cosine law) .²⁷ The formula, which was employed in calculating the sun’s altitude for every day of the year, was discovered by Peurbach with “God’s help.” ²⁸

John of Gmunden’s work in astronomy was of greater importance than his efforts in mathematics. It was through his teaching and writings that Vienna subsequently became the center of astronomical research in Europe. His astronomy lectures were entitled “Theoricae planetarum” (1406, 1420, 1422, 1423), “Sphaera meterialis” (1424), and “De astrolabio” 1434).²⁹ He probably did not himself make any systematic observations of the heavens, but his students are known to have done so. The instruments that he had constructed according to his own designs were used only in teaching and to determine time.³⁰ He was no astrologer, as can be seen from a letter of September 1432 to the prior Jacob de Clusa, who had made predictions on the basis of planetary conjunctions.³¹ Although his library contained numerous astrological writings, the stringent directions in his will pertaining to the lending of these dangerous works show what he thought of this pseudoscience. If occasionally he spoke of the properties of the zodiac signs and of bloodletting, it was because these subjects were of particular interest to purchasers of almanacs.³²

The great number of extant manuscripts of John of Gmunden’s works attests his extensive literary activity, which began in 1415 and steadily intensified until his death. Many of the manuscripts are in his own hand, such as those in Codex Vindobonensis 2440, 5151, 5144, and 5268.³³ Most of those done by students and other scribes date from the fifteenth century. Only a small portion (about twenty of the total of 238 manuscripts that Zinner located in the libraries of Europe) are from a later period. This indicates that his works were superseded by those of peurbach and Regiomontanus. John’s writings can be divided into tables, calendars, and works on astronomical instruments.

John of Gumunden produced five versions of his tabular works, which contained tables of the motions of the sun, the moon, and the planets, as well as of eclipses and new and full moons. They also included explanatory comments (tabulae cum canonibus). They are all enumerated in his deed of gift. Regiomontanus studied the first of the tabular works (in Codex Vindobonensis 5268) and found an error in it that he noted in the margin.³⁴ John was also the author of many individual writings on astronomy that were not made part of the tabular works. Shorter than the latter (with the exception of the tables of eclipses), they contained tables of planetary and lunar motions, of the true latitudes of the planets (with explanations), and of the true new and full moons, as well as tables of eclipses.³⁵ Further astronomical writings can be found in his works on astronomical instruments.

Along with elaborating and improving the values of his tables, John of Gmunden was especially concerned with the preparation of calendars, which provided in a more usable form the information contained in the tabular works. In addition to such astronomical data, they included the calendar of the first year of a cycle with saints’ days and feast days, dominical letter, and the golden numbers, so that the calendar could be used during all nineteen years of a cycle.³⁶ He brought out four editions of the calendar: the first covered 1415- 1434; the second, 1421- 1439; the third, 1425- 1443; and the fourth, 1439- 1514.³⁷ The fourth edition was printed on Gutenberg’s press in 1448.³⁸ Two other calendars bearing John’s name were published later. From one of them, a xylographic work, there remains only a woodblock; of the other, a peasants’s calendar, only a single copy is extant.³⁹

John of Gmunden’s third area of interest was astronomical instruments; he explained how they operated and gave directions for making them. In his deed of gift he mentioned two works in this field: a volume bound in red parchment containing Astrolabium Alphonsi and a little book written by himself, entitled Astrolabii quadrantes.⁴⁰ In his will he lists the following instruments: a celestial globe (sphaera solida), an “equatorium” of Campanus with models taken from the Albion (devised and written about by) Wallingford), an astrolabe, two quadrants, a spherea materialis, a large cylindrical sundial, and four “theorice lignee”⁴¹ He stipulated that these instruments should be kept well and seldom loaned out—the equatorium only very seldom (rarissime). Of all this apparatus nothing remains in Vienna. ⁴² On the other hand, about 100 manuscripts of his treatises on astronomical instruments have been preserved. To date no one has made a study of these manuscripts (which contain other works on astronomy) thorough enough to establish, in detail, what John took from his predecessors and what he himself contributed. The instruments he discussed, with regard to both their theoretical basis and their production and use, were the following:

1. The astrolabe. The text is composed of fourteen manuscripts.⁴³ The star catalog joined to one of them indicates that the first version dates from 1424.⁴⁴

2. The quadrant. John’s treatise on the quadrant exists in three versions.

Quadrant I: Fifteen manuscripts are extant of this version of the work, which dates from 1424- 1425.⁴⁵ Here he drew on a revision from 1359 of the Quadrans novus of Jacob Ibn Tibbon (also known as Jacob ben Mahir or Profatius Judaeus) from 1291- 1292 and on a revision from 1359.⁴⁶ To these John appended an introduction and remarks on measuring altitudes.⁴⁷ Several of the manuscripts also contain additional data that he presented in 1425: a table of the true positions of the sun at the beginnings of the months, star catalogs, and a table of the sun’s entrance into the zodiac signs.⁴⁸

Quadrant II: A second, more elaborate version of the work on the quadrant exists in only one manuscript; it no doubt stems from a student, who speaks of a “tabula facta a Johanne de Gmunden, 1425.” ⁴⁹

Quadrant III. This version, which is independent of John’s other writings on the topic, is known in thirteen manuscripts. ⁵⁰ One of them is dated 1439.⁵¹ Many of the manuscripts contain tables of solar altitudes for every half month and for various localities (calculated or taken from the celestial globe) as well as tables for the shadow curves of cylindrical sundials.⁵²

3. The albion. This universal device (Ldquo; all- by- one”), which combined the properties of the instruments used for reckoning time and location, was devised and built by Richard of Wallingford.⁵³ His treatise on it (1327) was revised by John of Gmunden, to whom it is often incorrectly attributed. Several manuscripts contain further additions by John, such as a star catalog for 1430 and also (most probably by him) instructions (1433) on the use of the albion in the determination of eclipses. ⁵⁴ He had earlier used the instrument for this purpose for 1415- 1432.⁵⁵

4. The equatorium. This instrument, made of either metal or paper, could represent the motions of the planets. ⁵⁵ It is found in the thirteenth century in the writings of Campanus and in those of John of Lignè res and Ibn Tibbon.⁵⁷ John called the device “instrumentum solemne” ⁵⁸ Following Campanus, he set forth its theoretical basis and described how to make and use it in a work published at the University of Vienna that was highly regarded by Peurbach and Regiomontanus.⁵⁶ The manuscripts occasionally also present tables of the mean motion of the sun and moon for 1428.⁶⁰

5. The torquetum. This instrument, whose origin is uncertain, was the subject of a treatise by a Master Franco de Polonia (Paris, 1284); it exists in manuscripts of the fourteenth and fifteenth centuries.⁶¹ John completed the treatise with an introduction and a conclusion. In the latter he stated that with the “turketum” one can determine the difference in longitude between two localities.

6. The cylindrical sundial. The origin of this instrument is likewise unknown; ir is described as early as the thirteenth century in an Oxford manuscript.⁶² John introduced it to Vienna. His work on it, Tractatus de compositione et usi cylindri, exists in nineteen manuscripts.⁶³ From these it can be inferred that he composed his treatise between 1430 and 1438. He calculated the shadow curves at Vienna, taking the latitude as π = 47° 46′; in the Oxford manuscript π is taken as 51° 50é. Shadow curves for other localities also appear in the manuscripts. ⁶⁴

A further work on sundials and nocturnals was written by John or by one of his students.⁶⁵

The study of mathematics and astronomy beyond what was offered in the quadrivium first became possible at Vienna through the efforts of Henry of Hesse, who brought back from Paris knowledge of the recent advances in mathematics (as is reported by Petrus Ramus ⁶⁶) The first evidence of this is found in the work of Nicolaus of Dinkelsbö hl, who taught, besides Sacrobosco’s astronomy and Euclid’s Elements, Oresme’s “latitudines formarum.” His last lecture (1405), on theories of the planets,⁶⁷ may have stimulated the young John of Gmunden to study astronomy. In any case, John studied the relevant available writings, transcribed them, and frequently added to them. Although he seldom mentioned his predecessors, his sources can be inferred to some extent. He was acquainted with the Alphonsine and Oxford tables and knew Euclid from the edition prepared by Campanus, to whose ideas on planetary theories he subscribed.⁶⁸ In his first tabular work John cited Robert the English- man.⁶⁹ Moreover, his Algorismus de minuciis phicis undoubtedly follows the account of John of John of Lignères. In 1433 he transcribed and completed John of Murs’s treatise on the tables of proportions; hence many manuscripts name him as the author.⁷⁰ It is not clear to what extent his treatise on the sine, arc, and chord depended on the work of his predecessors (Levi ben Gerson, John of Murs, John of Ligné res, and Dominic de Clavasio). His dependence on earlier authors is most evident in his writings on astronomical instruments (Campanus’ equatorium, Ibn Tibbon’s new quadrants, Richard of Wallingford’s albion, and Franco de Polonia’s torquetum). In his will he also mentions a work entitled Astrolabium Alphonsi.

John of Gmunden was influential both through his teaching and, long after his death, through his writings. Among his students Tannstetter mentions especially Georg Pruner of Ruspach. There are transcriptions (in London) by the latter of John’s works with remarks by Regiomontanus.⁷¹ “John’s co-workers included Johann Schindel, loannes Feldner, and Georg Miistinger, prior of the Augustinian monastery in Klosterneuburg”⁷² Fridericus Gerhard (d. 1464- 1465), of the Benedictine monastery of St. Emmeran in Regensburg, also had connections with the Vienna school; however, they were indirect, being based on his contacts with Master Reinhard of Kloster- Reichenbach, who worked at Klosterneuburg. Gerhard was a compiler of manuscript volumes that reflected the mathematical knowledge of the age; in them he included works by John of Gmunden, drawn partially from a lecture notebook of 1439.⁷³ Gerhard was particularly interested in geography; and geographical coordinates play a role in astronomy. Thus it is quite possible that his knowledge in this area also came from Vienna.⁷⁴

John of Gmunden greatly influenced Peurbach. To be sure, the latter cannot be considered a direct student of John’s since Peurbach was nineteen when John died. Nevertheless, he undoubtedly knew John personally, and he studied his writings thoroughly. The same is known of Regiomontanus, who in his student years at Vienna made critical observations in copies of John’s works.⁷⁵ He also bought a copy of the treatise on the albion.⁷⁶

The outstanding achievements of Peurbach and Regiomontanus resulted in John of Gmunden’s being overshadowed, as can be seen from the small number of manuscripts of his works that date from after the fifteenth century. Yet it was he who initiated the tradition that was established with Peurbach, whose scientific reputation caused the young Regiomontanus to come to Vienna (1450) instead of staying longer at Leipzing.

NOTES

1. Matrikel der Universitä t Wien, I, p. 57. Another John Sartoris de Gmunden is listed in the register for 14 April 1403 (I, p. 65). He cannot yet have been a master in 1406. He may have been the son of another tailor, or it may be a second matriculation.

2. Acta Facultatis artium Universitatis Viennensis 1385- 1416, p. 212. (Cited below as AFA.)

3. AFA, p. 261. John of Gmunden is the only one of all the candidates for whom no family is given; but if it is true (letter from P. Uiblein) that the lectures were distributed to the magistri legentes in the same sequence in which they received their magisterium, then John is identical with Krafft. Johannes Wissbier of Schwä bisch Gmü nd studied at Ulm in 1404. See M. Curtze, “ü ber Johann von Gmunden,” in Bibliotheca mathematica, 10 , no. 2 (1896), 4.

4. R. Klug, Johann von Gmunden, p. 14. Several variants exist in the MSS, including Gmund, Gmunde, Gmundt, and Gmundia. Concerning the two over the “u” in “Gmü nden” see ibid., 16. It is not an Umlaut but a vowel mark; on this point see H. Rosenfeld, in Studia neophilologica, 37 (1965), 126 ff., 132 f.

5. The name Nyder does not appear, as has been asserted, in the obituaries of the canons of St. Stephen. See J. Mundy, “John of Gmunden,” p. 198, n. 29. Regarding Schindel, it is a matter of a change of name in the Vienna MSS. See Klug, op. cit., p. 17, Curtze, op. cit., p.4. In 1407 this scholar left Prague for Vienna, where he taught privately for several years. He is mentioned with praise along with Peurbach and Regiomontanus by Kepler in the preface to the Rudolphine Tables. a Mster Johannes Krafft lectured on the books of Euclid around in 1407 (AFA, p. 281). Others also lectured on Euclid around this time (AFA, pp. 253, 292, 453).

6. AFA, pp. 292, 325, 365, 338, and 401.

7. AFA, p. 324: “magistro Iohanni de Gmunden data fuit regencia”; Klug, pp. 18 f.

8. J. Aschbach, Geschichte der Wiener Universitä t im ersten Jahrhundert ihres Bestehens, p. 457.

9. On his deanship see AFA, p. 405; Aschbach, op. cit., p. 458. On his posts as examiner see AFA, pp. 284, 370, 402. In 1413 he was also examiner of candidates for the licentiate (AFA, p. 391).

10. AFA, p. 345.

11. AFA, p. 421.

12. AFA, p. 472; Aschbach, loc. cit

13. AFA, p. 530.

14. Mundy, op. cit., p. 199; Klug, op. cit., pp. 20 f.

15. Aschbach, op. cit., p. 459 ; AFA, p. 530 .

16. Klug, op. cit., p. 18 . Even if he retired in 1434, he was already a clergyman in Laa.

17. AFA, p. 530 ; Klug, op. cit ., p. 21 .

18. The text of the will is given in Mundy, op. cit., p. 198; Klug, op. cit., pp. 90 ff.

19. E. Zinner, Leben und Wirken des Johann Müler von Königsberg, genannt Regiomontanus, p. 196, n. 16. (Cited below as ZR.

20. AFA, pp. 381, 430; Klug, op. cit., p. 18. See also Note 5.

21. G. Tannstetter, in Tabulate eclypsium Magistri Georgii Peurbachii, Wrote: “Libellum de arte calculandi in minuciis phisicis, Tabulas varias de parte proporcionali, Tractatum sinuum’ (fol. aa 3^v).

22. The title of the compendium is Contenta in hoc libello; on this point see D. E. Smith, Rara arithmetica, p. 118.

23. See Simon Stevin, De thiende, H. Gericke, K. Vogal, ed., in Oswald’s Klassiker der Exacten Wissenchaften, n.s. 1 (Braunchwieg, 1965), pp.47 ff.

24. Mundy, op. cit., p. 199.

25. The Treatise on tables of proportions, along with explanatory comments, is cited in E. Zinner, Verzeichnis der astronomischen Handschriften des deutschen Kulturgebietes, nos. 3585, 3586, 3695, and 3696 (dated for 1433 and 21 May 1440); two treatises on the sine, chord and arc are noted in this work by Zinner (cited below as ZA) as nos. 3591 and 3592; the letter (from Codex Vindobonensis 5268) was published by Busard (Vienna, 1971); then follows a work on Euclid, also by John of Gmunden.

26. Kludg, op. cit., p. 50.

27. Here H= meridian altitude, b= semidiurnal arc, and t= horary angle.

28. Codex Vindobonensis 5203, fol. 54r; ZR, p. 25—here cos b should be altered to (cos b—1).

29. Klug, op. 18.

30. ZR, p. 15. The Tabula de universo undoubtedly was also conceived for use at the university; on this point see Mundy, op. cit., p. 206; Klug, op.cit., pp.63 ff.

31. ZA, no. 3584.

32. ZA, no. 3732; Mundy, op, cit., p.204.

33. ZR, P. 16.

34. The MSS of the tabular works are as follows:

Version I: Codex Vendoboenesis 5268, fols. lv- 34r (ZA, no. 3587; ZA, no. 3588 is an extract), with explanation in ZA, no. 3589 (from 10 August 1437); the tables were valid for Vienna during 1433, 1436, and 1440, among other years. For a marginal notation by Refiomontanus (“non valet. Nam in alio circulo sumitur declinatio et in alio latitudo”) 0 see Zr, p. 43

Version II: ZA, no. 3709 (for 1400.)

Version III: ZA, nos. 3710- 3716, also ZA, no. 3719. MS 3711 (in Codex Vindobonensis 5151) was written by John of Gmunden himself, as was ZA, no. 3694 (of 20 May 1440).

Version IV: ZA nos. 3717, 3718. A student’s transcription in a MS at the British Museum is dated 1437; on this point see Munday, op. cit., p.202, n.73.

Version V: ZA, nos. 3691- 3693, with explanatory material at nos. 3688- 3690 (for 1440, 1444, 1446)

35. On planetary and lunar motions see ZA, nos. 3496, 3720, 3721, 3723; moreover, the tables in ZA, nos 11203, 11204, and 11207 “quamvis de motibus mediis. . .”) stem from John of Gmunden (ZA, p.523). On the true planetary latitudes see ZA. nos. 3697, 3699 (from 21 and 25 may 1440), 3698, 3700. On the true new and full moons see ZA, nos. 3702, 3707, 3708, 3733, 3734. Tables of eclipses are at ZA, nos. 3498, 3701, 3703- 3706, 3735 (for 1433 and 1440). Further works by him are undoubtedly ZA, nos. 3590 (astronomy0, 3725 (position of the heavenly spheres), and 3729 (intervals between heavenly bodies0. Codex latinus, monaiensis 10662, fols. 99v- 102r, contains a “Tabula stellarum per venerabilem Joh. de Gmunda” for 1430, and Codex latinus monaiensis 8950, fols 81r- 92v, a treatise on the “radices” of the sun, moon, and “Caput draconis.” See Mundy, op.cit., p.201, n. 72

36. Tannstetter (fol. aa 3^v) records that he left to the library a “Kalendarium quod multis sequentibus annis utile erat et jucundissimum” (perhaps ZA, no.3606)

37. lst calendar: ZA, nos. 3499- 3502 (four MSS). 2nd calendar (nine MSS): ZA, nos. 3503- 3511, 5378. This was the calendar for whose publication John of Gmunden obtained permission. See Mundy, op. cit., p. 201, n. 71; Klug, op. cit., p. 91.

3rd calendar (fifteen MSS): Za, nos. 3512- 3526; no. 3513 dates from 1431.

4th calendar: Exists in eighty MSS (Zinner, in 1938, knew of ninety-nine copies [ZR, p. 15]): ZA, nos. 3606- 3687, of which three date from 1439; this calendar was announced at the University of Vienna (ZA, p. 425). An extract with explanation by John of Gmunden exists in MS 12118 (ZA, p. 536).

38. J. Bauschinger and E. Schrö der, “Ein neu endeckter astronomischer Kalender fö r das Jahr 1448.”

39. Mundy. oop. cit., p. 203; Klug. op. cit., pp. 79 ff. (with illustrations on p. 81 and plates VIII and IX); Aschbach, op. cit., pp. 465 ff.

40. Each astronomical instrument served to “grasp the stars”(λαεβáν€Lν Tà ǎσTρα)

41. Tannstetter (fol. aa 3°) simply groups Campanus’ instrument (“equatorium motuum planetarum ex Campano transsumptum”) and almost all the others as “Compositio Astrolabii & utilitates eiusdem & quorundam aliorum instrumentorum.”

42. An ivory quadrant in the Kunsthistorisches Museum in Vienna was undoubtedly designed by John of Gmunden. See ZA, p. 16; Klug, op. cit., p. 26.

43. ZA, nos. 3593, 3593a- 3605.

44. ZA, no. 3593; ZA, no. 3602 contains still another star catalog as well as tables for the rising of the signs and for the entrance of the sun into the sings for the year 1425.

45. ZA. nos. 3555- 3569.

46. ZA, p. 424. On Ibn Tibbon see R. T. Gunther, Early, Science in Oxford, II, 164.

47. ZA, p. 468.

48. ZA, nos. 3556, 3557, 3569 for the table of true positions; ZA, nos. 3557, 3559, 3561, 3562 for the star catalogs; and for the sun’s entrance into the zodiac signs ZA, nos. 3357 and 3564 for 1424; no. 3568 for 1425; and no. 3569.

49. ZA. no. 3570; additional material in ZA, no. 3724 contains tables of equatorial altitudes. In the same volume of MSS there is also an essay on solar quadrants that is by either John of Gmunden or a student of his (ZA, no. 3731).

50. ZA, nos. 3571- 3583.

51. ZA, no. 3578.

52. For solar attitudes see ZA, nos. 3572, 3578, 3580 for Vienna, Nuremberg, Klosterneuburg, Prague, Venice, Rome, and the town of “Cö ppt; for tables of shadow curves see ZA, no. 3578 for the places named and for Regensburg. In addition, nos. 3577, 3579, 3583 have tables for Vienna; and no. 3576 has them for Vienna and Nuremberg.

53. Gunther, op. cit., pp. 49 f., 349ff.; ZA, nos. 11584- 11586; p. 52g.

54. ZA, nos. 11590- 11593, 11596, and p. 529.

55. Za, no. 3498.

56. Perhaps the wooden instruments (theorice lignee) named in the will are such equatoria.

57. Gunther, op. cit., p. 234.

58. Za, no. 3527- 3535; s. 2A p. 423.

59. John of Gmunden’s explanation of Campanus’s work is Za, no. 1912; construction and use of the instrument is at ZA, p. 423.

60. ZA, nos. 3527- 3531.

61. ZA, nos. 2787- 2800 and p. 416. See also G. Sarton, Introduction to the History of Science, II, 1005 and III, 1846; and Gunther, op, cit., pp. 35, 370 ff.

62. Gunther, op. cit., p. 123. There had been a MS in Germany since the fourteenth century.

63. ZA, nos. 3536- 3554.

64. Venice, Rome, Nuremberg, Prague, and Klosterneuburg; they stem in part from John of Gmunden and in part from Prior Georg. Moreover, tables of solar altitudes for Vienna, Nuremberg, and Prague are appended to some of the MSS (za, p. 424).

65. Za, nos. 3722, 3726, 3727, 3730.

66. Petrus Ramus, Mathematicarum scholarum, libri duo (Basel, 1569), p. 64: “Henricus Hassianus. . . primo mathematicas artes Lutetia Viennam transtulit” (Aschbach, op. cit., p. 386, n. 3).

67. AFA, p. 253.

68. A star catalog is completed in ZA, no 452. Sec also ZA, p. 390. The tables of the planetary latitudes in ZA, no. 3700 were taken from the Oxford tables (ZA, p. 426). On Campanus’ edition of Euclid, see ZA, no. 1912 and p. 405.

69. ZR, p. 14; Mundy, op. cit., p. 200.

70. ZA, no. 7423; on his being considered the author, see ZA, p. 475.

71. ZR, pp. 15, 43; Mundy, op. cit., pp. 197, 202, n. 73.

72. ZA, p. 529: “selder”; Mundy, op. cit., p. 197. Tannstetter, fol. aa 3^v: Schinttel.

73. ZR, pp. 50 f.; see also ZA, nos. 3565, 3578, 10979, 11198, 11205, 11206.

74. D. B. Durand, “The Earliest Modern Maps of Germany and Central Europe,” p. 498; Mundy, Eine Schrift iiber Orts koordinaten, in ZA, 3728.

75. ZR, p. 43.

76. ZR, pp. 53, 218; here one can find references to other works by John of Gmunden that Regiomontanus studied (and in part copied).

BIBLIOGRAPHY

I. Original Works. The only works by John of Gmunden to be printed, except for the posthumous calendars mentioned above, were the treatise on the sine (see below on Busard) and the “Algorithmus Magistri Joannis de Gmunden de minuciis phisicis,” which appeared in a compendium prepared by Joannes Sigrenius, entitled Contenta in hoc libello (Vienna, 1515). A facsimile of the title page is given in Smith, Rara arithmetica, p. 117. An extract on the finding of roots is in C. J. Gerhardt, Geschkhte der Mathematik in Deutschland (Munich, 1877), pp. 7 f. John of Gmunden,’ will is published in Mundy, “John of Gmunden,” p. 198; and in Klug, “Johann von Gmunden,” p. 90 ff., with a facsimile of the first page. All his other writings are preserved only in MS; they were complained by Zinner in ZA.

Three MSS are obtainable in microfilm or photostatic reproduction (Document 1645 of the American Documentation Institute, 1719 N Street, N.W. Washington, D. C.; in this regard see Mundy, op. cit., p. 196):

1. Codex latinus Monaiensis 7650, fols. Ir- 8r (the calendar cited at ZA, no. 3524).

2. Cod. lat. Mon. 8950, fol. 81v ( “proprietates signorum,” ZA, no. 3732).

3. Cod. st. Flor. XI, 102, Irv (letter to Jacob de Clusa, ZA, no. 3584; this letter was published in Klug, op. cit., pp. 61 ff.).

II. Secondary Literature. See Acta Facultatis artium Universitatis Viennensis 1385- 1416, P. Uiblein, ed. I (Vienna, 1968), a publication of the Institut fü r Österreichische Geschichtsforschung, 6th ser., Abt. 2; J. Aschbach, Geschichte der Wiener Universität im ersten Jahrhundert ihres Bestehens (Vienna, 1865), pp. 455- 467; J. Bauschinger and E. Schrö der, “Ein neu entdeckter astronomischer Kalender fü r das Jahr 1448,” in Verö ffentlichungen der Gutenberg-Gesellschaft, 1 (1902), 4- 14; D. B. Durand, “The Earliest Modern Maps of Germany and Central Europe,” in Isis19 (1933), 486- 502; R. T. Gunther, Early Science in Oxford, II, Astronomy (Oxford, 1923); R. Klug, “Johann von Gmunden, der Begrü nder der Himmelskunde auf deutschem Boden. Nach seinen Schriften und den Archivalien der Wiener Universität,” Akademie der Wissenschaften Wien, Phil.- hist. Kl., Sitzungsberichte, 222 , no. 4 (1943), 1- 93; Die Matrikel der Universität Wien, i, 1377- 1435, ed. by the Institut fü r ö sterreichische Geschichtsforschung (Vienna, 1956); J. Mundy, “John of Gmunden,” in Isis, 34 (1942- 1943), 196- 205; G. Sarton, Introduction to the History of Science, III (Baltimore, 1948), 1112 f.: D. E. Smith, Rara arithmetica (BostonLondon, 1908, 117, 449); G. Tannstetter, Tabulae eclypsium Magistri Georgii peurbachii. Tabula primi mobilis Joannis de Monteregio (Vienna, 1514), which contains Viri mathematici quos inclytum Viennense gymnasium ordine celebres habuit, fol. aa 3^v; L. Thorndike and P. Kibre, A Catalogue of Incipits of Mediaeval Scientific Writings in Latin (London, 1963), index, p. 1838; and E. Zinner, Verzeichnis der astronomischen Handschriften des deutschen Kulturgebietes (Munich, 1925), 119- 126; and Leben und Wirken des Johann Midler von Kü nigsberg, genannt Regiomontanus (Munich, 1925), pp. 14 ff. and index, p. 284.

See also H. H. Busard, “Der Traktat De sinibus, chordis et arcubus von Johannes von Gmunden,” in Denkschriften der Akademie der Wissenschaften, 116 (Vienna, 1971), 73-113.

Kurt Vogel