The Reception and Application of Arabic Science in Twelfth-Century Computistics: New Evidence from Bavaria

For a convenient overview, see Pedersen O. , “The ecclesiastical calendar and the life of the Church”, in Gregorian Reform of the Calendar , ed. by Coyne G. V. Hoskin M. A. Pedersen O. (Vatican City, 1983), 17–74. On the early medieval computus, see Wallis F. (transl.), Bede: The reckoning of time (Liverpool, 1999; repr. 2004); Warntjes I. , The Munich Computus: Text and translation. Irish computistics between Isidore of Seville and the Venerable Bede and its reception in Carolingian times (Stuttgart, 2010). The twelfth and thirteenth centuries will be the subject of Moreton J. M. (†), Compotus ecclesiasticus: A thirteenth-century calendar treatise in its context, which is currently being prepared for publication in the series Auctores Britannici Medii Aevi.

Two editions of this text exist: van Wijk W. É. (ed.), Le comput emendé de Reinherus de Paderborn (1171) (Amsterdam, 1951); Herold W. (ed.), Reinher von Paderborn: Computus emendatus. Die verbesserte Osterfestberechnung von 1171 (Paderborn, 2011). It is discussed in C. P. E. Nothaft, Dating the Passion: The life of Jesus and the emergence of scientific chronology (200–1600) (Leiden, 2012), 128–46. On the medieval Christian reception of the Jewish calendar, see Nothaft , Medieval Latin texts on the Jewish calendar: A study with five editions (Leiden, forthcoming 2014). Note that the present-day Jewish calendar, in use since roughly the tenth century, differs considerably from its ancient predecessors. For more information, see Stern S. , Calendar and community: A history of the Jewish calendar, second century BCE — Tenth century CE (Oxford, 2001).

See Bergmann W. , Innovationen im Quadrivium des 10. und 11. Jahrhunderts: Studien zur Einführung von Astrolab und Abakus im lateinischen Mittelalter (Stuttgart, 1985), 175–215.

On al-Khwānzmī, see Folkerts Menso Kunitzsch Paul , Die älteste lateinische Schrift über das indische Rechnen nach al-Hwārizmī (Munich, 1997) and ref. 37 below. For useful introductions to the general subject and references to further literature, see the articles assembled in C. Burnett, Numerals and arithmetic in the Middle Ages (Farnham, 2010), as well as R. Lemay, “The Hispanic origin of our present numeral forms”, Viator, viii (1977), 435–62; Folkerts M. , “Early texts on Hindu-Arabic calculation”, Science in context, xiv (2001), 13–38, repr. as chap. 2 in Folkerts, The development of mathematics in medieval Europe (Aldershot, 2006); Berggren J. L. , “Medieval arithmetic: Arabic texts and European motivations”, in Word, image, number, ed. by Contreni J. J. Casciani S. (Florence, 2002), 351–65; Burnett C. , “The semantics of Indian numerals in Arabic, Greek and Latin”, Journal of Indian philosophy, xxxiv (2006), 15–30.

Moreton J. , “John of Sacrobosco and the calendar”, Viator, xxv (1994), 229–44. The innovative role of the computus in the reception of Arabic science has also been noted by other scholars, including Faith Wallis, “The Church, the world, and the time: Prolegomena to a history of the medieval computus”, in Normes et pouvoir à la fin du moyen âge, ed. by Déprez-Masson M.-C. (Montreal, 1990), 15–29, p. 25; Lejbowicz M. , “Des tables pascales aux tables astronomiques et retour: Formation et réception du comput patristique”, Methodos: Savoirs et textes, vi (2006), http://methodos.revues.org/documents538.html.

Hermann's Abbreviatio compoti and Prognostica are jointly printed in N. Germann, De temporum ratione: Quadrivium und Gotteserkenntnis am Beispiel Abbos von Fleury und Hermanns von Reichenau (Leiden, 2006), 314–50. A new critical edition, prepared by Arno Borst (†) and completed for publication by Immo Warntjes, will appear in 2014 in the MGH series Quellen zur Geistesgeschichte des Mittelalters. The edition will be accompanied by an introduction by Immo Warntjes, who provides the first fully adequate account of the contents of these texts. I am very grateful to Dr Warntjes for letting me see his manuscript in advance. For previous attempts at an analysis, see Germann , De temporum ratione, 177–232; Cordoliani A. , “Le computiste Hermann de Reichenau”, Miscellanea storica ligure, iii (1963), 165–90; Borst A. , “Ein Forschungsbericht Hermanns des Lahmen”, Deutsches Archiv, xl (1984), 379–477; Bergmann W. , “Chronographie und Komputistik bei Hermann von Reichenau”, in Historiographia mediaevalis, ed. by Berg D. Goetz H.-W. (Darmstadt, 1988), 103–17.

The text, still unedited, is preserved in MS Oxford, Bodleian Library, Auct. F.1.9, fols 86ra–96ra. See McCluskey S. C. , Astronomies and cultures in early medieval Europe (Cambridge, 1998), 180–82, for an account of Walcher's achievement.

An edition of Gerland's text will soon be published by Alfred Lohr in the series Sudhoffs Archiv: Beihefte, vol. lxi. See also Moreton J. , “Before Grosseteste: Roger of Hereford and calendar reform in eleventh- and twelfth-century England”, Isis, lxxxvi (1995), 562–86.

On the general context of what is to follow, see Mercier R. , “Astronomical tables in the twelfth century”, in Adelard of Bath , ed. by Burnett C. (London, 1987), 87–118, repr. as chap. 7 in Mercier, Studies on the transmission of medieval mathematical astronomy (Aldershot, 2004); Mercier, “East and West contrasted in scientific astronomy”, in Occident et Proche-Orient, ed. by Draelants I. Tihon A. van den Abeele B. (Turnhout, 2000), 325–42; Chabás J. Goldstein B. R. , A survey of European astronomical tables in the late Middle Ages (Leiden, 2012). See also the pioneering study by Haskins C. H. , Studies in the history of mediaeval science (2nd edn, Cambridge, MA, 1927; repr. New York, 1960), 82–140.

See Samsó J. , Las ciencias de los antiguos en al-Andalus (Madrid, 1992), 84–93. A tenth-century commentary on these tables by Ibn al-Muthanna was translated into Latin by Sanctallensis Hugo . See Vendrell Millás E. , El comentario de Ibn al-Muṭannā a la Tablas Astronómicas de al-Jwārizmī: Estudio y edición crítica del texto latino, en la versión de Hugo Sanctallensis (Madrid, 1963), 96–102.

See Tolan J. , Petrus Alfonsi and his medieval readers (Gainesville, 1993), 55–61; Metlitzki D. , The matter of Araby in medieval England (New Haven, CT, 1977), 16–26; Casulleras J. , “Las tablas astronómicas de Pedro Alfonso”, in Estudios sobre Pedro Alfonso de Huesca, ed. by Lacarra M. J. (Huesca, 1996), 349–66.

MS Oxford, Bodleian Library, Auct. F.1.9, fols 96ra–99ra. I take the translation of the title from Burnett C. , The introduction of Arabic learning into England (London, 1997), 39. See Vallicrosa Millás J. M. , “Pedro Alfonso's contribution to astronomy”, Aleph, x (2010), 139–68; Tolan , Petrus Alfonsi (ref. 11), 61–6; McCluskey , Astronomies (ref. 7), 182–5.

Adelard's translation was edited by Suter H. , Die astronomischen Tafeln des Muḥammed ibn Mūsā al-KHwārizmī (Copenhagen, 1914; repr. Frankfurt/Main, 1997), and translated into English by Neugebauer O. , The Astronomical Tables of Al-KHwārizmī (Copenhagen, 1962). See further Cochrane L. , Adelard of Bath: The first English scientist (London, 1994), 73–84; Burnett C. , “The works of Petrus Alfonsi: Questions of authenticity”, Medium aevum, lxvi (1997), 42–79, pp. 52–5. The commonly cited date of 1126 is uncertain, as Neugebauer, The Astronomical Tables, 232, and Mercier, “Astronomical tables” (ref. 9), 99 n34, point out.

For an early reaction to this problem, see the entry for 1138 in John of Worcester's chronicle, which (incorrectly) states the beginning of the Arabic year in 1138 “so that the work which in Arabic is called ‘Ezich’ and which the learned al-KHwārizmī wrote most carefully on the course of the seven planets, and laid out in tables, is not consigned to oblivion”. See the English translation in The Chronicle of John of Worcester, iii: The annals from 1067 to 1140, ed. by McGurk P. (Oxford, 1998), 259–61.

Mercier , “Astronomical tables” (ref. 9), 95.

Suter , Die astronomischen Tafeln (ref. 13), 3.

Ibid., 110.

Cited according to Neugebauer, The Astronomical Tables (ref. 13), 12.

Ibid., 14; Mercier, “Astronomical tables” (ref. 9), 99. The only table that comes close is Tab. 3 in Suter, Die astronomischen Tafeln (ref. 13), 113.

See Pedersen F. S. , The Toledan Tables (4 vols, Copenhagen, 2002), i, 216–30; ii, 382–400, 505–7, 588–602; iii, 884–7, 901–21, 929–35, 940–4.

Ibid., i, 11–20.

Ibid., i, 37–43.

Raymond of Marseille, Liber cursuum planetarum, in Opera omnia, i, ed. by d'Alverny M.-T. Burnett C. Poulle E. (Paris, 2009), 126–341.

Burnett C. , “Hereford, Roger of (fl. 1176–1198)”, Oxford dictionary of national biography , http://www.oxforddnb.com/view/article/23955, accessed 2 May 2013. Haskins, Studies (ref. 9), 124–6; Russell J. C. , “Hereford and Arabic science in England about 1175–1200”, Isis, xviii (1932), 14–25; Metlitzki , The matter (ref. 11), 39–40; French R. , “Foretelling the future: Arabic astrology and English medicine in the late twelfth century”, Isis, lxxxvii (1996), 453–80.

MS London, British Library, Arundel 377, fol. 86vb: “Maluimus enim haec quam annos arabum et eorum menses propter difficultatem sequi, eo quod inusitata sint apud nostrates.”.

See Moreton , “Before Grosseteste” (ref. 8), 581–4.

For the general background, see Kaltenbrunner Ferdinand , “Die Vorgeschichte der gregorianischen Kalenderreform”, Sitzungsberichte der kaiserlichen Akademie der Wissenschaften (Wien), phil.-hist. Kl., lxxxii (1876), 289–414, and North John , “The Western calendar — ‘intolerabilis, horribilis, et derisibilis’: Four centuries of discontent”, in Gregorian Reform of the calendar, ed. by Coyne Hoskin Pedersen (ref. 1), 75–113, repr. in North, The universal frame (London, 1989), 39–77. See also ref. 70 below.

Grosseteste Robert , Computus correctorius (c. 4–5), ed. in Steele R. , Opera hactenus inedita Rogeri Baconi, vi (Oxford, 1926), 232–40.

See the third book of Giles of Lessines's Summa de temporibus (c. 1.18; 3.1–3, 6), which was erroneously edited as the Compotus of Roger Bacon, in Steele, ibid., 73–6, 151–9, 167–79.

Bacon Roger , Opus majus , ed. by Bridges J. H. (3 vols, Oxford, 1897–1900), i, 275–85; Bacon, Opus tertium (c. 70), in Opera quaedam hactenus inedita, i, ed. by Brewer J. S. (London, 1859), 281–95.

Campanus of Novara, “Computus maior” (c. 15), in Sphera mundi noviter recognita cum commentariis (Venice, 1518), fols 167va–69rb.

See Thorndike L. Kibre P. , A catalogue of incipits of mediaeval scientific writings in Latin (rev. edn, London, 1963), 241, 486, 1234; Benjamin F. S. Jr Toomer G. J. , Campanus of Novara and medieval planetary theory (Madison, 1971), 16, n58; Pedersen , The Toledan Tables (ref. 20), ii, 505–6; iii, 919–20.

The Tractatus de vero ciclo lunari was printed in Pierre d'Ailly, Tractatus de imagine mundi et varia ejusdem auctoris et Joannis Gersonis opuscula (Louvain, c. 1483), sigs. h2v—h5v.

Elements of the Muslim lunar calendar could also be gleaned from numerous other tables and texts that became newly available in the twelfth century. One example is chap. 1 of al-Farghānī's Elements of astronomy, first translated in 1135 by John of Seville. See Al Farghani Differentie scientie astrorum, ed. by Carmody F. J. (Berkeley, CA, 1943). Another translation, made in 1175 by Gerard of Cremona, was published as Il “Libro dell' aggregazione delle stelle”, ed. by Campani R. (Città di Castello, 1910), 56–7. See further Vallicrosa Millás J. M. (ed.), El libro de los fundamentos de las Tablas astronómicas de R. Abraham Ibn Ezra (Madrid, 1947), 98–9; Vallicrosa Millás , “Una obra astronomica desconocida de Johannes Avendaut Hispanus”, Osiris, i (1936), 451–75, pp. 461–5; Vendrell Millás , El comentario (ref. 10), 96–102.

See Lemay , “The Hispanic origin” (ref. 4), Figs 1a and 1b. See also Burnett C. , “Indian numerals in the Mediterranean Basin in the twelfth century, with special reference to the ‘Eastern forms’”, in From China to Paris: 2000 years transmission of mathematical ideas, ed. by Dold-Samplonius Y. (Stuttgart, 2002), 237–88, pp. 241, 270, repr. as chap. 5 in Burnett, Numerals (ref. 4). See also the comparison of numeral forms in A. Allard, “L'époque d'Adélard et les chiffres Arabes dans les manuscrits latins d'arithmétique”, in Burnett, Adelard (ref. 9), 37–43, p. 40.

This codex has been described many times, e.g., in Curtze M. , “Ueber eine Algorismus-Schrift des XII. Jahrhunderts”, Abhandlungen zur Geschichte der Mathematik , viii (1898), 1–27, pp. 1–8; Folkerts M. (ed.), ”Boethius” Geometrie II: Ein mathematisches Lehrbuch des Mittelalters (Wiesbaden, 1970), 9–12; Schmitz H.-G. , Kloster Prüfening im 12. Jahrhundert (Munich, 1975), 117–21; Toneatto L. , Codices artis mensoriae (3 vols, Spoleto, 1994–95), iii, 1123–6. Based on the identification of the main scribe with the monk Sigboto of Prüfening, the earliest parts have been dated to the years 1163 to 1168, but this was rejected on palaeographic grounds by H. von Fichtenau, “Wolfger von Prüfening”, Mitteilungen des österreichischen Instituts für Geschichtsforschung, li (1937), 313–57, p. 318. See also Pedersen, The Toledan Tables (ref. 20), i, 135–6.

The text was edited in full by Dickey B. G. , “Adelard of Bath: An examination based on heretofore unexamined manuscripts” (Ph.D. diss., University of Toronto, 1982), 251A–328. The first three books on arithmetic are also found in A. Allard, Muhammad Ibn Mūsā Al-KHwārizmī: Le calcul indien (Algorismus) (Paris, 1992), 23–61. See further Ambrosetti N. , L'eredità arabo-islamica nelle scienze e nelle arti del calcolo dell' Europa medievale (Milan, 2008), 197–213.

For descriptions, see Dickey , “Adelard” (ref. 37), 231–43; Allard, Muhammad (ref. 37), pp. xxxvi–xxxviii; Pedersen, The Toledan Tables (ref. 20), i, 135–6, 165–6 (MSS MOP only).

The version of the Liber ysagogarum in Clm 18927 only encompasses three books and is considerably different. See Allard, Muhammad (ref. 37), pp. xxi–xxiii.

MS A, where the designation is changed to Liber ysgogarum Alchoarismi ad totum quadrivium, does not have such tables.

See, e.g., De divisionibus temporum liber, in Migne J.-P. (ed.), Patrologia Latina, xc (Paris, 1850), cols 653–5, where the instantia are referred to as atomi, which is the more common term.

Allard, Muhammad (ref. 37), 24–5: “Sunt tamen uniuscuiusque planetae anni proprii, unde Arabes quos immitaturi sumus annos lunae secuntur. Egyptii vero pro compendio partes temporis alia denominatione sexagenaria scilicet habentes plurimes partes utpote secundam, tertiam, quartam, quintam, sextam hoc modo vocantes minuta, secunda, tertia, quarta excogitavere. Quantocunque enim aliquis numerus plures partes habuerit, tanto melius dividitur, ut ĪĪDXX. Et quia omnium numerorum praetermissa doctrina scientia nulla procedit ab ipsis nostri tractatus initium ratione Indorum sumatur.”.

See, e.g., Haskins , Studies (ref. 9), 24; Lemay, “The Hispanic origin” (ref. 4), 446, n46; Dickey, “Adelard” (ref. 37), 81–2, 96–7; Mercier, “Astronomical tables” (ref. 9), 95–6; Allard, Muhammad (ref. 37), p. ix; Burnett, “The works” (ref. 13), 51–2.

See Neugebauer , The Astronomical Tables (ref. 13), 137–45; Casulleras, “Las tablas” (ref. 11), 350–1.

Neugebauer , The Astronomical Tables (ref. 13), 139, 212.

See Dickey , “Adelard” (ref. 37), 77–9, 110–11; Cochrane, Adelard of Bath (ref. 13), 84, n32; Burnett, “The works” (ref. 13), 51–2; Allard, Muhammad (ref. 37), pp. viii–xxi.

I am greatly indebted to Immo Warntjes for bringing this manuscript to my attention.

For reproductions of MS P, fol. 70va, see Fig. 2 in Lemay, “The Hispanic origin” (ref. 4), and C. Burnett, “Algorismi vel helcep decentior est diligentia: The arithmetic of Adelard of Bath and his circle”, in Mathematische Probleme im Mittelalter, ed. by Folkerts M. (Wiesbaden, 1996), 221–331, p. 315, repr. as chap. 3 in Burnett, Numerals (ref. 4).

Robinson I. S. , The Papacy, 1073–1198: Continuity and innovation (Cambridge, 1990), 487–8; Feuchtner M. , “Erzbischof Eberhard I. von Salzburg (1089–1164)”, Beiträge zur Geschichte des Bistums Regensburg, xix (1985), 139–284, pp. 149–53, 225–59.

On the early medieval argumenta and the history of their transmission, see Warntjes I. , “The Argumenta of Dionysius Exiguus and their early recensions”, in Computus and its cultural context in the Latin West, AD 300–1200 , ed. by Warntjes I. Cróinín D. Ó (Turnhout, 2010), 40–111.

Vienna MS , ÖNB, cod. lat. 2453, fols 6r: “Sed secundum epactas in occasu solis, ut in presenti anno MCLV….” Ibid., fol. 6v: “Pro concurrentibus vero et indictionibus numerentur articuli manus usque ad presentem annum, que est XVI, initio sumpto a primo anno residui, nunc habente primum concurrentem cum bissexto et tertia indictione.” Ibid., fol. 7ra: “In anno domini MCLV invenitur epacta VII primi mensis anni lunaris tunc incipientis III nonarum Martii, eo acervandum regularibus mensium ad ipsorum secundum naturalem cursum initia semper invenienda.”.

The complete text of these argumenta is edited below in the Appendix.

Vienna MS , ÖNB, cod. lat. 2453, fol. 6v: “Et de annis a principio mundi lunaribus quidem IIII. DCCCCXVI et IIII menses et VI dies, in quibus quotiens XVIIII sunt totiens septeni menses in annos reducendi, cum VI etiam mensibus sumendi, sunt.”.

Munich MS , BSB, Clm 13021, fol. 30rb—va: “Hec est autem regula inventionis annorum Arabum per Hebraicos. Est quidem primo notandum Hebreos diem in 25920 minuta dividere. Et mensem in dies 29 et 12 horas ac 793 [Ms.: 739] minuta partiri. Hac quoque ratione annum ex diebus 354 et minutis 9516 constare. Annorum perfectorum summa per 19 dividatur convenit. Cuius divisionis denominatio quotiens unitate habuerit, totiens 7 [Ms.: 172] menses predicte summe tribuantur oportet. Nam denominatio per septenarium multiplicanda est. Inde est autem ad annos constituendos 12 dividenda. Cumque menses inde resterint cum residuis a prima divisione, si 12 compleverunt, predictis annis copulentur. Si vero non, imperfecti anni menses erunt, quorum principium est a Nisan consimili Martio. Et sciendum est quod si de eadem prima divisione 3 remanserint, 1 menses; si 6, 2; si 9, 3; si 11, 4; si 14, 5; si vero 17, 6 sumuntur. Ad presentem vero diem inveniendum, tota summa per septenos dies est dividenda. Quod residuum fuerit diebus est partiendum additis tamen duobus diebus et 5 horis ac 204 minutis et initio sumpto a die Saturni in quo dierum numerus defecerit. Ipsum presentem esse notandum est. Ad annos autem Arabum inveniendos, differentia tantum videlicet 4516 anni et 11 menses, de Hebraicis subtrahantur. Residuum itaque et annos et menses et dies Arabum manifeste demonstrant.” Cf. Dickey, “Adelard” (ref. 37), 314–15.

Vienna MS , ÖNB, cod. lat. 2453, fol. 7ra: “Sic ergo singulis annis adiectis XI, cum super XXX creverit tercenarius pro unitate numero qui excreverit presenti anno connumeratur, sed in subsequenti, sicut assignavimus, nequaquam computatur, sicque post annos XXX redeunt ad initia sua.”.

Note that the orderly sequence of ‘full’ and ‘hollow’ months indicated in the column to the right (marked demptiones) conceals the fact that there are two lunar months in June, one 30-day month ending on 1 June and another 29-day month ending on 30 June. As a result, the second half of the list has the month lengths in the wrong order, since the lunation assigned to July must consist of 30 days, the one for August must have 29 days etc. The text on fol. 7rb states that that “the regulars of the first month are II, of the second month I, and so forth [regulares autem primi mensis sunt II, secundi I etc.]”, which makes no sense in the context of the table.

Vienna MS , ÖNB, cod. lat. 2453, fol. 7rb: “Sciendum quoque est initium anni lunaris moveri infra solarem antequam transeat primum initium per XXXIII annos solares. Tunc ultra primum initium incipit iterum quinta die, unde etiam eius epacta numeranda est. Tenenda est quoque cuiusque anni nota.”.

See Lemay , “The Hispanic origin” (ref. 4), 458 and Fig. 4b. See also Curtze , “Ueber eine Algorismus-Schrift” (ref. 36), 3, 9. Another instance of this numeral form appears in the fragment of the Liber ysagogarum found in MS Genova, Biblioteca Universitaria, E.III.28, fol. 231v (s. XV).

Arrighi G. , “La numerazione ‘arabica’ degli Annales Ratisbonenses”, Physis, x (1968), 243–57.

Fichtenau , “Wolfger” (ref. 36), 321–4. See also Probst R. , “Die Regensburger und die Prüfeninger Annalen: Reflexion des Forschungsstandes und textkritische Untersuchungen”, Beiträge zur Geschichte des Bistums Regensburg, xxxiii (1999), 7–12.

Reproduced in Menninger K. , Zahlwort und Ziffer: Eine Kulturgeschichte der Zahl (2nd edn, 2 vols, Göttingen, 1958), ii, 239. See further Nagl A. , “Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande”, Part 1, Zeitschrift für Mathematik und Physik, hist.-lit. Abt., xxxiv (1889), 129–46, p. 134; Mazal O. Nemeth I. , Wissenschaft im Mittelalter: Ausstellung von Handschriften und Inkunabeln der Österreichischen Nationalbibliothek. Prunksaal 22. Mai bis 18. Oktober 1975 (Vienna, 1975), 190–1.

As already argued by Fichtenau, “Wolfger” (ref. 36), 320, n4.

The year is expressly mentioned on fol. 29r. See Unterkircher F. , Die datierten Handschriften der österreichischen Nationalbibliothek bis zum Jahre 1400 (Vienna, 1969), 18.

See, e.g., Borst A. , “Computus: Zeit und Zahl im Mittelalter”, Deutsches Archiv, xliv (1988), 1–82, p. 44. The text is habitually referred to as the ‘Salzburger Computus’ in the literature, despite the fact that there is no evidence linking it to Salzburg.

This confirms the doubts expressed by Fichtenau , “Wolfger” (ref. 36), 320, n6.

Vienna MS , ÖNB, cod. lat. 275, fol. 29v: “Lunaris autem mensis a coniunctione solis ad coniunctionem ex 29 diebus et 12 horis ac 792 partibus unius hore divise in 1080 partes constat. Sed annus eius in 354 diebus et 5a 76a unius diei completur.”.

The term ‘archaic version’ for these canons reflects their strong reliance on ‘KHwārizmīan’ material, but also the fact that they are rarely found in manuscripts after the twelfth century. For details, see Pedersen , The Toledan Tables (ref. 20), ii, 571–81; iii, 813–14, 826–7. See also Pedersen F. S. , “Alkhwarizmi's Astronomical Rules: Yet another Latin version?”, Cahiers de l'Institut du Moyen-Âge Grec et Latin, lxii (1992), 31–75.

MSS P, fol. 102r; M, fol. 32v; O, fol 18r; Pedersen, The Toledan Tables (ref. 20), ii, 598–600; iii, 843, 902, 914–15.

The only exception is the number of days in the last line, which is given as 29 instead of 22.

See most recently Nothaft C. P. E. , “Reforming the calendar at the University of Salamanca ca. 1468: Pedro Martinez de Osma and his Disputatio de anno…”, Humanista, xxiii (2013), 522–56; Nothaft, “Duking it out in the arena of time: Chronology and the Christian-Jewish encounter (1100–1600)”, in Religious criticism and the growth of knowledge, ed. by Hames Harvey (= Medieval encounters, special volume; forthcoming, 2014). I am currently preparing a large-scale survey on the history of calendar reform during the Middle Ages, which will also take into account the sources discussed in the present article.

A follow-up study, which will also discuss the evidence of MS Leipzig, Universitätsbibliothek, 328, will be undertaken in collaboration with Warntjes Immo .

See “Rhythmi computistici”, in MGH Poetae, iv/2/3, ed. trecker Karl (Berlin, 1923), 671.