Experimenting with Paper Instruments in Fifteenth-and Sixteenth-Century Astronomy: Computing Syzygies with Isotemporal Lines and Salt Dishes

Munich , Clm 14583, ff. 384v–5r, 383v; Gotha , Chart. A 472, f. 1v; Stöffler Johann , Tabulae astronomicae, verarum mediarumque coniunctionum & opppositionum solis & lunae exactissima supputatio pro omni tempore (Tübingen, 1514), sig. e2v—e3v; Schöner Johannes , Aequatorium astronomicum (Bamberg, 1521), sig. B5r.

North J. D. , “The Alfonsine tables in England”, in Prismata, naturwissenschaftsgeschichtliche Studien: Festschrift für Willy Hartner, ed. by Maeyama Y. Satzer W. G. (Wiesbaden, 1977), 269–301, p. 270.

North J. D. , (ed.), Richard of Wallingford: An edition of his writings, with introductions, English translation and commentary (3 vols, Oxford, 1976), i, 248–9; ii, 137–8.

Cf. Price Derek J. , The equatorie of the planetis (Cambridge, 1955); Zinner Ernst , Deutsche und niederländische astronomische Instrumente des 11. — 18. Jahrhunderts (Munich, 1956); Kennedy E. S. , The planetary equatorium of Jamshīd Ghiyāth al-Dīn al-Kāshī (d. 1429) (Princeton, 1960); Benjamin Francis S. Jr , and Toomer G. J. , Campanus of Novara and medieval planetary theory: Theorica planetarum (Madison, 1971); North , Richard (ref. 3), ii, 249–65; Poulle Emmanuel , Les instruments de la théorie des planètes selon Ptolémée: Équatoires et horlogerie planétaire du XIIIe au XVIe siècle (Geneva, 1980); Comes Mercè , Ecuatorios andalusíes: Ibn al-Samḥ, al-Zarqālluh y Abū-l-Ṣalt (Barcelona, 1991).

Guillermus Aegidius [Willem Gillisz of Zeeland], Liber desideratur super celestium motuum indagatione sine calculo (Lyon, 1494); Apian Peter , Astronomicum Caesareum (Ingolstadt, 1540), sig. B1r, “… sine numeris & calculis ad instrumenta redigeretur …”; Apian Peter , Eine grüntliche Ausßlegung des Buchs Astronomici Caesarei und seiner Instrument, darinne deß gantzen Hymmelslauff on alle Rechnung und Kopfbrechen, zu ewigen Zeytten, mit sampt den Finsternussen gefunden wirdt (Ingolstadt, 1540).

Poulle , Équatoires (ref. 4), 38.

Poulle , Équatoires (ref. 4), 35–6; Zinner , Instrumente (ref. 4), 36.

Cf. Sibum Heinz Otto , “Reworking the mechanical value of heat: Instruments of precision and gestures of accuracy in early Victorian England”, Studies in history and philosophy of science, xxvi (1995), 73–106; Sibum Heinz Otto , “Experimental history of science”, in Museums of modern science, ed. by Lindqvist Svante (Canton, 2000), 77–86; Taylor Katie , “A ‘pratique discipline’? Mathematical arts in John Blagrave's The mathematical jewel (1585)”, Journal for the history of astronomy, xli (2010), 2010–53.

Chabás José Goldstein Bernard R. , “Nicholaus de Heybech and his table for finding true syzygy”, Historia mathematica, xix (1992), 265–89; Chabás José Goldstein Bernard R. , “Computational astronomy: Five centuries of finding true syzygy”, Journal for the history of astronomy, xxviii (1997), 1997–105; Goldstein Bernard R. Chabás José , “Transmission of computational methods within the Alfonsine corpus: The case of the tables of Nicholaus de Heybech”, Journal for the history of astronomy, xxxix (2008), 2008–55.

North , Richard (ref. 3), ii, 209. Cf. ii, 174–76, 205–9; for illustrations of Richard's instrument, see iii, 33, 39.

I borrow this term from North, who however applied it only to the syzygy instrument of Peter Apian (see below). Yet Richard's lines are also isotemporal, even if they do not resemble a modern relief map or weather map as do Apian's. Cf. North , Richard (ref. 3), ii, 282–3.

This manuscript also contains an important collection of fifteenth-century geographical texts. Cf. Durand Dana Bennett , The Vienna-Klosterneuburg map corpus of the fifteenth century (Leiden, 1952), 71–6, 174–9; Chlench Kathrin , “Johannes von Gmunden — Handschriftenverzeichnis”, in Johannes von Gmunden (ca. 1384–1442): Astronom und Mathematiker, ed. by Simek Rudolf Chlench Kathrin (Vienna, 2006), 195–223.

Munich , Clm 11067, ff. 186r–7v. I have examined additional copies in Munich, Clm 25004, ff. 137r–8r; Munich , Universitätsbibliothek 4° 738, f. 92v (incomplete); and Leipzig, Universitätsbibliothek Ms. 1469, ff. 214r–15v. Other copies were preserved in Königsberg 2° 1735, Vienna 5258.

Melk 51, Tambach E355, Jena El 2° 73, and Zwickau XXII, VIII, 10, according to Zinner Ernst , Verzeichnis der astronomischen Handschriften des deutschen Kulturgebietes (Munich, 1925), nos. 3022–9, 3036, 3042.

Vienna, ÖNB cod. 5203, ff. 66v–9v; the title of the text is in the hand of Johannes Schöner, who later acquired this manuscript, according to Monika Maruska, “Johannes Schöner — ‘homo est nescio qualis’: Leben und Werk eines fränkischen Wissenschaftlers an der Wende vom 15. zum 16. Jahrhundert”, Ph.D. dissertation, Universität Wien, 2008, 97. Additional copies of this text, according to Zinner, Verzeichnis (ref. 13), nos. 7717–21, are in Heiligenkreuz 302, Melk 367, ÖNB cod. 5303, and Munich, Clm 19689, ff. 282r–4r (this latter manuscript, which I have seen, also attributes the text to Peurbach).

Vienna , ÖNB cod. 5203, f. 65v; Schöner Andreas (ed.), Opera mathematica Ioannis Schoneri (3 vols, Nuremberg, 1551), iii, f. 24v.

In addition to the seven manuscripts identified by Chabás Goldstein , “Nicholaus” (ref. 9), 271–2, I have noticed additional copies of Heybech's tables and/or canons in Munich, Clm 14111, ff. 133v–5r and 343r–4v; Clm 26666, ff. 21v–4v; Vienna , ÖNB cod. 2440, ff. 74v–5v; cod. 5425, ff. 37v–8r; Cracow , Biblioteka Jagiellońska 609, f. 234v; 610, ff. 334r–5v; 613, ff. 31r–2v; 1852, pp. 349–52; 1865, f. 68v, 144v; and Toruń , Biblioteka Uniwersytecka, Rps 74, ff. 85v–7r. In these fifteenth- and sixteenth-century codices, Heybech's tables invariably accompany materials from the Alfonsine corpus.

Gmunden's widely circulated Tabulae maiores, composed c. 1437, lists the maximal and minimal lunar velocities as 0;36,53,21 and 0;29,37,11°/hr, which yield a ratio of 60/48. Cf. de Mateo Beatriz Porres , “Les tables astronomiques de Jean de Gmunden: Édition et étude comparative”, Ph.D. dissertation, École Pratique des Hautes Études, Paris, 2003, Tab. 50.

Cf. Poulle , Équatoires (ref. 4), 375–404, 810–12. TN instruments (but not the syzygy instrument) can also be found in Gotha, Chart A 472, and Leipzig, Universitätsbibliothek Ms. 1479.

Poulle , Équatoires (ref. 4), 393–4, 404.

Cf. Biblioteca Vaticana, Pal. lat. 1396, ff. 30v–2r and Pal. Lat. 1489, ff. 239v–44r (the 1487 example is on ff. 243v–4r), both Virdung autographs. In the late 1480s and early 1490s, Virdung had studied at the universities of Cracow and Leipzig, presumably where he became acquainted with the TN. Cf. Thorndike Lynn , “Johann Virdung of Hassfurt, again”, Isis, xxv (1936), 363–71; Thorndike Lynn , “Another Virdung manuscript”, Isis, xxxiv (1943), 1943–3; Steinmetz Max , “Johannes Virdung von Haßfurt, sein Leben und seine astrologischen Flugschriften”, in ‘Astrologi hallucinati’, Stars and the end of the world in Luther's time, ed. by Zambelli Paola (Berlin, 1986), 195–214.

Munich , Clm 3001, f. 62r.

Poulle , Équatoires (ref. 4), 396–9.

In the TN construction, if f is the angle to any point F on the epicycle and g is the angle to any point G on the same hour line FG, ET is the distance from the centre of the epicycle to the Earth where the oblique lines converge, and r is the radius of the epicycle, it can be shown that.

ET / r = cos f sin g — Cos g sin f /. sin f — Sin g.

The problem of the direction of slant for the lunar oblique lines may have been recognized by whomever drafted Clm 3001, f. 62v (Fig. 4), where the centre of the eccentric lunar circle is shifted toward rather than away from the zero point of the calibrated outer circle for the arguments so that the lunar and solar lines now slant in the same direction. However for this orientation, the zero point of the αmacr should start at the bottom, not the top, of the instrument!

Gotha, A Chart 472, f. 1v, entitles its chapter on the TN syzygy instrument: “Jn dem 16 capitel, wie du das jnstrument, das man nennet das saltzuaß oder küffel der waren jncensio vnd opposicio der sonnen vnd mons, machen vnd die darjnnen suchen vnd erkennen solt.” Unfortunately, this manuscript, which preserves the only known German text on the TN dated by internal evidence to 1460 in Nuremberg, does not include Chap. 16 or an image of the syzygy instrument. Other examples of sixteenth-century, triangular Nuremberg salt dishes can be found in Sotheby's catalogue of fine European silver, Geneva sale, 6 May 1982 (Geneva, 1982), no. 200; Kohlhaussen Heinrich , Nürnberger Goldschmiedekunst des Mittelalters und der Dürerzeit, 1240 bis 1540 (Berlin, 1968), 499–500. For the piece in Fig. 6, see Tebbe Karin , Nürnberger Goldschmiedekunst, 1541–1868 (3 vols, Nuremberg, 2007), i/1, 207. As very little Nuremberg secular silver from the fifteenth century has been preserved, I have been unable to locate examples of earlier triangular salt dishes.

For Peurbach's values of f and g (see Table 1), ET/r = 11.62. For the Alfonsine lunar parameters at mean syzygy (Ee = 60 + 13,39, r = 6,20), ET/r = 11.63.

Although not noticed by the bibliographers (VD 16, ZV 13995), Schöner printed two versions of the 1521 Aequatorium, identical except for the radices. Although both state their meridians as Bamberg, the initial version sets Bamberg at 76 minutes of time, the second version at 55 minutes, east of Toledo. Canons for these equatoria appeared the next year in Schöner Johannes , Equatorii astronomici omnium ferme uranicarum theorematum explanatorii canones (Nuremberg, 1522).

Poulle , Équatoires (ref. 4), 83–8. Unfortunately, given his importance, Schöner is still awaiting major scholarly investigation. See Schottenloher Karl , “Johann Schöner und seine Hausdruckerei”, Zentralblatt für Bibliothekswesen, xxiv (1907), 1907–55; Solleder F. , “Herzog Ottheinrichs Frage an die Sterne: Zur Lebensgeschichte der Johannes Schöner und der Herren von Croaria”, in Staat und Volkstum, neue Studien zur bairischen und deutschen Geschichte und Volkskunde: Festgabe für Karl Alexander von Müller (Diessen, 1933), 281–317; Telle Joachim , “Das Arzneibuch Johannes Schöners und seine mittelhochdeutschen Quellen”, Centaurus, xvii (1972), 1972–41; Hauschke Sven , “Kurfürst Johann Friedrich von Sachsen und der Astronom und Mathematiker Johannes Schöner: Das Globenpaar von 1533/1534 in Weimar”, Der Globusfreund 2003/4, no. 51/52, 9–19; Maruska , “Schöner” (ref. 14).

One copy of a TN canon, Vienna, ÖNB cod. 5228, ff. 57v–61v, is an autograph by Schöner, containing other texts he had copied in 1499 and 1502 while at the university in Erfurt. It seems plausible to assume that Schöner became aware of the TN during those years. See Maruska , “Schöner” (ref. 14), 99.

Maruska , “Schöner” (ref. 14), 229.

Cf. Moll J. C. Albert , Johannes Stoeffler von Justingen (Lindau, 1877); Burmeister Karl Heinz , Sebastian Münster: Versuch eines biographisches Gesamtbildes (Basel, 1963), 27–9; Oestmann Günther , Schicksalsdeutung und Astronomie: Der Himmelsglobus des Johannes Stoeffler von 1493 (Stuttgart, 1993); Oestmann Günther , “Johannes Stoeffler, Melanchthons Lehrer in Tübingen”, in Philipp Melanchthon in Südwestdeutschland: Bildungsstationen eines Reformators, ed. by Rhein Stefan Schlechter Armin Wennemuth Udo (Karlsruhe, 1997), 75–85.

Wolkenhauer August , “Sebastian Münsters handschriftliches Kollegienbuch aus den Jahren 1515–18 und seine Karten (Cod. lat. 10691 der königlichen Hof- und Staatsbibliothek zu München)”, Abhandlungen der Gesellschaft der Wissenschaft zu Göttingen, phil.-hist. Kl., N.F. xi, no. 3 (Berlin, 1909); Poulle , Équatoires (ref. 4), 71–3, 322–35.

Starting at 0;07,48h for a solar longitude of 0° and reaching maxima of 0;20,44h and 0;32;52h, Stöffler's equation of time differs from the readily available versions in the editio princeps of the Alfonsine Tables (1483), Gmunden's tables, and Peurbach's eclipse tables (1514); I have not been able to identify a source for Stöffler's equation of time. Cf. Alphonso X, Tabulae astronomicae (Venice, 1483); Porres , “Gmunden” (ref. 17); Collimitius Georg Tanstetter (ed.), Tabulae eclypsium Magistri Georgij Peurbachij, Tabula primi mobilis Joannis de Monte Regio (Vienna, 1514).

Stöffler , Tabulae (ref. 1), sig. d6v. Stöffler compared these astrologers to the blind man who, according to Luke 18:41, beseeched Jesus, “Lord, I want to see”.

In about half of the copies I have examined, Stöffler's instrument has been assembled, usually with the pointer attached to the centre of the semicircle. The Augsburg physician and calendar-maker A. P. Gasser removed the bifolium from his book, mounted the paper on a backing of parchment, and bound the single sheet between his copies of Schöner's 1521 and Apian's 1540 equatoria. See Biblioteca Vaticana, Pal. Lat. S 30. Many sixteenth-century books of practical mathematics include figures, printed on only one side of a sheet, for readers to cut out and assemble. Cf. Taylor , “Mathematical arts” (ref. 8), 333, 336.

Munich, Clm 10961, ff. 112r–14r, 204r—v. Interestingly, Münster's instrument is not assembled, although elsewhere in this manuscript, ff. 14v–21r, he constructed and assembled small equatoria (after Campanus), beautifully coloured and delicately drawn, that still function smoothly. Given their small size (diameters of 8 cm or less), Münster's equatoria could not have been used for computation; he divided the arguments of the syzygy instrument to only 5 degrees. Münster's manuscript also contains copies of all the tables included in Stöffler's 1514 Tabulae (albeit arranged in different order) plus a full set of tables in the Alfonsine tradition, with mean motions collected by years from 1 to 2000 (radices are announced for Tübingen, 64 time minutes east of Toledo). The set includes eclipse tables from the 1483 editio princeps of the Alfonsine Tables and from Peurbach, as printed by Tanstetter in 1514. Cf. Wolkenhauer , “Kollegienbuch” (ref. 32), 33–5; Remak-Honneff Elisabeth Hauke Hermann , Katalog der lateinischen Handschriften der bayerischen Staatsbibliothek München: Die Handschriften der ehemaligen Mannheimer Hofbibliothek, Clm 10001–10930 (Wiesbaden, 1991), 204–8.

Since the lunar corrections vary only by up to 0;04h as the mean solar anomaly shifts from 0 to 180°, Stöffler's assumption of linear relationships is much more exact for the lunar part of the computation.

Calendar-makers generally needed times of true syzygy for an entire year, i.e., for 24 or 25 syzygies. Interestingly, the times of true syzygy (which are corrected by the equation of time) in Stöffler's well-known 1499 ephemerides do not appear to have been computed with the tables he would publish in 1514; and neither do they exactly match Regiomontanus's times. My comparison of the three sources for the 25 syzygies of 1514 indicates that the Almanach nova better matches Regiomontanus (σ = 0;01,26h) than it does times I compute from Stöffler's Tabulae (σ = 0;02,58h). Both Regiomontanus and Stöffler in the Almanach nova probably used John of Saxony's iterative method to compute the time corrections. Cf. Stöffler Johann Pflaum Jacob , Almanach nova plurimis annis venturis inservientia (Ulm, 1499); Zinner Ernst (ed.), Der deutsche Kalendar des Johannes Regiomontan, Nürnberg, um 1474 (Leipzig, 1937); Kremer Richard L. , “Thoughts on John of Saxony's method for finding true syzygy”, Historia mathematica, xxx (2003), 2003–77.

Kremer Richard L. , “Wenzel Faber's table for finding true syzygy”, Centaurus, xlv (2003), 305–29.

Münster Sebastian , Organum uranicum (Basel, 1536). For earlier manuscript versions of Münster's text and instruments (Munich, Clm 10691, Biblioteca Vaticana, Pal. Lat. 1368), see Poulle , Équatoires (ref. 4), 299–329. According to Poulle, a disk for true syzygies is not found in Clm 10691 but does appear in Pal. Lat. 1368, a manuscript I have not seen. For Münster , cf. Knapp Martin , Zu Sebastian Münsters ‘astronomischen Instrumenten’ (Basel, 1920); Burmeister , Münster (ref. 31); McLean Matthew , The cosmographia of Sebastian Münster (Aldershot, 2007).

Most copies of Münster's Organum that I have examined have no strings attached and no evidence of strings ever having been attached. I would guess that few sixteenth-century readers ever tried to use Münster's disks for computation. Gasser's copy, Biblioteca Vaticana, Pal. Lat. II.146, has threads attached to disks for the planetary equations, but the instruments for planetary latitudes and true syzygies lack threads.

Münster's two worked examples in his canon read the solar and lunar corrections to 0;05h. See Münster , Organum (ref. 40), 54–5.

The maximal correction at quadrature (15h) is greater than at the syzygies (9;40h) because, according to Ptolemaic lunar theory, the lunar epicycle is closer to the Earth at quadrature (R — e) than at syzygy (R + e) and more time is thus required to ‘correct’ for a given sweep of the lunar argument around the epicycle.

I enter Münster's solar circle at 5° intervals of the argument, reading for each the maximal — Minimal values, which are compared to Nicholaus's c₂. To compute this standard deviation, I drop four outliers of the 71 values, for arguments of = 45°, 50°, 55° and 230°.

Poulle , Équatoires (ref. 4), 325, 29; Burmeister , Münster (ref. 31), 185–90.

Price , Equatorie (ref. 4), 132.

Cf. Wattenberg Diedrich , Peter Apianus und sein Astronomicum Caesareum (Leipzig, 1967); Röttel Karl (ed.), Peter Apian: Astronomie, Kosmographie und Mathematik am Beginn der Neuzeit, 2nd edn (Eichstatt, 1997); Hamel Jürgen , “Peter Apian als Popularisator und Didaktiker der Mathematik und der Naturwissenschaften”, Acta Historica Leopoldina, liv (2008), 2008–75.

Apian , Ausßlegung (ref. 5), sig. A1v. North, Richard (ref. 3), ii, 279 suggested, without citing any evidence, that the blocks were “probably cut by Georg Apian”.

For a description based on a personal autopsy of over 100 extant copies, see Gingerich Owen , “A survey of Apian's Astronomicum Caesareum”, in Röttel (ed.), Apian (ref. 47), 113–22.

The instruments in Apian's Astronomicum have been oft described, albeit with varying degrees of perspicacity. Cf. Günther Siegmund , “Peter und Phillipp Apian, zwei deutsche Mathematiker und Kartographen”, Abhandlungen der königlich böhmischen Gesellschaft der Wissenschaften, 6th ser., xi (1882), 51–3; Ionides S. A. , “Caesar's astronomy”, Osiris, i (1936), 1936–89; Wattenberg , Apianus (ref. 47); Gingerich Owen , “Apianus's Astronomicum Caesareum and its Leipzig facsimile”, Journal for the history of astronomy, ii (1971), 1971–77; Poulle Emmanuel , “L'équatorie de l'empereur”, Archives internationales d'histoire des sciences, xxv (1975), 1975–36; Poulle , Équatoires (ref. 4), 113–22; Schmeidler Felix , “Die Scheiben in Peter Apians Astronomicum Caesareum”, in Röttel (ed.), Apian (ref. 47), 107–12; Heitzmann Christian , Die Sterne lügen nicht: Astrologie und Astronomie im Mittelalter und in der frühen Neuzeit (Wiesbaden, 2008), 95–7.

North , Richard (ref. 3), ii, 280; Poulle , Équatoires (ref. 4), 113.

North , Richard (ref. 3), ii, 282.

North , Richard (ref. 3), ii, 282–3, suggested that Apian derived the polar-coordinate design from Richard's albion; for this claim, however, North offered no evidence apart from the structural similarities of the two instruments, admitting that “the whole historical problem deserves closer attention than can be given to it here”. North's subsequent return to this question, “Coordinates and categories: The graphical representation of functions in medieval astronomy”, in Mathematics and its applications to science and natural philosophy in the Middle Ages: Essays in honor of Marshall Clagett, ed. by Grant Edward Murdoch John E. (Cambridge, 1987), 173–88, supplied no additional evidence for links between Richard and Apian and remained unaware of the tradition explicated in this paper.54. North , Richard (ref. 3), ii, 282–3.

North , Richard (ref. 3), ii, 282–3.

Tanstetter , (ed.), Tabulae (ref. 33), sig. a3v—d3r. Peurbach's table of time corrections is an expanded version of a table at 6-degree intervals prepared by John of Gmunden in the 1430s, who apparently took his from an identical table prepared in 1321 by John of Murs and Firmin of Beauval that circulated under the title of Tabulae permanentes. See Chabás Goldstein , “Five centuries” (ref. 9), 100–1; Porres Beatriz Chabás José , “John of Murs's Tabulae permanentes for finding true syzygies”, Journal for the history of astronomy, xxxii (2001), 2001–72, p. 64; Kremer Richard L. , “John of Murs, Wenzel Faber and the computation of true syzygy in the fourteenth and fifteenth centuries”, in Mathematics celestial and terrestrial: Festschriftfür Menso Folkerts zum 65. Geburtstag, ed. by Dauben Joseph W. (Stuttgart, 2008), 147–60.

In 1520–22, Apian had attended Tanstetter's mathematics lectures at the University of Vienna; Apian dedicated his 1528 edition of Peurbach's Novae theoricae planetarum to Tanstetter; in 1535 the two men jointly published an edition of Witelo's Optics. Surely Apian would have had easy access to Peurbach's eclipse tables. See Schöner Christoph , Mathematik und Astronomie an der Universität Ingolstadt im 15. und 16. Jahrhundert (Berlin, 1994), 359; Graf-Stuhlhofer Franz , Humanismus zwischen Hof und Universität: Georg Tannstetter (Collimitius) und sein wissenschaftliches Umfeld im Wien des frühen 16. Jahrhunderts (Vienna, 1996); Kühne Andreas , “Peter Apian als Herausgeber der ‘perspective commuynis’ von Witelo”, in Röttel (ed.), Apian (ref. 47), 233–8; Hayton Darin , “Instruments and demonstrations in the astrological curriculum: Evidence from the University of Vienna ca. 1500”, Studies in history and philosophy of biological and biomedical science, xli (2010), 2010–34.

Bassantin Jacques , Astronomique discours (Lyon, 1557), 233–4. Although Bassantin copied most of his instruments directly from Apian, their syzygy instruments differ fundamentally. Cf. Gingerich Owen , “Astronomical paper instruments with moving parts”, in Making instruments count, ed. by Bennett J. A. Ryan F. W. (Cambridge, 1993), 63–74, p. 72.

In De revolutionibus, IV, 28–9, Copenicus presented a new computational algorithm for the syzygy correction that has been called his “most valuable contribution to practical astronomy”. Like Bessantin's, Copernicus's method requires finding the solar and lunar corrections. See Swerdlow N. M. Neugebauer O. , Mathematical astronomy in Copernicus's De revolutionibus (New York, 1984), 276.

Pantin Isabelle , “L'illustration des livres d'astronomie à la renaissance: L'évolution d'une discipline à travers ses images”, in Immagini per conoscere: Dal rinascimento alla rivoluzione scientifica, ed. by Meroi Fabrizio Pogliano Claudio (Florence, 2001), 3–41.

Kremer Richard L. , The practice of Alfonsine astronomy in the fifteenth century: A survey of incunable calendars and almanacs (forthcoming).

Gingerich , “Leipzig facsimile” (ref. 50). I thank Owen Gingerich for allowing me to calculate with his personal copy of this facsimile.

My spreadsheet results generally agree, to the nearest minute, with results computed by John of Saxony's laborious, interative method. Cf. Kremer , “John of Saxony's method” (ref. 38), 263–77.

Kepler Johannes , New astronomy [1609], transl. by Donahue William H. (Cambridge, 1992), 234, from Kepler Johannes , Gesammelte Werke (Munich, 1937–), iii, 142. Cf. Bialas Volker , “Die rudolphinischen Tafeln von Johannes Kepler: Mathematische und astronomische Grundlagen”, Abhandlungen der bayerischen Akademie der Wissenschaften, math-naturwiss. Kl., N.F. cxxxix (Munich, 1969).