Peurbach and Marāgha Astronomy? The Ephemerides of Johannes Angelus and Their Implications

Angelus Johannes , Almanach novum atque correctum … calculatum super anno domini 1510 (Vienna, [1510]); idem, Almanach novum atque correctum … calculatum super anno domini 1512 (Vienna, 1512). Both almanacs are extremely rare: The 1510 is extant in Gniezno, Cathedral Chapter Library, Inc. 43e; Vienna, University Library, I. 233605; Munich, Bavarian State Library, Res. 4° Eph. astr. 155/2 (incomplete); the 1512 in Vienna, Austrian National Library, 72.J.114(3); Graz, University Library, I 4070 (incomplete); Munich, Res. 4° Eph. astr. 155/3. Both almanacs contain a preface, an annual ephemeris in twelve folios, the canon from Johannes Regiomontanus's ephemerides published in 1474, and a three-folio abridgment of some rules for astrological prognostication that Erhard Ratdolt had appended to his 1484 edition of Regiomontanus's ephemerides. Angelus's 1510 work presents only planetary longitudes as had Regiomontanus; the 1512 almanac, however, also lists planetary latitudes at 10-day intervals, following the pattern established by Stöffler Johann and Pflaum Jacob , Almanach nova plurimis annis venturis inservientia (Ulm, 1499). For typographical descriptions of Angelus's almanacs, see Dolsch Walther , Bibliographie der österreichischen Drucke des XV. und XVI. Jahrhunderts, i/1: Trient-Wien-Schrattenthal (Vienna, 1913), 69–70, 73–74; Verzeichnis der im deutschen Sprachbereich erschienenen Drucke des XVI. Jahrhunderts (Stuttgart, 1983–), E1196–7.

Appendices 1 and 2, below.

Aurifaber Stanislaus , Ephemerides anni Christi MDCII (Cracow, 1512), sigs. ai^v−aii^v; Collimitius Tanstetter Georg (ed.), Tabulae eclypsium magistri Georgij Peurbachij. Tabula primi mobilis Joannis de Monteregio (Vienna, 1514), sig. aa6^r; Rheticus Joachim Georg , De libris revolutionum Copernici narratio prima (Gdańsk, 1540), sig. Ii^r. None of the recent translators of the Narratio prima explored what Rheticus or Copernicus might have known about Angelus's work. See Rheticus Joachim Georg , Erster Bericht über die 6 Bücher des Kopernikus von den Kreisbewegungen der Himmelsbahnen, transl. by Zeller Karl (Munich, 1943), 184; Rosen Edward , Three Copernican treatises, 2nd edn (New York, 1959), 192; Rheticus Joachim Georg , Narratio prima: Edition critique, traduction fran&çaise et commentaire , ed. by Hugonnard-Roche Henri and Verdet Jean-Pierre (Studia Copernicana, xx; Wrocław, 1982), 192.

For recent reviews of the secondary literature on Angelus's life and work, see Haage Bernhard D. (ed.), Das Heidelberger Schicksalbuch: Das ‘Astrolabium planum’ deutsch aus CPG 832 der Universitätsbibliothek Heidelberg, Kommentar (Frankfurt, 1981), 34–59; Knobloch Eberhard , “Astrologie als astronomische Ingenieurkunst des Hochmittelalters: Zum Leben und Wirken des Iatromathematikers und Astronomen Johannes Engel (vor 1472–1512)”, Sudhoffs Archiv, lxvii (1983), 129–44; Grössing Helmuth , “Angelus, Johannes”, Archiv der Geschichte der Naturwissenschaften, xvi (1986), 789–92; and especially Schöner Christoph , Mathematik und Astronomie an der Universität Ingolstadt im 15. und 16. Jahrhundert (Berlin, 1994), 195–200, 223–32, 466–73, who offers a new and detailed account of Angelus's years in Ingolstadt. For the role of mathematics in a sixteenth-century medical career, see Westman Robert S. , “Humanism and scientific roles in the sixteenth century”, in Humanismus und Naturwissenschaften, ed. by Schmitz Rudolf and Krafft Fritz (Boppard, 1980), 83–99.

For the Celtis circle, see Stuhlhofer Franz , “Georg Tanstetter (Collimitius): Astronom, Astrologe und Leibarzt bei Maximilian I. und Ferdinand I.”, Jahrbuch des Vereins für Geschichte der Stadt Wien, xxxvii (1981), 7–49; Grössing Helmuth , Humanistische Naturwissenschaft: Zur Geschichte der Wiener mathematischen Schulen des 15. und 16. Jahrhunderts (Baden-Baden, 1983), 170–92; Schöner , Mathematik und Astronomie (ref. 4), 202–15, 257–80.

The only autograph of Angelus we have found is a signed receipt for five medical books that he deposited with the Ingolstadt faculty in 1483 as security for a loan. Munich, Ludwig-Maximilian-University Archive, O-V-I, 34a^r, transcribed in Ruf Paul , Mittelalterliche Bibliothekskataloge Deutschlands und der Schweiz, iii/2: Bistum Augsburg (Munich, 1932), 231–2 (see Figure 2).

See Grössing , “Angelus” (ref. 4), 790–4, for a bibliography of Angelus's published works. Seethaler Josef , “Das Wiener Kalenderwesen von seinen Anfängen bis zum Ende des 17. Jahrhunderts”, Ph.D. dissertation, University of Vienna, 1982, pp. 748–9, attributes to Angelus two single-sheet calendars for 1512 and 1513, printed in Vienna and recently found at the Hungarian Academy of Sciences in Budapest. We have been unable to examine these calendars, cited by neither Grössing nor standard bibliographies of publishing or astronomical history.

In Angelus's 1484 German calendar (Gesamtkatalog der Wiegendrucke (Leipzig, 1925–), 1892; hereafter GW), the times for new and full moons exceed those in Regiomontanus's ephemerides by 0;02° in all but two cases, and a conjunction of Saturn and Jupiter, predicted by Angelus at 233;44° on 25 November at 6:52 p.m., is set by Regiomontanus for 7:00 p.m. on the same day at 233;43°. Angelus's 1489 Latin calendar (GW 1896) presents verbatim Regiomontanus's times for the new and full moons, and for a lunar eclipse on 7 December. Angelus's 1490 German calendar (GW 1899) also exactly matches Regimontanus's times for the new and full moons in all but two cases. And the 1497 German practica (GW 1904) increases Regiomontanus's times for the new and full moons by 0;02° in all but two cases. To construct these calendars, Angelus undoubtedly employed Regiomontanus's ephemerides, occasionally shifting the lunar positions to a different meridian.

According to Knobloch , “Astrologie” (ref. 4), 136, Claude Fran&çois Milliet Dechales, Cursus seu mundus mathematicus , 2nd edn (Lyon, 1690), 1:88, first attributed a 1494 ephemerides to Angelus. Johann Friedrich Weidler, Historia astronomiae (Wittenberg, 1741), 327, continued this attribution, and a “1494 ephemerides” remained firmly entrenched in the bibliographical literature through Houzeau J. C. and Lancaster A. , Bibliographie générale de l'astronomie (Brussels, 1882–89), #14509. Yet neither GW (ref. 8), ii:267, nor Ernst Zinner, Geschichte und Bibliographie der astronomischen Literatur in Deutschland zur Zeit der Renaissance , 2nd edn (Stuttgart, 1964), 541, could confirm the existence of a such a treatise, and after considerable searching we have concluded that Dechales probably misread Tanstetter , Tabulae eclypsium (ref. 3), and created a ghost.

Appendices 1 and 2, below.

Appendices 1 and 2, below.

To sample longitudes at five-day intervals starting with 5 January 1510 and at two-day intervals for Mercury starting with 1 January 1510, we corrected 13 typographical errors in Angelus and 7 in Stöffler and Pflaum. Each data set (ignoring the lunar nodes) includes a total of 1442 longitudes; hence, the rate of obvious typographical errors is below one percent in both almanacs.

In their table of geographical longitudes, Stöffler and Pflaum list Toledo as 20.5°W of Ulm. For sixteenth-century readers' complaints about inaccuracies in this table, see Kremer Richard L. and Dobrzycki Jerzy , “Disputationes inter Viennensem et Cracoviensem, II” (forthcoming).

Appendix 2, below.

According to Stöffler's and Pflaum's chart of geographical longitudes, Vienna is 5;30°E of Ulm. Angelus apparently decided to displace the meridian for his 1512 almanac from Ulm to Vienna, but erred by a factor of 2 (perhaps confusing minutes of time with minutes of lunar motion in longitude, both of which were included in Stöffler's and Pflaum's table of geographical longitudes). See Kremer and Dobrzycki , op. cit. (ref. 13).

Leipzig, 5-day (Sun, Moon, Mercury), 15-day (Venus, Mars), and 30-day (Jupiter, Saturn) ephemerides for 1440–69, although filled with egregious scribal errors, is Alfonsine to±0;15° (Nuremberg, City Library, Cent VI 16, 149^r–95^r; cf. Zinner Ernst , Leben und Wirken des Joh. Müller von Königsberg genannt Regiomontanus, 2nd rev. edn (Osnabrück, 1968), 14–15); Regensburg, daily ephemerides for 1463, Alfonsine to±0;10° for all planets except Mars (±0;20°) (Munich, Bavarian State Library, Clm 14504, 140^r–5^v); Leipzig and Vienna, Regiomontanus's daily ephemerides for 1448, 1451, 1453–63, Alfonsine to±0;10° for all planets except Mars (± 0;20° when near the second stationary point) (Vienna, Austrian National Library, lat. 4988, 1^r–188n^r); Bamberg, daily ephemerides for 1464–84, Alfonsine to±0;10° (ibid., 189^r–314^r); Nuremberg, Regiomontanus's daily ephemerides for 1472, Alfonsine to±0;10° (13 handwritten folios bound with Regiomontanus , Ephemerides (ref. 1), Munich, Bavarian State Library, Rar. 299a); Nuremberg, Regiomontanus's daily ephemerides for 1475–1506, Alfonsine to±0;10° ( Regiomontanus , Ephemerides (ref. 1)). We do not know which versions of the Alfonsine Tables were used to compute these various ephemerides; yet in all cases, they generally match longitudes generated from Alfonso X, Tabule astronomice (Venice, 1483) to ±0;10°.

To compute planetary longitudes according to modern theory, we have used computer programs kindly provided by Owen Gingerich, which follow the procedures of Bryant Tuckerman, Planetary, lunar, and solar positions, A.D. 2 to A.D. 1649 (Philadelphia, 1964), but include additional subroutines written by Peter Huber to implement perturbation theory more exactly.

Over the two years in question, Angelus's longitudes for Saturn deviate by up to 1° from the modern positions, Stöffler's and Pflaum's by only half that amount. For Jupiter, the respective maximal deviations are about 2° and 1°; for Mars, nearly 7° and 4°; for Venus, 2.5° and 1°; and for Mercury, both almanacs differ from modern positions by a maximum of 14° near the stationary points.

Goldstein Bernard R. , “Levi ben Gerson's lunar model”, Centaurus, xvi (1972), 257–84; idem, “Levi ben Gerson's preliminary lunar model”, Centaurus, xviii (1974), 275–88; idem, “A new set of fourteenth century planetary observations”, Proceedings of the American Philosophical Society, cxxxii (1988), 371–99; Swerdlow Noel M. , “Regiomontanus on the critical problems of astronomy”, in Nature, experiment, and the sciences, ed. by Levere T. H. and Shea W. R. (Dordrecht, 1990), 165–95, pp. 172–3; Neugebauer O. , A history of ancient mathematical astronomy (Berlin, 1975), 145.

Levi ben Gerson explored this latter possibility, an arrangement Goldstein Bernard R. has called the “skew equant model” in The astronomy of Levi ben Gerson (New York, 1985), 114–29, 192–7.

We employ the notation of Olaf Pedersen, A survey of the Almagest (Odense, 1974). For useful discussions of Alfonsine procedures, see Poulle Emmanuel and Gingerich Owen , “Les positions des planètes au moyen âge: Application du calcul életronique aux tables alphonsines”, Académie des Inscriptions & Belles-Lettres, Comptes rendus, 1968, 531–48; North J. D. (ed.), Richard of Wallingford: An edition of his writings, with introductions, English translation and commentary (3 vols, Oxford, 1976), iii, 168–200; and Poulle Emmanuel (ed.), Les tables alphonsines avec les canons de Jean de Saxe: Édition, traduction et commentaire (Paris, 1984).

An exact tabular solution for the mechanisms of Equations [2] and [3] would require modifying the diversitas diametri. Yet for the values of the cofficients determined below, such modifications would not shift final longitudes by more than 0;10°. Furthermore Angelus in the prefaces did not mention altering any Alfonsine columns other than the equations of the centre and argument. It seems quite likely, therefore, that he employed unmodified equations for the diversitas diametri, as we have done for our computations.

The two columns in bold type implement the harmonic mechanisms of Equations [1] and [3]; the remaining columns are taken unchanged from the Alfonsine Tables.

In such cases, the true argument must be further modified, so that Equation [5] becomes.

Adjusting radices in planetary tables was not uncommon. A copy of John of Lignères's table of mean planetary motions, written c. 1470 at the St Egidien Monastery in Nuremberg, bears the following annotations in the hand of the copyist (Frater Laurentius): Secundum magistrum Iohannem 3 gradus addendi 1473 [for Saturn]; uno gradu subtrahendo secundum Ioh Kunig [for Jupiter]; addendi 4 gradus secundum magistrum Iohannem Kunics [for Mars] (Munich, Ludwig-Maximilian-University Library, 2° Cod. ms. 593, 12^r, 14^v, 17^r). Apparently the scribe thought that Regiomontanus in 1473 had introduced several constant modifications into the radices of planetary tables, alterations which do not appear, however, in Regiomontanus's extant ephemerides ( Zinner , Regiomontanus (ref. 16), 198; Schott Gerhard , Die Handschriften der Universitätsbibliothek München, iii/2, Die lateinischen mittelalterlichen Handschriften (Wiesbaden, 1979), 102–5). Likewise, a mid-fifteenth-century Jewish scholar in Mantua, Mordekhai Finzi, suggested changing the Alfonsine lunar radices ( Langermann Tzvi Y. , “The scientific writings of Mordekhai Finzi”, Italia: Studi e ricerche sulla storia, la cultura e la letteratura degli ebrei d'Italia, vii (1988), 8–44, pp. 15–20).

Examination of the daily longitudinal increments in both almanacs suggests that Stöffler and Pflaum often computed positions at 10-day (Mars), 5-day (Venus) or 3-day (Mercury) intervals, and linearly interpolated between these epochs. Angelus's daily increments, however, vary more smoothly and reveal no obvious computational epochs longer than one day.

Poulle and Gingerich , Les positions des planètes (ref. 21), 542, 544; see above, ref. 16.

Already in 1456, Peurbach and Regiomontanus were computing ephemerides together, using Giovanni Bianchini's tables. By 1457, they began systematically observing the motion of Mars, noting that its longitude might deviate by up to 1;00° from the Alfonsine predictions. In 1458, Regiomontanus found that his own computed longitudes for Mars differed dramatically from those in an almanac calculated by one Master Purkhard Nestler of Salzburg, and wrote into his almanac for that year: “Corrige Martem.” At some point, Peurbach apparently asked Nestler to observe Mars, noting that the predicted locations for that planet could err by 1;30°. See Peurbach to Johann Nihil of Bohemia, n.d. [1456], in Czerny Albin , “Aus dem Briefwechsel des grossen Astronomen Georg von Peurbach”, Archiv für österreichische Geschichte, lxxii (1888), 281–304, p. 302; Regiomontanus's unpublished ephemerides for 1457–58, in Vienna, Austrian National Library, lat. 4988, 90^v, 116^v, 117^v, 118^r, 118^v, 119^r, 120^r; Ziegler Jacob , In C. Plinii De naturali historia librum secundum commentarius (Basel, 1531), 446; Zinner , Regiomontanus (ref. 16), 44, 56. Extant sources record 19 planetary observations by Regiomontanus from which longitudes can be determined; 13 of these were of Mars. Bernard Walther, the Nuremberg merchant who continued Regiomontanus's observational activity starting in 1475, devoted half of the nearly 100 planetary observations he made through 1479 to Mars. See Schmeidler Felix (ed.), Joannis Regiomontani Opera collectanea (Osnabrück, 1972), 645–71.

See Aiton E. J. , “Peurbach's Theoricae novae planetarum: A translation with commentary”, Osiris, 2nd ser., iii (1987), 5–43, p. 8.

We shall ignore more esoteric geometries for the inner planets, such as placing an equant point and eccentric within the epicycle. Although such an arrangement can produce the ellipse drawn out by Equations [2] and [3], we know of no ancient or medieval astronomer who sought to generate the second anomaly with Ptolemaic techniques developed for the first anomaly.

See Roberts Victor , “The solar and lunar theory of Ibn ash-Shāātir: A pre-Copernican Copernican model”, Isis, xlviii (1957), 428–32; idem, “The planetary theory of Ibn al-Shāātir: Latitudes of the planets”, Isis, lvii (1966), 208–19; Kennedy E. S. and Roberts Victor , “The planetary theory of Ibn al-Shāātir”, Isis , 1 (1959), 227–35; Abbud Fuad , “The planetary theory of Ibn al-Shāātir: Reduction of the geometric models to numerical tables”, Isis, liii (1962), 492–9; Swerdlow Noel M. , “The derivation and first draft of Copernicus's planetary theory: A translation of the Commentariolus with commentary”, Proceedings of the American Philosophical Society, cxvii (1973), 423–512.

See Swerdlow , “Commentariolus” (ref. 31), 470, for a derivation of this property of the double epicycle model.

See Ragep F. J. , “The two versions of the Ṭūsī Couple”, in From deferent to equant: A volume of studies in the history of science in the ancient and medieval Near East in honor of E. S. Kennedy , ed. by King David A. and Saliba George (New York, 1987), 329–56, pp. 332–40; idem, Nasir al-Din al-Ṭūsī's Memoir on astronomy (New York, 1993), 194–212; Swerdlow , “Commentariolus” (ref. 31), 499–509; idem and Neugebauer O. , Mathematical astronomy in Copernicus's De revolutionibus (New York, 1984), 46–47; Saliba George , “The role of the Almagest commentaries in medieval Arabic astronomy: A preliminary survey of Ṭūsī's redaction of Ptolemy's Almagest”, Archives internationales d'histoire des sciences, xxxvii (1987), 3–20; di Bono Mario , “Copernicus, Amico, Fracastoro and Ṭūsī's device: Observations on the use and transmission of a model”, Journal for the history of astronomy, xxvi (1995), 133–54.

The literature on medieval theories of precession and trepidation is vast; for useful surveys, see Dobrzycki Jerzy , “Teoria precesji w astronomii średniowiecznej”, Studia i materialy z dziejów nauki polskiej, ser. c, xi (1965), 3–47, and most recently, Goldstein Bernard R. , “Historical perspectives on Copernicus's account of precession”, Journal for the history of astronomy, xxv (1994), 187–97.

Aiton , “Peurbach's Theoricae” (ref. 29), 36–43; Swerdlow , “Regiomontanus” (ref. 19).

Toomer G. J. , Ptolemy's Almagest (London, 1984), 599–601; Riddell R. C. , “The latitudes of Venus and Mercury in the Almagest”, Archive for history of exact sciences, xix (1978), 95–111; Goldstein Bernard R. , al-Biṭtrūjī on the principles of astronomy (2 vols, New Haven, 1971).

This same objection can be made against another means of producing angular harmonic motion by uniformly rotating spheres, viz., the combination of two concentric spheres rotating in opposite directions on non-parallel axes which Ragep, al-Ṭūsī's Memoir (ref. 33), 451–2, felicitously has labelled the “Eudoxan couple”.

Aiton , “Peurbach's Theoricae” (ref. 29), 38; Regiomontanus to Speier, 1465, in Curtze Maximilian , “Der Briefwechsel Regiomontan's mit Giovanni Bianchini, Jacob von Speier und Christian Roder”, Abhandlungen zur Geschichte der mathematischen Wissenschaften, xii (1902), 187–336, p. 303; Swerdlow , “Regiomontanus” (ref. 19), 180–4; Shank Michael H. , “The ‘Notes on al-Biṭrūjī’ attributed to Regiomontanus: Second thoughts”, Journal for the history of astronomy, xxiii (1992), 15–30.

Florence, Bib. Naz. Cen., MS Magl. XI 144, 16^r–17^v. This text, discovered by Ernst Zinner in 1953, has not been published. See Shank , “The ‘Notes on al-Biṭrūjī’” (ref. 38), 19–4 and note 32, and Shank Michael H. and Kremer Richard L. , “Regiomontanus's homocentric astronomy” (forthcoming). We thank Menso Folkerts for allowing us to use his microfilm of this manuscript, and Noel Swerdlow for sending us his working transcription and translation of the text.

MS Magl. XI 144 (ref. 39), 16^r, transl. by Shank , “The ‘Notes on al-Biṭrūjī’” (ref. 38), 20–4. See Shank and Kremer , “Regiomontanus's homocentric astronomy” (ref. 38).

Swerdlow , “Regiomontanus” (ref. 19), 174. For Regiomontanus's program of reform, see Zinner , Regiomontanus (ref. 16), 279–80; Rose Lawrence Paul , The Italian Renaissance of mathematics: Studies on humanists and mathematicians from Petrarch to Galileo (Geneva, 1975), 90–117; Swerdlow and Neugebauer , Mathematical astronomy (ref. 33), 52–54.

Swerdlow , “Regiomontanus” (ref. 19), 172–4. All of Regiomontanus's extant almanacs, however, are strictly Alfonsine (see ref. 16), as is Peurbach's Tabulae eclypsium (completed 1459, published in Tanstetter, Tabulae eclypsium (ref. 3)) and an anonymous perpetual almanac for 1455–1540 (copies in Munich, Bavarian State Library, Clm 3001, 2–52; Clm 19550, 93–148; Clm 18778, 79–97), attributed by Zinner , Regiomontanus (ref. 16), 35, to Peurbach.

See Kennedy E. S. , “A survey of Islamic astronomical tables”, Transactions of the American Philosophical Society, n.s., xlvi (1956), 121–77.

Aiton , “Peurbach's Theoricae” (ref. 29), 36. “Propter dictas epicyclorum inclinationes atque reflexiones orbes parvi epicyclos intra se locantes a quibusdam ponuntur ad quorum motum eaedem contingunt” (Vienna, Austrian National Library, lat. 5203, 21^r; Schmeidler (ed.), Opera (ref. 28), 787). None of the early commentators on the Theoricae remarks on this passage. See Birkenmajer L. A. (ed.), Albertus de Brudzewo: Commentariolum super theoricas novas planetarum Georgii Purbachii [1482] (Cracow, 1900), 143–5; Capuani Franciscus , Theoricarum novarum textus Georgij Purbachij cum utli ac preclarissima expositione (Paris, 1515), 52^r–53^r; Reinhold Erasmus , Theoricae novae planetarum Georgij Peurbacchii (Wittenberg, 1542), sig. dv^r.

See Ragep , al-Ṭūsī's Memoir (ref. 33), 214–16, 450–4.

See Hellman Doris C. and Swerdlow Noel M. , “Peurbach, Georg”, in Dictionary of scientific biography (New York, 1970–80), xv, 473–9, p. 475; Langermann Tzvi Y. , Ibn al-Haytham's On the configuration of the world (New York, 1990).

Ragep , al-Ṭūsī's Memoir (ref. 33), 450. Another work by al-Haytham, replying to a critic of his text on latitude theory, is extant and has been edited by Sabra A. I. , “Ibn al-Haytham's treatise: Solutions of difficulties concerning the movement of Iltifāf”, Journal for the history of Arabic science, iii (1979), 388–422. But apparently this latter text also was never translated into Latin.

Ragep , al-Ṭūsī's Memoir (ref. 33), 58–62, lists eight Arabic commentaries on the Tadhkira written before 1460, none of which is known to have been translated into Latin.

Mancha J. L. , “Ibn al-Haytham's homocentric epicycles in Latin astronomical texts of the XIVth and XVth centuries”, Centaurus, xxxiii (1990), 70–89.

Ragep , al-Ṭūsī's Memoir (ref. 33), 58.

Cf. Ragep , al-Ṭūsī's Memoir (ref. 33), 48–51; Swerdlow , “Commentariolus” (ref. 31), 433–5. See Saliba George , “The astronomical tradition of Marāgha: A historical survey and prospects for future research”, Arabic sciences and philosophy, i (1991), 67–99, pp. 74–81; and Bono Di , “Copernicus, Amico, Fracastoro” (ref. 33), for surveys of research on links between the Marāgha School and Copernicus.

Figure 7 is produced according to Levi's lunar models and parameters, as decribed by Goldstein , “Lunar model” (ref. 19); idem, “Preliminary lunar model” (ref. 19). Apparently, Levi never completed the planetary modified models. See idem, “Planetary observations” (ref. 19).

For Werner's highly original model of precession which attracted no followers, see Dobrzycki , “Teoria precesji” (ref. 34), 29–4; idem, “Astronomical aspects of the calendar reform”, in Gregorian reform of the calendar , ed. by Coyne G. V. Hoskin M. A. , and Pedersen O. (Vatican City, 1983), 117–26, p. 122. Werner allotted the trepidational motion of “Thabit's” and Peurbach's models to the solstitial points of two concentric spheres. Two circles of trepidation, of equal radii and centred on the solstitial points of the next higher sphere, rotate in opposite directions so that trepidational variations in longitude do not introduce shifts in the obliquity of the ecliptic. Werner thus managed to generate linear harmonic motion by the uniform motions of two circles.

See Kren Claudia , “The rolling device of Nasir al-Din al-Ṭūsī in the De spera of Nicole Oresme?”, Isis, lxii (1971), 490–8; Alfonso [Abner of Burgos], Meyashsher 'Aqov, fac. edn with Russian transl. by Gluskina G. M. and commentary by Gluskina Lurie S. Y. and Rosenfeld B. A. (Moscow, 1983), 85–86, 107; Langermann , “Mordekhai Finzi” (ref. 25), 38; idem, “Medieval Hebrew texts on the quadrature of the lune”, Historia mathematica, xxiii (1996), 31–53. We thank Y. Tzvi Langermann for providing a preprint of this latter article, and for drawing our attention to Alfonso's text.

Swerdlow , “Commentariolus” (ref. 31), 424; see Saliba , “Marāgha” (ref. 51), 98–4. Furthermore, our reconstruction of Angelus's tables weakens Di Bono's recent claim, in “Copernicus, Amico, Fracastoro” (ref. 33), that Copernicus reinvented the Ṭūsī couple rather than borrowing it from Arabic traditions via unknown sources.

Punctuation and spelling have been standardized, and the text has been divided into paragraphs.

For the Alfonsine tables, see Appendix 4. The Tabulae astronomicae of Giovanni Bianchini (d. c. 1469), written around 1440 and published in 1495 in Venice, include a large set of planetary tables based on Alfonsine parameters and theory, but arranged in a double-entry format for easier use. See Birkenmajer Louis , “Flores Almagesti: Ein angeblich verloren gegangener Traktat Giovanni Bianchini's”, Bulletin international de l'Académie des Sciences de Cracovie, Cl. des sciences mathématiques et naturelles, sér. A, 1911, 268–78; Swerdlow , “Regiomontanus” (ref. 19), notes 6–7.

Dorn Hans (c. 1430–1509), a Viennese Dominican and student of Peurbach's, might have brought one of his teacher's manuscripts into the cloister library ( Grössing , Humanistische Naturwissenschaft (ref. 5), 145, 278). For relations between the cloister and the Viennese university in the fifteenth century, see Frank Wilhelm Isnard , Hausstudium und Universitätsstudium der Wiener Dominikaner bis 1500 (Vienna, 1968). According to a 1513 catalogue of the Dominican library, several manuscripts were then extant which may have contained the remarks Angelus here attributed to Peurbach: “Theoricam planetarum, textus cum tabulis faciendi tabulas” (shelfmark 147); “Almanach Purkardi ad meridianum Salczeburgensem” (O19); “Theorice planetarum textus”, which included “Tabula Alphonsij cum canonibus” (S5); “Tabula de motibus planetarum” (S7); and “Canones tabulares magistri Iohannis de Lyneriis primi et secundi mobilis et alia plura ad astronomiam deserviencia” (S9). Of these manuscripts, only S7 and S9 can be found currently in the Convent library (Cod. 141/111 and Cod. 189/155), neither of which contains any comments about the reliability of tables ( Gottlieb Theodor , Mittelalterliche Bibliothekskataloge Österreichs, i: Niederösterreich (1915; fac. reprint edn, Aalen, 1974), 357, 384, 400; Czeike Felix , Verzeichnis der Handschriften des Dominikanerkonvents in Wien bis zum Ende des 16. Jahrhunderts (Vienna, 1952), 128, 170). In observational records published in 1544, and in 1512 apparently still in Nuremberg, Peurbach and Regiomontanus criticized the accuracy of the Alfonsine Tables after finding that the predicted times for eclipses and planetary conjunctions did not match the observed times ( Schmeidler (ed.), Opera (ref. 28), 645–60; Czerny , “Aus dem Briefwechsel” (ref. 28), 302; Zinner , Regiomontanus (ref. 16), 354).

Regiomontanus in a 1471 letter to Christian Roder (d. 1478), rector of the university in Erfurt, chastized astronomers who used the Alfonsine Tables as a “gift of heaven” without considering that these tables did not explain how the celestial sphere could have a double motion (precession and trepidation); did not reflect the fact that the solar and other eccentricities had changed since Antiquity; and yielded predictions that agreed neither with ancient nor modern observations ( Curtze , “Der Briefwechsel Regiomontan's” (ref. 38), 326–7; cf. Folkerts Menso , “Conrad Landvogt, ein bisher unbekannter Algebraiker um 1500”, in Amphora: Festschrift für Hans Wussing zu seinem 65. Geburtstag , ed. by Demidov Sergei S. (Basel, 1992), 229–59, pp. 233–4). In earlier letters to Bianchini and Jacob Speier, Regiomontanus had harshly criticized the Alfonsine Tables for employing a false model for the motion of the eighth sphere; yielding Martian longitudes that deviated by varying amounts (up to 2°) from the observed positions; predicting longitudes for Venus that exceeded by 0;45° the observed motion, and giving false latitudes for this planet; and incorrectly predicting the times and extents of lunar eclipses ( Curtze , “Der Briefwechsel Regiomontan's” (ref. 38), 263–6, 303–4; cf. Swerdlow , “Regiomontanus” (ref. 19)). Angelus probably had access to all these letters, since Regiomontanus himself had aggregated his correspondence with Bianchini, Speier and Roder, and the letters (currently Nuremberg, City Library, Nür Cent app. 56c) remained in Nuremberg in the library of Bernard Walther through at least 1523 ( Curtze , “Der Briefwechsel Regiomontan's” (ref. 38), 187–9; Zinner , Regiomontanus (ref. 16), 259–60, 324).

Probably Johannes Tolhopf (d. 1503), a mathematician who lectured at the university in Ingolstadt from 1472 until 1475 when he went to Rome to work on calendar reform for Pope Sixtus IV and probably met Regiomontanus ( Thorndike Lynn , Science and thought in the fifteenth century (New York, 1929), 298–300). Thereafter, Tolhopf returned only briefly to Ingolstadt in 1479 and in 1482, when he was prevented from rejoining the Arts Faculty by a group of young masters that included Angelus. Citing this latter incident, Schöner , Mathematik und Astronomie (ref. 4), 162–89, argued that Tolhopf could not have been close to Angelus. Yet the only other Ingolstadt professor who could have introduced Angelus to astrology and astronomy, the physician Erhard Windsberger, had first acquired his degree in Paris c. 1475, and hardly could have known Regiomontanus.

Stiborius (d. 1515) received his Master of Arts degree in 1484 in Ingolstadt, joined the circle of mathematicians around Conrad Celtis, and followed the latter to Vienna in 1497, where he held a chair for mathematics at the university from 1501 to 1503 and also lectured on mathematics in Celtis's Collegium poetarum et mathematicorum. With Georg Tanstetter and Stephanus Rosinus, Stiborius founded the so-called “second Viennese mathematical school”. He wrote at least sixteen texts on astronomical instruments, cartographic projection and spherical trigonometry. Even before moving to Vienna, Angelus had become acquainted with Stiborius. See Angelus to Celtis , 29 March 1498, in Rupprich Hans (ed.), Der Briefwechsel des Konrad Celtis (Munich, 1934), 322; Bonorand Conradin , Joachim Vadian und der Humanismus im Bereich des Erzbistums Salzburg (St Gallen, 1980), 213–14; Grössing , Humanistische Naturwissenschaft (ref. 5), 174–81, 197–8.

John of Gmunden may have observed from this well-known tower, and Stiborius mentioned it in a text on spherical trigonometry, in Munich, Bavarian State Library, Clm 24103, 5^v, 6^v. For a fourteenth-century illustration of the tower, see Grössing , Humanistische Naturwissenschaft (ref. 5), 325.

Probably either Regiomontanus, Ephemerides (ref. 1), or Stöffler and Pflaum , Almanach nova (ref. 1). Several editions of both ephemerides had been published before 1510.

This humanist rhetoric about the need to restore science after its long period of decline frequently appears in Regiomontanus's prose, less so in Peurbach's. See Curtze , “Der Briefwechsel Regiomontan's” (ref. 38), 327; Rose , The Italian Renaissance (ref. 41), 90–117; Pedersen Olaf , “The decline and fall of the Theorica planetarum”, in Science and history: Studies in honor of Edward Rosen , ed. by Hilfstein Erna Czartoryski Paweł , and Grande Frank D. (Wrocław, 1978), 157–85; Grössing , Humanistische Naturwissenschaft (ref. 5), 84–91, 117–21.

Appendix 3 provides details concerning Angelus's comparisons (Cases 1 to 15) of the ‘new’ and ‘old’ almanacs.

In both the 1510 and 1512 almanacs, Angelus appears to have computed the solar and lunar positions with unmodified Alfonsine theory. See p. 192 above.

The ‘ruler’ of a horoscope, or that celestial body whose influence is most pronounced on the chart.

For the use of the armillary sphere, an unwieldy and not widely used observational instrument of pre-modern astronomy, see Kremer Richard L. , “Bernard Walther's astronomical observations”, Journal for the history of astronomy, xi (1980), 174–91; Włodarczyk Jarosław , “Observing with the armillary sphere”, Journal for the history of astronomy, xviii (1987), 173–95.

“And he [Ptolemy] has erred in the determination of the station of the planet and the time of its retrogradation, so that it is possible that he is wrong in determining the time of retrogration of Mars by almost 18 days, and in the retrogradation of Venus by almost two and a half days, and this always by an excess of time.” Apian Peter , Instrumentum primi mobilis: Accedunt ijs Gebri filii Affla Hispalensis … libri IX de astronomia … per Giriardum Cremonensem latinitate donati (Nuremberg, 1534), sig. aa2^r. Jābir ibn Aflaḥ's Correction of the Almagest (early twelfth century), best known for its critique of the classical ordering of the planets, had been translated into Latin by Gerard of Cremona. Regiomontanus owned and annotated a copy of this treatise. Zinner , Regiomontanus (ref. 16), 76, 310–11; Lorch Richard P. , “The astronomy of Jābir ibn Aflaḥ”, Centaurus, xix (1975), 85–107.

Stanislaus Cracoviensis, called Aurifaber, was a member of the Faculty of Arts in Cracow, and by 1517 would become a Fellow of the Collegium Maius of the Cracow Academy. While still a young member of the faculty, he obtained the privilege to prepare almanacs for the university. His extant works include two printed almanacs for 1511 and 1512, almanacs in manuscript for 1513 and 1514, and Judicia for 1512 and 1513 ( Markowski Mieczysław , Astronomica et astrologica Cracoviensia ante annum 1550 (Florence, 1990), 189–91). Aurifaber , Ephemerides 1512 (ref. 3), sigs. ai^v–aii^v, contains a preface that sharply attacks Angelus's 1510 almanac albeit without naming its author, stresses the competence of Alfonso and Bianchini as authors of astronomical tables, and defends Aurifaber's value for the geographical longitude of Cracow. For an extended analysis of the dispute between Aurifaber and Angelus, see Kremer and Dobrzycki , “Disputationes” (ref. 13).

Tanstetter , Tabulae eclypsium (ref. 3), sig. aa4^r, listed among Peurbach's works a “Tabulae aequationum motuum planetarum novae, nondum perfectae et ultimum completae”. This text, undoubtedly identical to that described here, has not been found. If Angelus had these tables “for a long time”, he may have acquired them before moving to Vienna, perhaps from the materials Regiomontanus left in Nuremberg after his death (see Zinner , Regiomontanus (ref. 16), 245–65). Hellman and Swerdlow , “Peurbach” (ref. 46), 478, translated a note by Regiomontanus, written in his copy of the Almagest below the tables of planetary equations, asserting that Peurbach “had made more accurate equations” (Nuremberg, City Library, Nür Cent III 25, 80^r, actually: “Correctiones invenies in libro magistri Georgij”), and speculated that Peurbach's “tables” to which Tanstetter alluded were simply “recomputations of the planetary equations at 0;10° intervals using Alphonsine parameters”. A comparison of the tables of planetary equations in the copies of the Almagest owned by Peurbach (Vienna, Austrian National Library, lat. 4799, 65^v–67^v) and Regiomontanus (Nür Cent V 62, 219^v–22^v; Nür Cent III 25, 79^v–80^v), and Toomer , Almagest (ref. 36), indicates that all three manuscripts are filled with scribal errors, but that Peurbach's has considerably fewer than does Nür Cent III 25. It seems likely, therefore, that Regiomontaus's remark in Nür Cent III 25 refers to Peurbach's Almagest in Vienna lat. 4799 (as suggested by Zinner , Regiomontanus (ref. 16), 310) and not to some other independent set of tables. More recently, Grössing , Humanistische Naturwissenschaft (ref. 5), 108, suggested that Peurbach's incomplete tables might be in Vienna, Dominican Convent, Cod. 141/111, 113^v–47^v. Yet this MS contains an incomplete set of the Oxford Tables which do not appear to be modified in any way.

In notes acquired by Schöner Johann and published in 1544, Regiomontanus occasionally compared the longitudes of planets he observed (positions set essentially by the locations he assumed for the reference stars against which he measured the planets) with Alfonsine predictions. In six cases of explicit comparisons between 1461–71, the Alfonsine errors ranged from 0;40° to 3;00°. However Angelus probably is referring here to Regiomontanus's 1464 letter to Bianchini: “Mars was seen to differ in the heavens and in computation by two degrees in relation to the fixed stars and other observations” (transl. in Swerdlow , “Regiomontanus” (ref. 19), 172).

In 1497, Angelus probably spent some time in Würzburg. Zinner , Regiomontanus (ref. 16), 228, suggested that this “student” of Regiomontanus may have been the Würzburg jurist Friedrich Brogel, the Ingolstadt mathematician Johann Tolhopf, or the physician Eberhard Schleusinger. Cf. Knobloch , “Astrologie” (ref. 4), 132–3.

Regiomontanus had calculated his printed ephemerides for what he considered to be the meridian of Nuremberg (21.1°E of Toledo). See p. 193 above.

Regiomontanus, Disputationes contra Cremonensia in planetarum theoricas deliramenta (Nuremberg, c. 1474) criticized the early fifteenth-century translation of Ptolemy's Geography by Jacobus Angelus. In the 1474 inventory of works he wanted to publish in Nuremberg, Regiomontanus placed a new translation of the Geography third on the list. See Zinner Ernst , “Die wissenschaftlichen Bestrebungen Regiomontans”, Beiträge zur Inkunabelkunde, N.F., i (1935), 89–103; Pedersen , “The decline and fall” (ref. 64), 179. Shank Michael H. , who is preparing an edition of the Disputationes, kindly let us consult his translation of this text.

The Alfonsine Tables replaced Ptolemy's uniform precession with a scheme in which the motion of the eighth sphere is composed of secular and periodic changes. For Jābir ibn Aflaḥ's criticisms of Ptolemy, see Lorch , “Jābir ibn Aflaḥ” (ref. 69).

Apparently Angelus thought that “Alfonso” (d. 1284) and Jābir ibn Aflaḥ (fl. first half of twelfth century) were deceived by Campanus (fl. thirteenth century), even though the latter's Theorica planetarum once explicitly referred to Jābir ibn Aflaḥ. Angelus does not specify Campanus's alleged deceptions, but may be referring to the latter's misunderstanding of “Thabit's” model for trepidation. See Benjamin Francis S. Jr , and Toomer G. J. , Campanus of Novara and medieval planetary theory: Theorica planetarum (Madison, 1971), 35, 144–5, 332–3, 378–9.

Paraphrasing a ten-line verse in Aurifaber , Ephemerides 1512 (ref. 3), sig. Aii^v (Nunc mea nunc tenues reddit tibi fistula flatus. / Et sonat exiguos parua cicuta sonos. / … / Et ubi belligeros recinent mea classica cantus: / Edicentque graues tympana tacta vices.), these lines imply that Angelus wrote his preface after having read Aurifaber's text. See Kremer and Dobrzycki , op. cit. (ref. 13).

See above, pp. 192–3.

Beyond the comments at ref. 3 above, we have found few responses to Angelus's almanacs by contemporary readers. A sixteenth-century hand has lightly annotated the 1512 almanac now in Vienna (ref. 1), noting the geographical longitudes for several cities in Angelus's diatribe against Aurifaber and adding at the very end of the treatise a list of rules for astrological prognostication. Angelus's claims to have improved Alfonsine predictions, however, are passed over in silence. Beheim Lorenz (c. 1457–1521), the Bamberg canon well-known for his interest in astrology, wrote Willibald Pirckheimer in October of 1511 that he had seen a “new almanac and it seems good enough”. Reicke Emil (ed.), Willibald Pirckheimers Briefwechsel (2 vols, Munich, 1940–56), ii, 114–16, suggests Beheim was referring to Angelus's 1512 almanac. By the end of the century, Angelus's almanacs had become scarce. See Hagecius Thaddeus to Munnos Hieronymous , 22 July 1574, in Tychonis Brahe Dani Opera omnia , ed. by Dreyer J. L. E. (15 vols, Copenhagen, 1913–29), vii, 399–400.

In the 1512 preface, Angelus described how an armillary sphere might be employed to measure planetary longitudes. Apparently he had used this expensive, cumbersome instrument that would not have been available to most of his contemporary readers. Bernard Walther, probably the most skilled fifteenth-century astronomical observer, achieved a precision of about 0;05° with the armillary sphere but only after many years of practice. See Kremer , “Bernard Walther's observations” (ref. 68); Włodarczyk , “Observing with the armillary sphere” (ref. 68).

Bold type indicates prediction closest to modern configuration. All data in columns 1–3 not explicitly given in the prefaces are placed in square brackets. “Common” values are taken from Stöffler's and Pflaum's Almanach nova, unless otherwise noted. To identify Angelus's unnamed stars and his longitudes for them (column 1), coordinates from the star catalogue printed in the 1483 Alfonsine tables (but not normally considered part of the Alfonsine corpus; see Poulle , Les tables alphonsines (ref. 21), 224; Kunitzsch Paul , “The star catalogue commonly appended to the Alfonsine Tables”, Journal for the history of astronomy, xvii (1986), 89–98) were reduced to the epoch of the almanachs using Alfonsine motions of the eighth sphere, a procedure that Angelus followed, as can be seen explicitly in Case 1.

According to Stöffler and Pflaum , on May 22 Mars was at 201;05°, at its second stationary point.

According to Stöffler and Pflaum , the conjunction occurs on October 27 at 10 p.m.

According to Stöffler and Pflaum , the conjunction occurs on April 17 at 1 a.m.

Angelus probably means June 7, when his almanac places Mars at 268;52°.

The maximal difference between the almanacs for Venus occurs on January 19.

This value, not in the 1512 almanac, apparently was computed by Angelus especially for the preface.

See Poulle and Gingerich , “Les positions des planètes” (ref. 21); North J. D. , “The Alfonsine Tables in England”, in Prismata … Festschrift für Willy Hartner , ed. by Maeyama Y. and Satzer W. G. (Wiesbaden, 1977), 269–301; Poulle Emmanuel , “Jean de Murs et les tables alphonsines”, Archives d'histoire doctrinale et littéraire du Moyen Age, lv (1980), 241–71; idem, “The Alfonsine Tables and Alfonso X of Castille”, Journal for the history of astronomy, xix (1988), 97–113; Gingerich Owen , “The Alfonsine Tables in the age of printing”, in De astronomia Alphonsi Regis , ed. by Comes Mercè Puig Roser and Samsó Julio (Barcelona, 1987), 89–95; Dobrzycki Jerzy , “The ‘Tabulae resolutae’”, ibid., 71–77.

Poulle and Gingerich , “Les positions des planètes” (ref. 21), 541–4; North , Richard of Wallingford (ref. 21), iii, 195–7.

North , Richard of Wallingford (ref. 21), iii, 197.