Late Antique Calendrical Thought and its Reception in the Early Middle Ages 978-2-503-57710-4

Table of contents :
Abbreviations ix
Foreword xi
IMMO WARNTJES, Introduction: state of research on late antique
and early medieval computus 1
ALDEN A. MOSSHAMMER, Towards a new edition of the Computus
of AD 243 43
JAN ZUIDHOEK, The initial year of De ratione paschali and the
relevance of its paschal dates 71
DANIEL MC CARTHY, The paschal cycle of St Patrick 94
LUCIANA CUPPO, Felix of Squillace and the Dionysiac computus
II: Rome, Gaul, and the insular world 138
BRIGITTE ENGLISCH, Osterfest und Weltchronistik in den
westgotischen Reichen 182
DAVID HOWLETT, An addition to the Hiberno-Latin canon:
De ratione temporum 212
MARINA SMYTH, Once in four: the leap year in early medieval
thought 229
C. PHILIPP E. NOTHAFT, Chronologically confused: Claudius of
Turin and the date of Christ’s passion 265
LISA CHEN OBRIST, What a difference a day makes: the eighth
day of the week in book 10 of Hrabanus Maurus’ De rerum
naturis 293
DÁIBHÍ Ó CRÓINÍN, Archbishop James Ussher (1581–1656)
and the history of the Easter controversy 309
Bibliography 352
index 3

Citation preview

STUDIA TRADITIONIS THEOLOGIAE Explorations in Early and Medieval Theology

Theology continually engages with its past: the people, experience, Scriptures, liturgy, learning, and customs of Christians. The past is preserved, rejected, modified; but the legacy steadily evolves as Christians are never indifferent to history. Even when engaging the future, theology looks backwards: the next generation’s training includes inheriting a canon of Scripture, doctrine, and controversy; while adapting the past is central in every confrontation with a modernity. This is the dynamic realm of tradition, and this series’s focus. Whether examining people, texts, or periods, its volumes are concerned with how the past evolved in the past, and the interplay of theology, culture, and tradition.

STUDIA TRADITIONIS THEOLOGIAE Explorations in Early and Medieval Theology 26 Series Editor: Thomas O’Loughlin, Professor of Historical Theology in the University of Nottingham

EDITORIAL BOARD

Director Prof. Thomas O'Loughlin Board Members Dr Andreas Andreopoulos, Dr Nicholas Baker-Brian, Dr Augustine Casiday, Dr Mary B. Cunningham, Dr Juliette Day, Dr Johannes Hoff, Dr Paul Middleton, Dr Simon Oliver, Prof. Andrew Prescott, Dr Patricia Rumsey, Dr Jonathan Wooding, Dr Holger Zellentin

LATE ANTIQUE CALENDRICAL THOUGHT AND ITS RECEPTION IN THE EARLY MIDDLE AGES

Proceedings of the 3rd International Conference on the Science of Computus in Ireland and Europe Galway, 16–18 July, 2010 Edited by Immo Warntjes & Dáibhí Ó Cróinín

H

F

Cover illustration: Tabula Peutingeriana ©  ÖNB Vienna: Cod. 324, Segm. VIII + IX

© 2017, Brepols Publishers n.v., Turnhout, Belgium All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the publisher. D/2017/0095/215 ISBN 978-2-503-57709-8 e-ISBN 978-2-503-57710-4 DOI 10.1484/M.STT-EB.5.113899 Printed on acid-free paper

To the memory of Luciana Cuppo

TABLE OF CONTENTS

Abbreviations

ix

Foreword

xi

IMMO WARNTJES, Introduction: state of research on late antique and early medieval computus

1

ALDEN A. MOSSHAMMER, Towards a new edition of the Computus of AD 243

43

JAN ZUIDHOEK, The initial year of De ratione paschali and the relevance of its paschal dates

71

DANIEL MC CARTHY, The paschal cycle of St Patrick

94

LUCIANA CUPPO, Felix of Squillace and the Dionysiac computus II: Rome, Gaul, and the insular world

138

BRIGITTE ENGLISCH, Osterfest und Weltchronistik in den westgotischen Reichen

182

DAVID HOWLETT, An addition to the Hiberno-Latin canon: De ratione temporum 212 MARINA SMYTH, Once in four: the leap year in early medieval thought

229

C. PHILIPP E. NOTHAFT, Chronologically confused: Claudius of Turin and the date of Christ’s passion

265

LISA CHEN OBRIST, What a difference a day makes: the eighth day of the week in book 10 of Hrabanus Maurus’ De rerum naturis 293 DÁIBHÍ Ó CRÓINÍN, Archbishop James Ussher (1581–1656) and the history of the Easter controversy

309

Bibliography

352

index

375

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ABBREVIATIONS

CCCM Corpus Christianorum, Continuatio Mediaevalis CCSL Corpus Christianorum, Series Latina CLA Codices Latini Antiquiores CSEL Corpus Scriptorum Ecclesiasticorum Latinorum GCS  Die griechischen christlichen Schriftsteller der ersten Jahrhunderte LM Lexikon des Mittelalters MGH Monumenta Germaniae Historica Auct. ant. Auctores antiquissimi Epp. Epistolae (in Quart) LL nat. Germ. Leges nationum Germanicarum Poetae Poetae Latini medii aevi SS rer. Merov. Scriptores rerum Merovingicarum PG Patrologia Graeca PL Patrologia Latina

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FOREWORD

This volume of essays represents the Proceedings from the third International Conference on the Science of Computus in Ireland and Europe, which took place in Galway, 16–18 July, 2010. It follows (belatedly) a volume of papers from the previous Proceedings of the First Galway Conference of 2006, that had the sub-title Computus and its cultural context in the Latin West, AD 300–1200 (2010), which brought together papers by ten of the leading scholars in the field, whose papers ranged from the origins of the Anno Domini system of time-reckoning to the study of computus in Ireland c.AD 1100. All those present at that pioneering first event were agreed that a second, follow-up conference should be organised, which duly took place (also in Galway), 18–20 July 2008. The Proceedings of that volume too appeared subsequently, with the title, The Easter Controversy of Late Antiquity and the Early Middle Ages (2011), with eleven contributions ranging from the Easter-reckonings of Rome and Milan in the late fourth century down to the Liber de astronomia of the Irishman Dicuil c.AD 800. It took longer than Immo or I had expected to produce the papers of the third Galway Conference, but we hope that you will agree that the wait was worth it! This volume brings together papers that range in date from the Computus of AD 243 through the computistical writings of the ninth-century authors Claudius of Turin and Hrabanus Maurus down to the great seventeenth-century scholar and chronographer, Archbishop James Ussher (†1656). One of the contributions, by Leofranc Holford-Strevens on the Disputatio Chori, grew to monographlength and will be published separately in the Studia Traditionis Theologiae series. As Conference organisers, Immo and I were delighted to

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Foreword

welcome back to Galway many of the colleagues who had been with us for the first two gatherings, but also some younger scholars, such as Lisa Chen and Philipp Nothaft. One of the very gratifying aspects of the Galway Conferences, during those first years and in the years since, is the way that younger scholars have joined our ranks, and the enthusiasm with which the more established scholars have always welcomed them into their ranks. Several of these younger colleagues have since gone on to establish themselves among the first rank of scholars in the field. The Third Conference attracted speakers from Canada, England, Germany, Ireland, Italy, and the Netherlands. Sadly, however, Luciana Cuppo, one of our founding-members and a regular speaker at our previous gatherings, passed away before she could see the fruits of her present labours in print. Luciana did, however, devote her final weeks to ensuring that we received a clean copy of her paper; thanks to the generosity and collaboration of her family, that paper is presented here in the form in which she would have wanted to see it. We cherish fond recollections of Luciana’s good-humoured and erudite contributions to all the Galway Conferences, and we take this opportunity to mark our respect for her profound scholarship by dedicating this collection to her memory. A regular feature of the Galway Conferences has been the launch of (at least) one book related to our activities. At the 2010 Conference we had the pleasure of launching the Proceedings of the First Conference, Computus and its cultural context in the Latin West, AD 300–1200 (Studia Traditionis Theologiae 5), and Immo’s magnificent volume, The Munich Computus: text and translation. Irish computistics between Isidore of Seville and the Venerable Bede and its reception in Carolingian times (Sudhoffs Archiv 59). We look forward with anticipation to the Seventh Conference, which will take place—in Galway!—in July 2018, when we hope to be able to present these Conference Proceedings to our peers. Preparation is already under way to publish the papers of the intervening Conferences in the years to come. We would like to thank all those whose encouragement and help have made the Galway Computus Conferences to date such an outstanding success. These include the authorities in the National University of Ireland, Galway, which has hosted all of our events and provided generous funding for them; the Director and staff of the Moore Institute for the Humanities, where the Conferences have always taken place; Hilary Murphy in the Buildings Office, NUIG, for her unfailing kindness and efficiency, and last—but certainly not least—Maura Walsh (Ó Cróinín),

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Foreword

who once again designed the wonderful Conference poster and programme. The NUIG Medieval MA students were also a great help with the day-to-day running of the Conference. We owe a particular debt to George Janzen (University of Leipzig), who was an invaluable help with the editorial preparation of the Proceedings. To Immo, in particular, I owe special thanks for having once again taken on, almost single-handedly, the heavy burden of seeing the Proceedings through the editorial process. If it has taken longer than we anticipated to nurse the volume into print, that is no fault of his. I would also like to thank the contributors to the volume, for their patience and forbearance when it appeared that the volume might not see the light of day. Immo and I hope that they—and the other readers of the papers—will feel that the volume was worth the wait. We are grateful also to Prof. Thomas O’Loughlin (Nottingham), who has continued to offer a home in his series of Studia Traditionis Theologiae, and to Dr Bart Janssens, Brepols, who, by his steadfast support for the publication of these volumes, has encouraged us to believe in the value of what we have been doing. It remains only to thank those institutions and repositories that granted permission to reproduce images from manuscripts in their care: the British Library, London; the Pierpont Morgan Library, New York; the Archivo de la Catedral, León; the Bodleian Library, Oxford. Finally, it should be noted that the contributions to this collection (as in previous Proceedings of the Galway Computus Conference) are the responsibility of the individual authors and do not necessarily represent the views either of the editors or of the publishers. Dáibhí Ó Cróinín, Galway, 12 May 2017

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IMMO WARNTJES

INTRODUCTION: STATE OF RESEARCH ON LATE ANTIQUE AND EARLY MEDIEVAL COMPUTUS

The essays in this volume emerged from papers presented at the 3rd International Conference on the Science of Computus in Galway in July 2010. The conference was established in 2006 and has since been held biannually. It may be fair to say that before this date, the late antique and medieval science of computus was safely in the hands of only a few specialists.1 Computus is a decidedly Christian science of calendrical calculations, which evolved around the need to establish the date of Easter. Since the third century, Easter was supposed to fall on the first Sunday after the first full moon after the spring equinox, as is still the custom today. Therefore, its calculation depended, from a late antique and medieval perspective, on the mathematical modeling of the tropical year (the Julian calendar) and of the lunar phases—the synodic lunar months, i.e. the periods from one new moon to the next (84-year or 19-year lunar cycle). From the seventh century onwards, computus evolved from its core purpose of calculating Easter to become a more wide-ranging subject explaining the natural world as part of the quest of understanding God’s creation. In the 16th and 17th centuries, scholars like Scaliger, Petavius, and Bucherius used computistical texts for their study of historical chronology The following will also be published separately on , and will be updated on an annual basis. Writing a biographical essay is a difficult task, as it will never be fully comprehensive and it reflects, probably more than other academic publications, the interests and biases of the author. I am extremely grateful to Philipp Nothaft and Dáibhí Ó Cróinín for discussing an early draft with me and for pointing out some publications I had initially neglected. As this essay is designed as a survey of past and present publications on computus, it was agreed that, contrary to the procedure in the rest of this volume, full bibliographical details are provided in the notes. 1

Late Antique Calendrical Thought and its Reception in the Early Middle Ages, ed.  by Immo Warntjes and Dáibhí Ó Cróinín, Turnhout, Brepols, 2017 (Studia Traditionis Theologiae, 26), pp. 1-42 © BREPOLSHPUBLISHERSDOI 10.1484/M.STT-EB.5.114732

IMMO WARNTJES

and early church history.2 At the same time, James Ussher tried to reconstruct the Easter controversy (i.e. the dispute about the theological and scientific accuracy of conflicting lunar calendars) on this basis (as demonstrated by Dáibhí Ó Cróinín in the present volume).3 The title of founding father of the modern study of late antique and medieval computus, however, must go to the early 18th-century Dutch scholar Johannes van der Hagen, whose studies of the 1730s are still worth consulting.4 His lead was most prominently followed by Bruno Krusch. Initially, Krusch wanted to work on the Easter reckoning of Victorius of Aquitaine (published in AD 457) for his Ph.D. However, he soon realized that, in order to do so, he first needed to come to grips with the pre-mid-fifth century tradition.5 This study culminated in his celebrated Studien of 1880, which contains the standard edition of numerous key texts (though many of Krusch’s theories there expressed, like that of an older and newer Supputatio Romana, have since been superseded). Shortly before his death in 1940, Krusch eventually produced new editions of the writings of Victorius of Aquitaine and Dionysius Exiguus.6 In the meantime, the Irish Scaliger, J.  J. (1583) De emendatione temporum, Paris, 2nd  ed. Geneva 1629; Petavius, D. (1627) Opus de doctrina temporum, 2 vols, Paris; Bucherius, A. (1634) De doctrina temporum. Commentarius in Victorium Aquitanum […] aliosque antiques canonum paschalium scriptores, Antwerp. 3 Ussher, J. (1639) Britannicarum ecclesiarum antiquitates, Dublin. 4 van der Hagen, J. (1733) Observationes in Prosperi Aquitani chronicon integrum ejusque LXXXIV annorum cyclum, et in anonymi cyclum LXXXIV annorum a Muratorio editum, nec non in anonymi laterculum paschalem centum annorum a Bucherio editum, Amsterdam; van der Hagen, J. (1734) Observationes in veterum patrum et pontificium, prologos et epistolas paschales, aliosque antiquos de ratione paschali scriptores, Amsterdam; van der Hagen, J. (1736) Observationes in Heraclii imperatoris methodum paschalem; ut et in Maximi monachi computum paschalem; nec non in anonymi chronicon paschale, Amsterdam; van der Hagen, J. (1736) Dissertationes de cyclis paschalibus […], ut et de enneadecaeteridis […], nec non de computo solari, Amsterdam. 5 Krusch, B. (1880) Studien zur christlich-mittelalterlichen Chronologie: Der 84jährige Ostercyclus und seine Quellen, Leipzig, V; Krusch, B. (1938) ‘Studien zur christlich-mittelalterlichen Chronologie: Die Entstehung unserer heutigen Zeitrechnung,’ Abhandlungen der Preußischen Akademie der Wissenschaften, Jahrgang 1937, phil.-hist. Klasse, Nr. 8, Berlin, 5–6. 6 Krusch (1938); the 1892 edition of Victorius by Theodor Mommsen in MGH Auct. ant. 9, 667–735, was based on Krusch’s notes. Of the other numerous important publications by Krusch, see especially Krusch, B. (1884) ‘Die Einführung des griechischen Paschalritus im Abendlande,’ Neues Archiv der Gesellschaft für ältere deutsche Geschichtskunde 9, 99–169; Krusch, B. (1884) ‘Über eine Handschrift des Victurius,’ Neues Archiv der Gesell­ schaft für ältere deutsche Geschichtskunde 9, 269–81; Krusch, B. (1885) ‘Chronologisches aus Handschriften,’ Neues Archiv der Gesellschaft für ältere deutsche Geschichtskunde 10, 81–94; Krusch, B. (1910) ‘Das älteste fränkische Lehrbuch der dionysianischen Zeitrechnung,’ in Mélanges offerts à M. Émile Chatelain, Paris, 232–42; Krusch, B. (1933) ‘Neue Bruchstücke 2

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STATE OF RESEARCH ON COMPUTUS

scholar Bartholomew Mac Carthy and the German classicist Eduard Schwartz, in two seminal studies that still remain core readings on the subject (unfortunately both composed in somewhat inaccessible language), corrected many of Krusch’s youthful and extremely bold reconstructions.7 Mac Carthy introduced computus into the study of medieval Ireland, tackling the intricacies of what he believed to have been the native tradition (the 84 (14)-year Easter reckoning now known as the latercus). Another  true pioneer in the study of Irish computistics was Mario Esposito. He certainly was not afraid of tackling difficult texts. His first publication, of 1907, provides a study and transcript of Dicuil’s Liber de astronomia, one of the most interesting texts of the Carolingian age, which still awaits a full-scale study and critical edition.8 Esposito followed this up by two more articles on this famous Irish scholar in Carolingian Francia and his calendrical treatise, but also highlighted the interrelation between computus and exegesis in the Irish tradition of the seventh century by an analysis of De mirabilibus sacrae scripturae.9 If Esposito was an outsider in academia in Ireland, the same applied to Krusch within Germany. He was respected for his editorial rigour (but feared for his fierce temperament), and as a member of various Akademien, but as an archivist not within the academic inner circle. In German universities, computus received some attention from the philologists. Medieval Latin became a university subject in its own right, Ludwig Traube, in 1902, the first full professor of the subject. His extensive manuscript researches led him to a wonderful study of the transmission of Helperic’s De computo.10 His successor, Paul Lehmann, followed this up in 1912 with the standard edition of the Computus of AD 562 der Zeitzer Ostertafel vom Jahre 447,’ Sitzungsberichte der Preußischen Akademie der Wissenschaften, Jahrgang 1933, philosophisch-historische Klasse, 981–97. 7 Mac Carthy, B. (1901) Annala Uladh: The Annals of Ulster, vol. 4: introduction and sources, Dublin; Schwartz, E. (1905) ‘Christliche und jüdische Ostertafeln,’ Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, philologischhistorische Klasse, Band 8, Nr. 6, Berlin. 8 Esposito, M. (1907) ‘An unpublished astronomical treatise by the Irish monk Dicuil,’ Proceedings of the Royal Irish Academy 26C, 378–446. 9 Esposito, M. (1914) ‘An Irish teacher at the Carolingian court: Dicuil,’ Studies: an Irish quarterly review 3, 651–76; Esposito, M. (1920) ‘A ninth-century astronomical treatise,’ Modern philology 18, 1–12; Esposito, M. (1919) ‘On the pseudo-Augustinian treatise De mirabilibus sacrae scripturae written in Ireland in the year 655,’ Proceedings of the Royal Irish Academy 35C, 189–207. 10 Traube, L. (1893) ‘Computus Helperici,’ Neues Archiv der Gesellschaft für ältere deutsche Geschichtskunde 18, 73–105, repr. with editional notes in Traube (1909–20), iii 128–56.

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attributed to Cassiodorus.11 Lehmann’s successor, Bernhard Bischoff, in turn worked on ‘Ostertagstexte und Intervalltafeln’, published in 1940.12 During Krusch’s life-time, the late antique, early medieval Easter controversy received considerable attention. Two scholars stand out in this respect: In 1880, Louis Duchesne wrote an important article on the alleged founding moment of Easter calculations, the Council of Nicea of AD  325.13 In the very early 20th century, Joseph Schmid surveyed the Easter controversy in two short monographs, which, together with Krusch’s famous ‘Einführung des griechischen Paschalritus im Abendlande’ (‘The introduction of the Greek paschal reckoning in the Latin West’), must form the basis of any new inquiry into this field.14 In the inter-war years, the Low Countries became one of the most progressive regions for the study of the Middle Ages (the Belgian Henri Pirenne and the Dutch scholar Johan Huizinga being only the most prominent examples). Computistical research also profited from this more general trend. André van de Vyver in Ghent made a name for himself in the second half of the 1920s initially as an expert on Boethius. In the 1930s, however, he published two groundbreaking studies on Cassiodorus and Abbo of Fleury, which remain seminal readings to the present day.15 In the 1950s, he followed this up with a valuable study of late antique Easter reckonings.16 Walter Émile van Wijk became privaatdocent in de mathematische en technische tijdrekenkunde at the University of Leiden in 1924. Unlike his predecessors in the field, Ludwig Ideler and Friedrich Karl Ginzel, who produced monumental ‘handbooks’ on 11 Lehmann, P. (1912) ‘Cassiodorstudien,’ Philologus 71, 278–99, repr. in Lehmann (1959­–62), ii 47–55. 12 Bischoff, B. (1940) ‘Ostertagtexte und Intervalltafeln,’ Historisches Jahrbuch 60, 549–80, repr. in Bischoff (1966–81), ii 192–227. 13 Duchesne, L. (1880) ‘La question de la pâque au concile de Nicée,’ Revue des questiones historiques 28, 5–42. See also Daunoy, F. (1925) ‘La question pascale au concile de Nicée,’ Echos d’Orient 24, 424–44. 14 Schmid, J. (1904) Die Osterfestberechnung auf den britischen Inseln vom Anfang des vierten bis zum Ende des achten Jahrhunderts, Regensburg; Schmid, J. (1907) Die Osterfestberechnung in der abendländischen Kirche, Freiburg. See also Schmid, J. (1905) Die Osterfestfrage auf dem ersten allgemeinen Konzil von Nicäa, Wien. Also worth consulting is Grosjean, P. (1946) ‘Recherches sur les débuts de la controverse pascale chez les celts,’ Analecta Bollandiana 64, 200–45. 15 van de Vyver, A. (1931) ‘Cassiodore et son oeuvre,’ Speculum 6, 244–92; van de Vyver, A. (1935) ‘Les oeuvres inédites d’Abbon de Fleury,’ Revue bénédictine 47, 125–69. 16 van de Vyver, A. (1957) ‘L’évolution du comput alexandrin et romain du IIIe au e V siècle,’ Revue d’histoire ecclésiastique 52, 5–25.

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STATE OF RESEARCH ON COMPUTUS

mathematical and technical chronology,17 van Wijk’s primary interest was in twelfth- and thirteenth-century developments. His 1936 edition of the Massa compoti of Alexander de Villa Dei, with its substantial introduction, translation, and commentary, provides a good introduction into the field.18 His 1951 edition of Reinher of Paderborn’s Computus emendatus was as visionary as the 12th-century work itself.19 The 1930s saw the emigration of some of the brightest German minds, principally to the US. It is worth remembering that this phenomenon was not only due to the rise of the Nazis in Germany. Academia in general became more mobile and therefore global since the late 19th century, and especially scholars of languages (German / English / Celtic) often sought employment overseas. A good example is Heinrich Henel, who became professor of German at Queen’s University in Kingston, Canada in 1932. In 1934, Henel put Anglo-Saxon computistics on the map, with his brilliant Studien zum altenglischen Computus (interestingly written in German and published in Leipzig).20 This he followed up with an edition of Ælfric’s De temporibus anni in 1942.21 Otto Neugebauer, on the other hand, reacted to changed politics in Germany in his decision to leave the country in 1933 (though his story is different to the more traditional pattern of fleeing Nazi persecution). Neugebauer taught history of mathematics in the famous mathematics department of the University of Göttingen, many members of which were opposed to the activities of the Nazi party. Still, few followed Neugebauer’s lead and example. He was appalled by the Nazi policies that discriminated against his Jewish students and colleagues, and refused the oath of loyalty to the new regime. Anticipating persecution, he moved first to Copenhagen. When academic publishing collapsed in Germany in 1938 (to which he had continued to contribute as editor of the Zentralblatt 17 Ideler, L. (1825–26) Handbuch der mathematischen und technischen Chronologie, 2 vols, Berlin; Ginzel, F. K. (1906–14) Handbuch der mathematischen und technischen Chronologie, 3 vols, Leipzig. It is worth mentioning here also the studies of medieval chronology by the Oxford archivist and scholar of diplomatics, Reginal Lane Poole, collected in his 1934 Studies in chronology and history, Oxford. 18 van Wijk, W. E. (1936) Le nombre d’or: étude de chronologie technique suivie du texte de la Massa compoti d’Alexandre de Villedieu (c. 1170–1250), La Haye. 19 van Wijk, W. E. (1951) ‘Le computu emendé de Reinherus de Paderborn (1171), publié d’après le Ms V.P.L. 191-E de la Bibliothèque de l’Université de Leiden,’ Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen, afd. letterkunde 59, no. 3, Amsterdam. 20 Henel, H. (1934) Studien zum altenglischen Computus, Leipzig. 21 Henel, H. (1942) Ælfric’s De temporibus anni, London.

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für Mathematik, published by Springer), he had to move again, this time to the US, joining Brown University, where he stayed for the rest of his distinguished career. His principal interest lay in Babylonian and Egyptian astronomy and mathematics, but in the last few decades of his life he got more and more interested in medieval science, both Arabic and Latin. His monograph on Ethiopic computus contains some references to the Alexandrian reckoning, and he also produced an article on the Computus of AD 562.22 The Nazi destruction of the academic landscape hit mathematics particularly hard, but also in other areas the brightest minds, like Max Förster and Wilhelm Levison, had to emigrate to escape almost certain death (Levison’s famous England and the Continent in the eighth century derives from his 1943 Oxford lectures; Levison was one of Krusch’s students and collaborators, and Krusch continued correspondence with him right to the point of emigration). A significant turn in the study of computus was marked by the publications of the great American scholar Charles W. Jones in the 1930s and 1940s. Before Jones, the principal interest in computus lay in the first few centuries of Christianity. As a Bedan scholar (attracted to the field by his teacher, the English-born Max Laistner), Jones shifted the focus of research to the early medieval period. His grand tour of European libraries was, unfortunately, cut short by the outbreak of Word War II in 1939, but it had provided him with an unparalleled insight into the manuscript transmission history of early medieval computistical texts, which he published as Bedae pseudepigrapha in 1939.23 This preparatory work was followed in turn by his masterful edition of Bede’s scientific works in 1943, whose roughly 120-page introduction is still considered the seminal account of the development of computistical thought from the foundation of Christianity to the age of Bede in the eighth century.24 Jones had created an awareness that only through the systematic study of the thousands of computistical manuscripts that survive from before c.AD 1200 could one arrive at a thorough understanding of the late antique and medieval science of computus. This lead was followed by Alfred Cordoliani, who, from the 1940s to the 1960s, mined the libraries Neugebauer, O. (1979) Ethiopic Astronomy and Computus, Wien; Neugebauer, O. (1982) ‘On the Computus paschalis of ‘Cassiodorus’,’ Centaurus 25, 292–302. 23 Jones, C. W. (1939) Bedae pseudepigrapha: scientific writings falsely attributed to Bede, Ithaca. 24 Jones, C. W. (1943) Bedae opera de temporibus, Cambridge. See also the reviews of this work by Heinrich Henel in American Historical Review 49 (1944), 694–96; Journal of English and Germanic Philology 43 (1944), 411–16. 22

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of France, Switzerland, and Spain, from Mount St Michel to Tarragona, in search of new discoveries. His method was more descriptive than analytical, however, and his interpretations not always well founded. But in his more than 40 articles, he certainly identified key figures and themes for future research, from Martin of Braga to Gerland the Computist, from the Easter controversy to the re-calculation of the incarnation era.25 More importantly, Cordoliani made the first (and, to the present day, only) serious attempt at systematizing the thousands of short formulae found in hundreds of manuscripts (something that had not been achieved by Thorndike & Kibre’s famous Incipits of mediaeval scientific writings in Latin of 1937).26 Cordoliani’s broad manuscript research was complemented by Giles Meersseman and Edvige Adda’s detailed study of a ninth-century Computus attributed to Pacificus of Verona.27 Whether triggered by Jones’s survey of the early development of computistical thought in the Latin West, or simply following the established focus, Byzantinists discovered Latin computus of the first centuries of Christianity as a fruitful field of research in the 1950s and 1960s. Marcel Richard worked extensively on Hippolytus and the underlying lunar cycles.28 After a strong interest in chronology throughout his distinguished career (culminating in La chronologie of 1958), Venance Grumel turned to the early Easter controversy and the question of Anatolius’ paschal tract in particular.29 During this time, theologians also showed considerable interest in computus and its exegesis. George Ogg produced seminal studies on For a full list of Cordoliani’s publications on computus, see the Appendix to this introduction. 26 Cordoliani, A. (1960) ‘Contribution à la littérature du comput ecclésiastique au moyen âge,’ Studi medievali, Ser. 3, 1, 107–37; Cordoliani, A. (1961) ‘Contribution à la littérature du comput ecclésiastique au moyen âge,’ Studi medievali, Ser. 3, 2, 169–208. Thorndike, L. and P. Kibre (1937) A catalogue of incipits of mediaeval scientific writings in Latin, Cambridge. 27 Meersseman, G. G. and E. Adda (1966) Manuale di computo con ritmo mnemotecnico dell’arcidiacono Pacifico di Verona, Padua. 28 Richard, M. (1950) ‘Comput et chronographie chez saint Hippolyte,’ Mélanges de science religieuse 7, 237–68; Richard, M. (1951) ‘Comput et chronographie chez saint Hippolyte, 5. La durée de la vie du Christ,’ Mélanges de science religieuse 8, 19–50; Richard, M. (1953) ‘Encore le problème d’Hippolyte,’ Mélanges de science religieuse 10, 13–52, 145–80; Richard, M. (1966) ‘Notes sur le comput de cent-douze ans,’ Revue des études byzantines 24, 257–77; Richard, M. (1974) ‘Le comput pascal par octaétéris,’ Le muséon 87, 307–39. 29 Grumel, V. (1958) La chronologie, Paris; Grumel, V. (1960) ‘Le problème de la date paschale aux IIIe et IVe siècles,’ Revue des études byzantines 18, 163–78; Grumel, V. (1964) ‘La date de l’équinoxe vernal dans le canon pascal d’Anatole de Laodicée,’ in Mélanges Eugène Tisserant, 7 vols, Vatican, ii 217–40. Both the Easter controversy and Anatolius have constantly been the subject of individual publications, too many to list here. 25

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the Computus of AD 243, and later turned his attention to Hippolytus.30 In Germany, in more recent years, Wolfgang Huber and August Strobel devoted important monographs to the early Easter question.31 From the late 1950s onwards, excellent work was done in Germany on computistical chronology, or, as Joachim Wiesenbach would term it, critical computists. Interestingly, this work exclusively derived from exceptional Ph.D. theses of scholars who did not necessarily pursue an academic career afterwards, which may explain why this area of research remained occasional and never really entered the mainstream of Medieval Studies. Anna-Dorothee von den Brincken submitted a thesis entitled Studien zur lateinischen Weltchronistik bis in das Zeitalter Ottos von Freising to the University of Münster in 1956, which was published the following year.32 This she followed up with two important studies on Heimo of Bamberg and Marianus Scottus in the early 1960s, but when she became professor in Köln, her interest had moved more decidedly towards cartography and medieval chronicles more broadly.33 Joachim Wiesenbach identified the Liber decennalis of Sigebert of Gembloux, which was considered lost by previous scholarship, in Rome, Biblioteca Angelica, 1413.34 This discovery served as the foundation for his Ph.D. at the University of Frankfurt submitted in 1979 and published in 1986, Ogg, G. (1962) ‘Hippolytus and the introduction of the Christian era,’ Vigiliae Christianae 16, 2–18; Ogg, G. (1954) ‘The tabella appended to the Pseudo-Cyprianic De pascha computus in the Codex Remensis,’ Vigiliae Christianae 8, 134–44; Ogg, G. (1955) The Pseudo-Cyprianic De pascha computus, London. 31 Huber, W. (1969) Passa und Ostern: Untersuchungen zur Osterfeier der alten Kirche, Berlin; Strobel, A. (1977) Ursprung und Geschichte des frühchristlichen Osterkalenders, Berlin; Strobel, A. (1984) Texte zur Geschichte des frühchristlichen Osterkalenders, Münster. In this context, see also the later publication by Gerlach, K. (1998) The antenicene pascha: a rhetorical history, Leuven. 32 von den Brincken, A.-D. (1957) Studien zur lateinischen Weltchronistik bis in das Zeitalter Ottos von Freising, Düsseldorf. 33 von den Brincken, A.-D. (1960) ‘Die Welt- und Inkarnationsära bei Heimo von St Jakob: Kritik an der christlichen Zeitrechnung durch Bamberger Komputisten in der ersten Hälfte des 12. Jahrhunderts,’ Deutsches Archiv für Erforschung des Mittel­ alters 16, 155–94; von den Brincken, A.-D. (1961) ‘Marianus Scottus, unter besonderer Berücksichtigung der nicht veröffentlichten Teile seiner Chronik,’ Deutsches Archiv für Erforschung des Mittelalters 17, 191–238. See also von den Brincken, A.-D. (1982) ‘Ma­ rianus Scottus als Universalhistoriker iuxta veritatem Evangelii,’ in Heinz Löwe, Die Iren und Europa im früheren Mittelalter, 2 vols, Stuttgart, ii 970–1009; von den Brincken, A.D. (1979) ‘Beobachtungen zum Aufkommen der retrospektiven Inkarnationsära,’ Archiv für Diplomatik 25, 1–20. 34 Wiesenbach, J. (1977) ‘Der Liber decennalis in der Hs. Rom, Biblioteca Angelica 1413, als Werk Sigeberts von Gembloux,’ Deutsches Archiv für Erforschung des Mittel­ alters 33, 171–81. 30

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a critical edition of the text with a solid introduction to the computistical background.35 Another edition of a crucial text was prepared by Hans Martin Weikmann for his 1984 Würzburg Ph.D., which focused on Heimo of Bamberg’s De decursu temporum. It is now available as a monograph of 2004.36 Inspired by Jones, Cyril Hart, in 1970, revived the study of late Anglo-Saxon computus, following on where Henel left off with his study of Aelfric and the anonymous tradition. Hart focused on Ramsey abbey, its principal computist, Byrhtferth, and (as he believed) its key scientific manuscript, Oxford, St. John’s College 17, eventually culminating in a 2003 Survey of the development of mathematical, medieval, and scientific studies in England before the Norman conquest.37 Peter Baker continued this work with a preliminary study on Byrhtferth in Anglo-Saxon England in 1982, and then the critical edition (together with Michael Lapidge) of Byrhtferth’s Enchiridion in 1995.38 St John’s College 17 was analysed in detailed by Faith Wallis for her 1985 Ph.D.,39 and she has since published widely on the (principally English) computistical tradition. Most notably, she opened up the study of (not only Anglo-Saxon) computistics to a wider audience by providing a translation of Bede’s De temporum ratione, which she has followed up since, in collaboration with Calvin Kendall, with translations of Bede’s De natura rerum and De temporibus.40 Wiesenbach, J. (1986) Sigebert von Gembloux: Liber decennalis, Weimar. Weikmann, H. M. (2004) Heimo von Bamberg: De decursu temporum, Hanno-

35 36

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37 Hart, C. J. R. (1970) ‘The Ramsey computus,’ English Historical Review 85, 29– 44; Hart, C. J. R. (1970) ‘Byrhtferth and his manual,’ Medium aevum 41, 95–109; Hart, C. J. R. (2003) Learning and culture in late Anglo-Saxon England and the influence of Ramsey Abbey on the English monastic schools, vol. 2: a survey of the development of mathematical, medical, and scientific studies in England before the Norman conquest, Lewiston. 38 Baker, P.  S. (1982) ‘Byrhtferth’s Enchiridion and the computus in Oxford, St  John’s College 17,’ Anglo-Saxon England 10, 22–37. Baker, P.  S. and M.  Lapidge (1995) Byrhtferth’s Enchiridion, Oxford. For an older, pioneering study, see Kluge, F. (1885) ‘Angelsächsische Excerpte aus Byrhtferth’s Hanboc oder Enchiridion,’ Anglia 8, 298–337. 39 See the excellent website which arose from her Ph.D. interest: http://digital. library.mcgill.ca/ms-17/index.htm. 40 Wallis, F. (1999) Bede: The Reckoning of Time, Liverpool; Kendall, C. A. and F. Wallis (2010) Bede: On the nature of things and On times, Liverpool. See also Wallis, F. (1989) ‘The church, the world and the time: prolegomena to a history of the medieval computus,’ in M.-C. Déprez-Masson, Normes et pouvoir à la fin du moyen âge, Montreal, 15–29; Wallis, F. (2015) ‘What a medieval diagram shows: a case study of computus,’ Studies in iconography 36, 1–40.

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The renewed interest in Bede’s scientific texts may have been triggered by Charles W. Jones’s second (re-)edition of the works in the Corpus Christianorum Series Latina in 1975–80, this time including Bede’s Chronicles, but lacking the substantial introduction and detailed commentary of the 1943 edition.41 This publication also marked an important turn towards the early medieval computistical tradition and its texts. Jones’s student, Wesley Stevens, produced a critical edition of Hrabanus Maurus’ De computo in Corpus Christianorum Continuatio Mediaevalis, and has since worked extensively on Hrabanus’ pupil Walahfrid Strabo.42 In the 1980s, Dáibhí Ó Cróinín put early medieval Irish computistics on the map. First, his discovery of the long-lost 84-year Easter table used in the regiones Scottorum until AD 716, in Briton Wales until the later eighth century, provided, for the first time, a solid basis for the study of the early medieval Easter controversy, and it was pivotal in the reconstruction of the chronological apparatus of the Irish annals.43 In effect, this discovery provided, for the first time, a series of reliable dates for the first three centuries of Irish history.44 Together with his wife, Maura Walsh, he produced in 1988 an edition and translation of the key text for the Irish side of the Easter controversy, Cummian’s letter of AD 632.45 In the same volume, he edited the most sophisticated of the Irish computistical textbooks (De ratione conputandi), a model of its time, not only rivaling but surpassing Bede’s rather wordy accounts on the subject.46 41 Jones, C. W. (1975–80) Bedae Venerabilis opera, pars VI, Corpus Christianorum Series Latina CXXXIIIA–C, Turnhout. 42 Stevens, W.  M. (1979) ‘Rabani Mongontiacensis episcopi De computo,’ in CCCM 44, 163–323. 43 Mc Carthy, D. P. and D. Ó Cróinín (1987–88) ‘The ‘lost’ Irish 84-year Easter table rediscovered,’ Peritia 6–7, 227–42, repr. in Ó Cróinín (2003), 58–75; Mc Carthy, D. P. (1993) ‘Easter principles and a fifth-century lunar cycle used in the British Isles,’ Journal for the History of Astronomy 24, 204–24; Mc Carthy, D. P. (1998) ‘The chronology of the Irish annals,’ Proceedings of the Royal Irish Academy 98C, 203–55. 44 http://www.irish-annals.cs.tcd.ie/. 45 Walsh, M. and D. Ó Cróinín (1988) Cummian’s letter De controversia paschali, together with a related Irish computistical tract, De ratione conputandi, Toronto. 46 The discovery was announced and its historical context discussed in Ó Cróinín, D. (1982) ‘A seventh-century Irish computus from the circle of Cummianus,’ Proceedings of the Royal Irish Academy 82C, 405–30. For Ó Cróinín’s other contributions to early medieval computus, see the essays collected in his Early Irish history and chronology, Dublin 2003. Ó Cróinín’s scholarship was complemented by the contemporary studies by Kenneth Harrison: Harrison, K. P. (1973) ‘The Synod of Whitby and the beginning of the Christian era in England,’ Yorkshire Archaeological Journal 45 (1973), 108–14; Harrison, K. P. (1976) The framework of Anglo-Saxon history to A.D. 900, Cambridge; Harrison, K. P. (1977–78) ‘Epacts in Irish chronicles,’ Studia Celtica 12–13, 17–32; Har-

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In the late 1980s and early 1990s, Joan Gómez Pallarès, in various articles that evolved out of his 1986 Barcelona Ph.D. thesis, aimed at reconstructing Visigothic computistics, providing diplomatic editions of the key tracts.47 Though not entirely successful in the end, this was the first serious attempt at a comprehensive study of the computistical tradition of the Iberian peninsula from the seventh to the eleventh centuries. The 1990s saw an increased interest in Carolingian computus. Two Aachen-based scholars, Paul Butzer and Dietrich Lohrmann, utilised the local dynamics in the former centre of the Carolingian Empire triggered by the 1400th anniversary of the Admonitio generalis (AD 789) and of the AD 790s, the formative decade in what is termed the Carolingian Renaissance, to organize two international conferences on the wider context of Carolingian science, a truly pioneering undertaking.48 Out of this context emerged the only substantial study on Alcuin’s contribution to the computistical endeavour in the Carolingian age by Kerstin Springsfeld.49 The study of Carolingian diagrams and cosmology was taken up anew by Barbara Obrist, culminating in her La cosmologie médiévale of 2004,50 which, in many ways, provides the cultural background to Bruce Eastwood’s more technical pioneering work on astronomical diagrams from the 1980s onwards. In 1998, Stephen McCluskey provided the first survey of early medieval astronomy in his Astronomies and cultures in early medieval Europe,51 while in 1994 Brigitte Englisch attempted to

rison, K. P. (1978) ‘Easter cycles and the equinox in the British Isles,’ Anglo-Saxon England 7, 1–8; Harrison, K. P. (1979) ‘Luni-solar cycles: their accuracy and some types of usage,’ in M. H. King and W. M. Stevens, Saints, scholars and heroes, 2 vols, Collegeville, ii 65–78; Harrison, K. P. (1982) ‘Episodes in the history of Easter cycles in Ireland,’ in D. Whitelock, R. McKitterick, and D. Dumville, Ireland in early medieval Europe, Cambridge, 307–19; Harrison, K. P. (1984) ‘A letter from Rome to the Irish clergy, AD 640,’ Peritia 3, 222–29. 47 Gómez Pallarès, J. (1986) Estudios sobre el Computus Cottonianus, diss. Barcelona, www.tdx.cbuc.es/handle/10803/5248; his articles are collected in Gómez Pallarès, J. (1999) Studia chronologica: estudios sobre manuscritos latinos de cómputo, Madrid. 48 Butzer, P. L. and D. Lohrmann (1993) Science in Western and Eastern civilization in Carolingian times, Basel; Butzer, P. L. et al. (1998) Karl der Große und sein Nachwirken. 1200 Jahre Kultur und Wissenschaft in Europa, 2: Mathematical Arts, Turnhout. 49 Springsfeld, K. (2002) Alkuins Einfluß auf die Komputistik zur Zeit Karls des Großen, Stuttgart. 50 Obrist, B. (2004) La cosmologie médiévale. I: Les fondements antiques: textes et images, Firenze. 51 McCluskey, S.  C. (1998) Astronomies and cultures in early medieval Europe, Cambridge.

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define the place of computus within the late antique, early medieval concept of the quadrivium.52 The turn of the millennium saw a brief resurgence of interest in the incarnation era. George Declercq published a stimulating short volume on Anno Domini, both in French and English, which he supplemented with a more detailed study in Sacris Erudiri.53 The main focus of Anno Domini lies on the Easter reckonings of Victorius of Aquitaine and Dionysius Exiguus, and it therefore provides the context for Krusch’s 1938 editions of these works. The seventh International Medieval Congress in Leeds in 2000 also took the turn of the millennium as an excuse to address questions of time and eternity in the Middles Ages, which culminated in a volume on the theme in 2003.54 Anna-Dorothee von den Brincken’s introduction to historical chronology was also published appropriately in 2000.55 The last in the list of scholars who promoted the study of computistics before 2006 is Arno Borst. One of Borst’s trademarks was to work on the history of the place he was living in. When he received the Ruf to the newly founded University of Konstanz with its ambitious educational reform program, he immediately started to work on the monastic culture of the Lake Constance region. He got particularly interested in the Reichenau monk Hermannus Contractus (†1054), arguably one of the most brilliant and wide-ranging intellectuals of the eleventh century. From 1975, Borst pursued the project of writing a biography of Hermann. For this, he needed to understand Hermann’s oeuvre, which, to Borst’s dismay, was not readily accessible. In particular, Hermann’s computistica and astrolabica still remained largely unexplored in numerous manuscripts, with no edition, for many texts not even a transcript available in print (Cordoliani had worked on Hermann’s computistica in the early 1960s, Bergmann in the 1980s56). Englisch, B. (1994) Die Artes liberales im frühen Mittelalter (5–9. Jh.), Stuttgart. Declercq, G. (2000) Anno domini: les origins de l’ère chrétienne, Turnhout; in English: Anno Domini: the origins of the Christian era, Turnhout 2000; Declercq, G. (2002) ‘Dionysius Exiguus and the introduction of the Christian era,’ Sacris Erudiri 41, 165–246. 54 Jaritz, G. and G.  Moreno-Riaño (2003) Time and eternity: the medieval discourse, Turnhout. 55 von den Brincken, A.-D. (2000) Historische Chronologie des Abendlandes: Kalenderreformen und Jahrtausendrechnungen – eine Einführung, Stuttgart. 56 Cordoliani, A. (1963) ‘Le computiste Hermann de Reichenau,’ Miscellanea sto­ rica ligure 3, 161–90; Bergmann, W. (1988) ‘Chronographie und Komputistik bei Her52 53

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Therefore, Borst embarked on producing editions of Hermann’s texts before tackling the broader goal of a biography. But Hermann’s computistical writings, in particular, proved difficult to edit, as the Reichenau monk obviously based his knowledge on a centuries-old tradition, which had not received adequate attention in modern scholarship. Borst tried to fill that lacuna in 1990 with a sweeping overview of the history of computus, which must have made him aware of all the work that still needed to be done.57 Certainly, after this he concentrated on Hermann’s potential sources. First, in 1994, he produced a full-scale study on the reception of Pliny in the Middle Ages.58 Then he turned for a decade to the early medieval calendar tradition, culminating in his monumental edition of the so-called Reichskalender.59 Finally, he engaged with computistical texts. His massive 2006 editions of 20 computistical texts from Francia of the period AD 721–818 is a milestone in the modern study of early medieval computus, and we were proud to honour this achievement by launching the 3-volume work at the First Galway Computus Conference.60 From Jones to Borst, that is from the 1930s to the early 21st century, the study of early medieval computus focused principally (with few exceptions) on works by authors known by name. Borst, however, created an awareness that for any proper understanding of this calendrical science, it is essential to analyse the much more numerous anonymous texts. The three Carolingian encyclopediae (Lib. ann. of AD 793, Lib. comp. of AD 809/12, and Lib. calc. of AD 818) had previously received some attention, but a critical edition of each of these was still wanting.61 mann von Reichenau,’ in D. Berg and H.-W. Goetz, Historiographia mediaevalis: Studien zur Geschichtsschreibung und Quellenkunde des Mittelalters, Darmstadt, 103–17. 57 Borst, A. (1990) Computus: Zeit und Zahl in der Geschichte Europas, Berlin; it was translated into English by Andrew Winnard as The ordering of time: from the ancient computus to the modern computus, Chicago 1993; see also Borst, A. (1988) ‘Computus: Zeit und Zahl im Mittelalter,’ Deutsches Archiv für Erforschung des Mittelalters 44, 1–82. 58 Borst, A. (1994) Das Buch der Naturgeschichte: Plinius und seine Leser im Zeitalter des Pergaments, Heidelberg. 59 Borst, A. (1998) Die karolingische Kalenderreform, Hannover; Borst, A. (2001) Der karolingische Reichskalender und seine Überlieferung bis ins 12. Jahrhundert, 3 vols, Hannover; Borst, A. (2004) Der Streit um den karolingischen Kalender, Hannover. 60 Borst, A. (2006) Schriften zur Komputistik im Frankenreich von 721 bis 818, 3 vols, Hannover. 61 E.g. Neuß, W. (1941) ‘Eine karolingische Kopie antiker Sternzeichenbilder im Codex 3307 der Biblioteca Nacional zu Madrid,’ Zeitschrift des deutschen Vereins für Kunstwissenschaft 8, 113–40; Boschen, L. (1972) Die Annales Prumiensis, Düsseldorf; Springsfeld (2002); and numerous articles by Eastwood.

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Other crucial texts (like Prol. Aquit. of AD 721, Dial. Burg. of AD 727, Dial. Neustr. of AD 737, Cap. comp. of AD 809) had only been transcribed or mentioned in passing in the literature, but remained sidelined by a focus on Bede, Alcuin, and Hrabanus Maurus. However, other extremely important texts had remained hidden in the manuscripts (like Dial. Langob. of c.AD 750, Quaest. Austr. of AD 764, Quaest. Langob. of c.AD 780, Epist. Rat. of AD 809, Arg. Aquens. of AD 816), and part of Borst’s achievement was to bring these texts to the attention of scholars. To be sure, Borst’s corpus has its problems, especially in the representation of texts. But it still represents a landmark publication in the field of computistical studies. Schriften zur Komputistik im Frankenreich marked the culmination and high-point of Borst’s impressive career. He was fully aware that Schriften would be his last major publication. In fact, he postponed a crucial medical operation in order to see the three volumes through the press, as he feared that he would not be able to continue his normal, highly intense level of work afterwards. Unfortunately, he died in the following year. But the end of Borst’s scholarly endeavor heralded a new, vibrant, and exceptionally productive interest in what had previously been a neglected area of research. Whether the establishment of the Galway Conference also contributed to this more general trend remains for others to judge. Certainly, publications on late antique and early medieval computistics have spiraled since 2006. In the following, I can only name the most prominent publications, surveying the literature chronologically according to content from AD 200 to 1200. First place in such a survey must be given to Alden Mosshammer. In 2008, Mosshammer produced his Easter Computus and the origins of the Christian era, the first monograph in English on the beginnings of the computistical tradition, c.AD 200–600.62 In effect, this updates (but does not replace) Eduard Schwartz’s Christliche und jüdische Ostertafeln published a century earlier. In subsequent years, Mosshammer has followed up this survey with more detailed studies. In the present volume, he outlines prolegomena to a new edition of the oldest computistical text that has survived, the Computus of AD  243. Another and more prominent focus of his is on the fifth century. This century is crucial for our understanding of the history of computus, as it saw the biggest production of new (and rivaling) Easter reckonings in the late antique and Mosshammer, A. A. (2008) The Easter Computus and the origins of the Christian era, Oxford. 62

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medieval period. Mosshammer concentrated on the laterculus of Augustalis as outlined in the Carthaginian computus of AD 455 (preserved in Lucca, Biblioteca Capitolare Feliniana, 490).63 Both the laterculus and the Carthaginian computus as a whole are a great reminder of the active role played by the Vandal kingdom in late antique culture, and especially its scientific strand, which deserves a full-scale investigation in the future. Besides the Easter reckonings described in the Computus Cartha­ ginensis (the laterculus of Augustalis, the circulus primus, and the circulus secundus), the fifth century witnessed the creation of three more prominent methods of calculating Easter: the latercus (the reckoning followed in Britain and Ireland until the eighth century), probably invented by Sulpicius Severus c.AD 410; the famous Zeitz table of AD 447; and the 532-year table established by Victorius of Aquitaine, which dominated western Europe until the mid-eighth century. Since the discovery of the only known copy of the latercus in Padua, Biblioteca Antoniana, I 27, this system has been very much in the focus of modern research. Most notably, Dan Mc  Carthy reconstructed the chronological apparatus of the Irish annals on this basis, which culminated in his monograph of 2008, The Irish annals: their genesis, evolution and history.64 In 2011, he rounded off a series of earlier publications on the history and cultural impact of this Easter reckoning.65 In 2006, Caitlin Corning reassessed the insular Easter controversy of the seventh century on the basis of the reconstructed data of the latercus and contemporary epistles and historiography (principally Bede).66 Surprisingly, however, Corning did not draw on the substantial computistical corpus of the seventh century, which would have refined, if not changed her picture. In 2010, Leofranc Holford-Strevens demonstrated, how our improved understanding of the latercus has helped to re-assess key episodes in the insular Easter controversy, in this case the Synod of Whitby. He was also instrumental in establishing technical details of this Easter reckoning, 63 Mosshammer, A.  A. (2011) ‘The Computus of 455 and the laterculus of Augustalis, with an appendix on the fractional method of Agriustia,’ in Warntjes and Ó Cróinín (2011), 21–47. 64 Mc  Carthy, D.  P. (2008) The Irish Annals: their genesis, evolution and history, Dublin. 65 Mc Carthy, D. P. (2011) ‘On the arrival of the latercus in Ireland,’ in Warntjes and Ó Cróinín (2011), 48–75. Of his earlier publications on this topic, see especially Mc Carthy, D. P. (1994) ‘The origin of the latercus paschal cycle of the insular Celtic churches,’ Cambrian Medieval Celtic Studies 28, 25–49. 66 Corning, C. (2006) The Celtic and Roman traditions: conflict and consensus in the early medieval church, New York.

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a process that began with Dan Mc Carthy’s 1993 reconstruction of the data of the Padua latercus, followed by Holford Strevens’s analyses in his 1999 Oxford companion to the year (together with Bonnie Blackburn) and in his seminal 2007 article.67 I then compared the Padua latercus data with the latercus information in the Munich Computus in 2007.68 The Zeitz table regained considerable prominence in recent times due to the rediscovery of some of its fragments in the Stiftsbibliothek in Zeitz, which made the German newspapers in 2005.69 Among German academics, however, this did not trigger any further curiosity.70 Dan Mc Carthy presented an impressive re-evaluation of the Zeitz table at the Fourth Galway conference in 2012 (to be published in the following proceedings). It can only be hoped that this will lead to a large-scale analysis of the reformation of the Roman 84-year reckoning (the Supputatio Romana) in the fifth century, which appears to provide the wider context for the latercus, the Zeitz table, and the tables described in the Carthaginian Computus. Victorius of Aquitaine still seems to suffer from Krusch’s scathing verdict of 1938: ‘Der calculator scrupulosus [Victorius] war ein ganz beschränker Kopf und außerdem nicht einmal ehrlich.’ (‘Victorius was a very limited individual, and not even an honest one.’)71 Columbanus, c.AD  600, was of a similar opinion, giving Victorius ‘bad press’ especially for his (diplomatic) use of alternative dates for Easter according to what he believed were the Alexandrian and the Roman traditions.72 The Visigothic monk Leo, in AD 627, wrote along the same lines but 67 Mc Carthy (1993); Blackburn, B. and L. Holford-Strevens (1999) The Oxford companion to the year, Oxford, 870–75; Holford-Strevens, L. (2008) ‘Paschal lunar calendars up to Bede,’ Peritia 20, 165–208: 178–87. 68 Warntjes, I. (2007) ‘The Munich Computus and the 84 (14)-year Easter reckoning,’ Proceedings of the Royal Irish Academy 107C, 31–85. 69 Frankfurter Allgemeine Zeitung, 4 November 2005. 70 At the time, and regularly since, the Zeitz fragments were and are on display in Berlin or Zeitz (or both), and an exhibition booklet was produced: Overgaauw, E. and F.-J. Steving (2005) Die Zeitzer Ostertafel aus dem Jahre 447, Petersberg. This, however, shows no interest either in the construction of the table itself or its scientific context. The standard accounts remain Mommsen, T. (1863) ‘Zeitzer Ostertafel vom Jahre 447,’ Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin, Jahrgang 1862, philosophisch-historische Klasse, 537–66; MGH Auct. ant. 9, 501–09; Krusch, B. (1933) ‘Neue Bruchstücke der Zeitzer Ostertafel vom Jahre 447,’ Sitzungsberichte der Preußischen Akademie der Wissenschaften, Jahrgang 1933, philosophisch-historische Klasse, 981–97. 71 Krusch (1938), 15. 72 Columbanus, Epistolae 1–2 (ed. by Walker (1957), 2–22).

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using less insulting language.73 This, however, should not lead us to the conclusion that Victorius is not worth studying. Quite the contrary, both Columbanus and Leo are important reminders of the relevance of Victorius still in the seventh century, and Dan Mc Carthy’s contribution in the present volume highlights the potential technical problems that followers of this Easter reckoning faced. One of the key factors in minimizing Victorius’ importance, no doubt, was Bede’s insistence on ‘Roman unity’ when there was none. The sixth to eighth centuries not only saw a conflict between ‘Celtic’ and ‘Roman’ tradition, as Bede, our only detailed witness, would want to make us believe. Quite the contrary, ‘Romans’, in Ireland, Anglo-Saxon England, Visigothic Spain, Frankish Gaul, Italy, and Rome, debated whether Victorius or the Alexandrian reckoning in the shape of Dionysius Exiguus’ translation was to be followed. Masako Ohashi has for years, since her 1999 Nagoya Ph.D. thesis, insisted on the centrality of this issue, and more recent publications of 2015 by E. T. Dailey and myself emphasize this argument more forcefully.74 The Easter controversy of the early Middle Ages needs to be rewritten on this basis. Alden Mosshammer also leads the way in the reception of the Alexandrian reckoning in the Latin West. The Alexandrian system of calculating Easter was formalized in the late third century. Until the midseventh century, Rome followed different practices, first 84-year tables, then Victorius of Aquitaine. The controversy between Rome and Alexandria is well documented especially for the fifth-century, but for the sixth century comparable insight is lacking. The earliest reception of the Alexandrian reckoning in the Latin West in the fifth and sixth centuries is directly connected to a vaguely defined corpus of texts that bears 73 Krusch (1880), 298–302. Only one manuscript transmitting the full letter was known to Krusch, Cologne, Dombibliothek 83-II, 184r–185v; a second copy has come to light in Bremen, Universitätsbibliothek, msc 0046, 41r–44r. A new edition of this central monument of the Iberian Easter controversy is currently prepared by José Carlos Martín Iglesias as ‘La Epistola de computo paschali (CPL 2300) del monje León: nueva edición y estudio de una obra probablemente hispano-visigoda’. 74 See especially Ohashi, M. (2005) ‘Theory and history: an interpretation of the paschal controversy in Bede’s Historia ecclesiastica,’ in S. Lebecq, M. Perrin, and O. Szerwiniack, Bède le Vénérable: entre tradition et postérité, Lille, 177–85; Ohashi, M. (2011) ‘The Easter table of Victorius of Aquitaine in early medieval England,’ in Warntjes and Ó Cróinín (2011), 48–75; Dailey, E. T. (2015) ‘To choose one Easter from three: Oswiu’s decision and the Northumbrian synod of AD 664,’ Peritia 26, 47–64; Warntjes, I. (2015) ‘Victorius vs Dionysius: the Irish Easter controversy of AD 689,’ in P. Moran and I. Warntjes, Early medieval Ireland and Europe: chronology, contacts, scholarship, Turnhout, 33–97.

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the name of the 17th-century Jesuit scholar Jacques Sirmond.75 Many of these texts have mistakenly been labeled Irish forgeries of the seventh century, but more recent research places them more securely in their predominantly fifth- and sixth-century context. Dan Mc Carthy and Aiden Breen produced in 2003 a new edition and translation of De ratione paschali ascribed to Anatolius of Laodicea, which they consider to be a faithful Latin translation of Anatolius’ third-century original tract.76 Jan Zuidhoek analyses, in the present volume, the potential base year of the Easter table incorporated in this text. Rick Graff reconsidered the Disputatio Morini in the proceedings of the first Galway conference in 2010.77 Alden Mosshammer analysed the Prologus Cyrilli and its recensions in Vigiliae Christianae in 2013, and a new edition of this text and of the Prologus Theophili by him has just been published.78 Only critical editions of all such texts will enable a solid understanding of the Easter question in the transition period from Late Antiquity to the early Middle Ages. Iberia and Italy were the first regions in the Latin West to embrace the Alexandrian reckoning. The recensions of the Prologus Cyrilli analysed by Mosshammer are transmitted principally in Visigothic manuscripts or those with Visigothic connections. He has followed this up by an edition of three computistical texts from Visigothic Spain to be published in the following Galway proceedings, and I will present an overview of sixth- to eighth-century Visigothic computistics in the same volume. The picture that emerges is one of discussion, debate, and conflict between the Victorian and the Alexandrian reckonings, which was solved in favour of Alexandria by the AD 640s (though, in the present volume, Brigitte Englisch suggests, rather controversially, that the Visigoths fol75 For the Sirmond corpus, see Jones, C. W. (1937) ‘The ‘lost’ Sirmond manuscript of Bede’s Computus,’ English Historical Review 52, 205–19, repr. in C. W. Jones, Bede, the schools and the computus, ed. by Wesley M. Stevens, Aldershot 1993, article X (omitting the crucial final pages); Jones (1943), 105–13; Ó Cróinín, D. (1983) ‘The Irish provenance of Bede’s computus,’ Peritia 2, 229–47, repr. in Ó Cróinín (2003), 173–90; Ó Cróinín, D. (2003) ‘Bede’s Irish computus,’ in Ó Cróinín (2003), 201–12; Wallis (1999), lxxii–lxxix; Springsfeld (2002), 64–80; Warntjes (2011). 76 Mc Carthy, D. P. and A. Breen (2003) The ante-Nicene Christian Pasch: De ratione paschali – The Paschal tract of Anatolius, bishop of Laodicea, Dublin. 77 Graff, E. (2010) ‘The recensions of two Sirmond texts: Disputatio Morini and De divisionibus temporum,’ in Warntjes and Ó Cróinín (2010), 112–42. 78 Mosshammer, A. A. (2013) ‘The Praefatio (Prologus) sancti Cyrilli de Paschate and the 437-year (not 418!) paschal list attributed to Theophilus,’ Vigiliae Christianae 67, 49–78; Mosshammer, A. A. (2017) The Prologues on Easter of Theophilus of Alexandria and [Cyril], Oxford.

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lowed their own system based on actual lunar observation). Isidore’s curious Alexandrian Easter table remains one of the central avenues not only into Visigothic computistics, but also into the transmission of the Etymologiae as a whole.79 From a manuscript perspective, the mid-ninth century Paris, Bibliothèque nationale de France, Lat. 609, certainly is our key witness for the early period in Spain, and Alden Mosshammer is currently preparing an edition of its oldest, seventh-century layer. A new, collaborative edition of the famous Antiphonary of León will shed more light on the various phases of Visigothic computistics right up to the eleventh century, and the Visigothic tradition will certainly be one of the most fruitful areas of computistical studies in the future. In Italy, the Alexandrian reckoning made an early appearance in a letter of Ambrose of Milan written in AD 386. Its authenticity has long been questioned, but Max Lejbowicz, in his analysis of the computistical importance of the letter presented at the first Galway conference, considers it authentic.80 The Alexandrian reckoning was popularized in AD 525 by Dionysius Exiguus’ translation from Greek into Latin of the table and accompanying explanatory material. The Alexandrian Easter table has a 532-year cyclic structure. However, it was first designed as a 95-year table, in which every fourth year (those designated bissextile) had to be recalculated in order to update it to the immediately following 95-year period. Dionysius’ table of AD 532 to 626 followed from Cyril’s of AD 437 to 531. The expiration of Dionysius’ table—and therefore 79 I have two articles on Isidore’s Easter table in preparation; the first will be published in Revue d’histoire des textes, n.s. 13 (2018) as ‘The continuation of the Alexandrian Easter table in seventh-century Iberia and its transmission to ninth-century Francia (Isidore, Etymologiae 6.17)’. For the reception of Isidore in seventh- and eighth-century cosmology and computistics, see the studies of Marina Smyth: Smyth, M. (1987) ‘Isidore of Seville and early Irish cosmography,’ Cambridge Medieval Celtic Studies 14 (1987), 69–102; Smyth, M. (1996) Understanding the universe in seventh-century Ireland, Woodbridge; Smyth, M. (2015) ‘Isidorian texts in seventh-century Ireland,’ in A.T. Fear and J. Wood, Isidore of Seville and his reception in the early Middle Ages: transmitting and transforming knowledge, Amsterdam, 111–30. 80 Lejbowicz, M. (2008) ‘Une étape contournée dans l’unification des pratiques computistes médiévales latines,’ Cahiers de recherches médiévales et humanistes 15, 277–305; Lejbowicz, M. (2010) ‘Les pâques baptismales d’Augustine d’Hippone: une étape contournée dans l’unification des pratiques computistes latines,’ in Warntjes and Ó Cróinín (2010), 1–39. See also his survey of the development and cultural importance of computus up to AD  1200 in Lejbowicz, M. (2006) ‘Des tables pascales aux tables astronomique et retour,’ Methodos 6, 1–67, as well as his earlier Lejbowicz, M. (1992) ‘Computus. Le nombre et le temps altimédiévaux,’ in B. Ribémont, Le temps, sa mesure et sa perception au moyen âge, Caen, 151–96. For Ambrose’s paschal letter, Lejbowicz’s study builds on Zelzer, M. (1978) ‘Zum Osterfestbrief des heiligen Ambrosius und zur römischen Osterfestberechnung des 4. Jahrhunderts,’ Wiener Studien 91, 187–204.

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its necessary continuation in AD  626—was a crucial moment in the history of early medieval computus, especially since the papal curia had not yet embraced the Alexandrian system. In Italy, this re-calibration of Dionysius’ table is ascribed to a certain Felix, and Luciana Cuppo’s most recent research focussed strongly on this key episode (both in the present volume and in the previous conference proceedings).81 Essential for this task of re-calculating the data of the Easter table was the set of nine argumenta or calendrical algorithms which formed part of Dionysius’ explanatory material. Before the eighth century, these argumenta were re-calibrated in AD  562 (in the circle of Cassiodorus), and expanded and added to in AD 581, 625, 675, 689, and 695. This presents the only traceable direct and continuous line of transmission of calendrical science from Late antiquity into the early Middle Ages. I  have sketched this development in the first proceedings of the Galway conference and I hope to provide an edition with translation and commentary of all of these texts shortly.82 In the eighth century, this genre of the computistical formulary (defined as a collection of principally calendrical algorithms) mushroomed and became one of the central features of Carolingian s­ cience. An equally important, third genre of computistical texts (after tables and formularies) was invented in the second half of the seventh-century in Ireland: the textbook. The process itself still needs to be studied in detail, but the reception of Isidore’s writings and the quest for understanding the liturgical calendar and God’s creation as manifested in the cosmos appear to have been the key triggers. Dáibhí Ó Cróinín led the way here with his edition of De ratione conputandi in 1988. This was followed by the edition of the Munich Computus in 2010, and a third fullscale textbook, discovered in 2006, still awaits publication (the Computus Einsidlensis).83 With these three major texts at hand, it is possible to 81 Cuppo, L. (2011) ‘Felix of Squillace and the Dionysiac computus I: Bobbio and Northern Italy (MS Ambrosiana H 150 inf.),’ in Warntjes and Ó Cróinín (2011), 110–36. 82 Warntjes, I. (2010) ‘The Argumenta of Dionysius Exiguus and their early recensions,’ in Warntjes and Ó Cróinín (2010), 40–111; Warntjes, I. (2011) ‘The Computus Cottonianus of AD 689: a computistical formulary written for Willibrord’s Frisian mission,’ in Warntjes and Ó Cróinín (2011), 173–212. 83 Warntjes, I. (2010a) The Munich Computus: text and translation. Irish computistics between Isidore of Seville and the Venerable Bede and its reception in Carolingian times, Stuttgart; Warntjes, I. (2005) ‘A newly discovered Irish computus: Computus Einsidlensis,’ Peritia 19, 61–64; Bisagni, J. and I. Warntjes (2008) ‘The Early Old Irish material in the newly discovered Computus Einsidlensis (c.AD 700),’ Ériu 58, 77–105.

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define the Irish contribution to this calendrical science and to identify further treatises that help to reconstruct the thought-world of the Irish ‘Golden Age’, c.AD  650–750. Particularly intriguing are the tract De comparatione epactarum of AD 689 (which draws attention to the fact that the systems ascribed to Victorius and Dionysius are irreconcilable), the Victorian Prologue of AD  699 (paralleling the three major linear time-lines, annus mundi, annus passionis, and AD), and the eclipse prediction of AD 754, all discovered since 2006.84 But our improved knowledge of early medieval Irish computistics also makes it possible to clearly assess the Irish contribution to the Carolingian educational reform.85 Ultimately, this will provide the context for Dúngal and Dicuil, two of the most important Irish peregrini of the Carolingian age. Werner Bergmann drew attention once more to the importance of Dicuil’s Liber de astronomia in the proceedings of the first Galway conference,86 and a critical edition of the text remains one of the main desiderata not only in the study of early medieval computus, but also of the Carolingian intellectual endeavour. Scholarship on Bede’s computistica has flourished over the past decade principally due to the translations by Faith Wallis and the combined efforts of Máirín MacCarron and Peter Darby, who, since 2011, have held sessions on Bede annually at the major medievalists’ congresses in Kalamazoo and Leeds (with a thorough eye on Bede’s contribution to early medieval scientifica).87 There is an apparent shift in Bedan research, systematically analyzing Bede’s oeuvre according to certain themes. Darby applied this approach to themes relating to time, with a 2012 monograph on Bede and the end of time, and a 2014 volume of collected essays on Bede and the future (ed. together with Faith Wallis).88 This turn from 84 Warntjes (2010a), CLII–CLVIII, 322–26, 328; Warntjes, I. (2010) ‘A newly discovered prologue of AD  699 to the Easter table of Victorius of Aquitaine in an unknown Sirmond manuscript,’ Peritia 21, 255–84; Warntjes, I. (2013–14) ‘An Irish eclipse prediction of AD 754: the earliest in the Latin West,’ Peritia 24–25, 108–15. 85 Cf.  Warntjes, I. (2013) ‘Seventh-century Ireland: the cradle of medieval science?,’ in M. Kelly and C. Doherty, Music and the stars: mathematics in medieval Ireland, Dublin, 44–72; Warntjes, I. (2016) ‘Computus as scientific thought in Ireland and the early medieval West,’ in R. Flechner and S. Meeder, The Irish in early medieval Europe: identity, culture and religion, New York, 158–78. 86 Bergmann, W. (2011) ‘Dicuils Osterfestalgorithmus im Liber de astronomia,’ in Warntjes and Ó Cróinín (2011), 242–87. 87 See their website: bedenet.com. 88 Darby, P. (2012) Bede and the end of times, Farnham; Darby, P. and F.  Wallis (2014) Bede and the future, Farnham, with essays by J. T Palmer and M.  MacCarron on Bede’s computistical texts. See also von den Brincken, A.-D. (2006) ‘Jahrtausend­

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focusing on individual works to the entire œuvre of an author is very laudable, but from an historian’s point of view, obviously still quite limited. Bede, like any other author, was the product of his time. It was only towards the end of the eighth century that he became standard reading, as Joshua Westgard’s recent work on the manuscript transmission of Bede’s texts has highlighted again.89 Throughout the eighth century, he was only one of many computists, many anonymous, but with equally interesting (and, in some cases, more advanced) ideas. MacCarron analyzed the Irish background to Bede’s computistical texts in 2015,90 and more such studies are needed in the future, focusing on Bede’s background and his immediate, eighth-century impact.91 This brings us right into the Carolingian age. Since the publication of Schriften, Borst’s methodology and main thesis has been under scrutiny. The late Hartmut Hoffmann questioned the corpus itself, arguing that the sermon Arn Serm. (the eleventh text of the volume) should not, strictly speaking, be considered computistical.92 Most other critical voices questioned Borst’s theory of the study (and teaching) of computus being a heavily centralized project at the heart of the Carolingian educational reform, personified by Alcuin and Charlemagne.93 Accordrechnung, christliche Ära und Eschatologie bei Beda Venerabilis,’ Sitzungberichte der geisteswissenschaftlichen Klasse der Akademie gemeinnütziger Wissenschaften zu Erfurt 5, 11–26. 89 Westgard, J. A. (2010) ‘Bede and the continent in the Carolingian age and beyond,’ in S. De Gregorio, The Cambridge companion to Bede, Cambridge, 201–15. See also Faith Wallis’s ‘Bede and science’ in the same volume, pp. 113–26. 90 MacCarron, M. (2015) ‘Bede, Irish computistica and annus mundi,’ Early Medieval Europe 23, 290–307. 91 The seventh- and eighth-century context is missing or is only briefly referred to in the studies of Francisca Plaza Picón and Antonio Gonzáles Marrero: del Mar Plaza Picón, F. and J. A. González Marrero (2004) ‘El vocabulario del cómputo en el De temporibus liber de Beda,’ Minerva 17, 125–37; del Mar Plaza Picón, F. and J. A. González Marrero (2006) ‘De computo uel loquela digitorum: Beda y el cómputo digital,’ Faventia 28, 115–23; del Mar Plaza Picón, F. and J. A. González Marrero (2006) ‘Un acercamiento a los tratados del cómputo de Beda,’ Fortunatae 117–25; del Mar Plaza Picón, F. and J. A. González Marrero (2011) ‘La Epistola ad Wicthedum: un apéndice del De temporum ratione de Beda,’ in J. M. Gázquez, Ó. de la Cruz Palma, and C. Ferrero Hernández, Estudios de latín medieval hispánico, Firenze, 579–88. Worth consulting is the Fribourg Ph.D. thesis by R.-P. Pillonel-Wyrsch, published in 2004, which is a commentary to De temporum ratione: Le calcul de la date de pâques au moyen âge: analyse et commentaires sur de temporum ratione de Bède, Fribourg. 92 Hoffmann, H. (2012) ‘Abisag calefaciente oder Der karolingische Traktat De sole et luna,’ Deutsches Archiv für Erforschung des Mittelalters 68, 445–78. 93 Cf. Dobcheva, I. (2013) ‘The umbrella of Carolingian computus,’ in M. J. Muñoz Jiménez, P.  Cañizares Ferriz, and C.  Martin, La compilación del saber en la edad

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ing to Borst, a canon of computistical knowledge was set, first in the key monastic centres, then in the Aachen palace school, from where it radiated out to the more peripheral educational institutions (monastic and cathedral schools). This approach is reflected in the editions proper: Especially for the Cologne formulary of AD  760/92 (Lect. comp.) and the three Carolingian encyclopediae of AD 793, 809/12, 817 (Lib. ann., Lib. comp., Lib. calc.), Borst listed and used numerous manuscripts which contain only very few of the passages / algorithms of these texts. This creates the misleading impression that those passages were directly copied from these, allegedly normative encyclopediae. Quite the contrary, those concepts and algorithms circulated widely in Western Europe, they were common knowledge and therefore did not need the push by a centralized authority. More importantly, Borst stripped individual passages of their immediate context, and therewith clouded the fact that they belonged to separate cohesive texts in their own right, like the Computus of AD 757, the Computus Rhenanus of AD 776, the unfinished Computus of AD 789, etc. These texts will be properly introduced into scholarship in a separate volume on Carolingian computistics arising out of the last Galway conference. Dating clauses are a good indicator of the vibrancy of calendrical science in the early Middle Ages, but the vast number of them preserved in hundreds of manuscripts of especially the ninth and tenth centuries still needs to be examined and systematized (which, no doubt, can only be achieve as a big project with considerable (wo)manpower). Ninth-century computistics have therefore been approached from other perspectives. The historians of science have focused specifically on diagrams. Most prominent in this area is the scholarship of Bruce Eastwood, who, since the 1980s, has analysed the planetary diagrams occurring in the Carolingian commentaries to Pliny, Macrobius, Martianus Capella, and Calcidius. His collected essays appeared in the Variorum series in 2003, and in the following year he published, together with Gerd Grasshoff, a slim but influential volume which neatly outlines the manuscript evimedia, Porto, 211–30; Palmer, J. T. (2011) ‘Computus after the paschal controversy of AD 740,’ in Warntjes and Ó Cróinín (2011), 213–41; Palmer, J. T. (2011) ‘Calculating time and the end of time in the Carolingian world, c. 740–820,’ English Historical Review 126, 1307–31; Warntjes, I. (2012) ‘Köln als naturwissenschaftliches Zentrum in der Karolingerzeit: die frühmittelalterliche Kölner Schule und der Beginn der fränkischen Komputistik,’ in H. Finger and H. Horst, Mittelalterliche Handschriften der Kölner Dombibliothek, Viertes Symposion, Köln, 41–96.

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dence.94 In 2007, i.e. in the period since 2006 that primarily interests us here, Eastwood produced a wonderfully detailed monograph in Ordering the heavens, effectively harvesting the fruits of 30 years of research.95 The book may not make for light reading for the faint-hearted, but it provides Carolingian scholars with an indispensable guide to planetary theory. In 2011, both Eastwood and Stephen McCluskey, the doyens of the study of early medieval astronomy, contributed to a fine collection of essays on the Carolingian reception of Martianus Capella (ed. by Mariken Teeuwen and Sinéad O’Sullivan).96 The second approach to this material is an art historical one. The Carolingian period continues to fascinate scholars and the interested public alike also because of its lavish depictions of constellation figures and celestial phenomena. The pioneering study here probably is Fritz Saxl’s 1915 Verzeichnis astrologischer und mythologischer illustrierter Handschriften des lateinischen Mittelalters. More recently, in 2003, Bianca Kühnel provided a sweeping survey in The end of times in the order of things,97 and, in the last few years of his productive life, since 2005, the Cologne art historian Anton von Euw turned his interest to computistical manuscripts.98 Chief among the early medieval representation of constellation figures is the so-called Aratus Latinus, to which Floren94 Eastwood, B. S. (2002) The revival of planetary astronomy in Carolingian and post-Carolingian Europe, Aldershot; Eastwood, B. S. and G. Graßhoff (2004) Planetary diagrams for Roman astronomy in medieval Europe ca. 800–1500, Philadelphia. 95 Eastwood, B. S. (2007) Ordering the heavens: Roman astronomy and cosmology in the Carolingian Renaissance, Leiden. 96 Eastwood, B.  S. (2011) ‘The power of diagrams: the place of the anonymous commentary in the development of Carolingian astronomy and cosmology,’ in M. Teeuwen and S.  O’Sullivan, Carolingian scholarship and Martianus Capella: ninth-century commentary traditions on De nuptiis in context, Turnhout, 193–220; McCluskey, S. C. (2011) ‘Martianus and the traditions of early medieval astronomies,’ in ibidem, 221–44. 97 Kühnel, B. (2003) The end of time in the order of things: science and eschatology in early medieval art, Regensburg. 98 Early studies, which presumably started his interest in astronomical depictions, are his descriptions of Cologne, Dombibliothek, 83-II and 103 in Plotzek, J. M. and U. Surmann (1998) Glaube und Wissen im Mittelalter, Die Kölner Dombibliothek, München, 129–56. See von Euw, A. (2005) ‘Astronomie und Zeitrechnung im Karolingerreich,’ in H. Finger, Mittelalterliche Handschriften der Kölner Dombibliothek: Erstes Symposium, Köln, 21–64; von Euw, A. (2006) ‘Artes liberales und artes technicae im Spiegel der antiken, früh- und hochmittelalterlichen Handschriftenüberlieferung,’ in C. Stiegemann and M. Wemhoff, Canossa 1077: Erschütterung der Welt, 2 vols, München, i 544–54; von Euw, A. (2010) ‘Alkuin als Lehrer der Komputistik und Rhetorik Karls des Großen im Spiegel der St Galler Handschriften,’ in E. Tremp and K. Schmuki, Alkuin von York und die geistige Grundlegung Europas, St Gallen, 251–62.

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tine Mütherich drew renewed attention in the late 1970s.99 It survives in various recensions in numerous manuscripts, and Ivana Dobcheva is currently preparing a full-scale study of these for her Ph.D. Dieter Blume, Mechthild Haffner, and Wolfgang Metzger have supplied, in 2012, a massive catalogue of 68 manuscripts containing depictions of the 42 constellations.100 Hot off the press is Eric Ramírez-Weaver’s A saving science, which analyses the pictorial program in the Libri Computi of 809 in its political, ecclesiastical, intellectual, and artistic context.101 The third approach to the ninth century lies in detailed analyses of selected manuscripts, particularly those attributed to famous Carolingian intellectuals. Most prominent among these (beside the art historical study by Ramírez-Weaver on the famous codex Madrid, Biblioteca Nacional, 3307 dedicated to Drogo of Metz) are the monograph-length studies on Walahfrid Strabo’s Vademecum (St  Gall, Stiftsbibliothek, 878) and related manuscripts. Wesley Stevens has worked on the subject since the 1980s, and his Rhetoric and reckoning in the ninth century: The vademecum of Walahfrid Strabo is due to appear this year.102 The idea of connecting the various parts of the manuscript to the biography of its compiler is an excellent one, though the details will surely prove controversial. In 2014, Richard Corradini submitted a monumental Habilitationsschrift to the University of Vienna, entitled Karolingische Gelehrsamkeit und Zeitforschung im Kompendium des Walahfrid Strabo, which will also appear shortly in two volumes. Very recently, Jacopo Bisagni in Galway has drawn attention to the potential of studying the transmission of computistical ideas from Ireland through Brittany to the western Carolingian Empire.103 Methodologically, the focus lies principally on Breton and Loire valley manu Mütherich, F. (1989) ‘Die Bilder des Leidener Aratus,’ in B.  Bischoff et  al., Aratea: Kommentar zum Aratus des Germanicus Ms.  Voss. Lat.  Q. 79, Bibliotheek der Rijksuniversiteit Leiden, Luzern, 31–68, repr. in F. Mütherich, Studies in Carolingian manuscript illumination, London 2004, 147–265. 100 Blume, D., M. Haffner, and W. Metzger (2012) Sternbilder des Mittelalters: der gemalte Himmel zwischen Wissenschaft und Phantasie, Bd. 1: 800–1200, 2 vols, Berlin. Unfortunately, this 2-volume work is neither up to date nor affordable for the interested individual. 101 Ramírez-Weaver, E. M. (2017) A saving science: capturing the heavens in Carolingian manuscripts, University Park. 102 Stevens, W. M. (2017) Rhetoric and reckoning in the ninth century: the vademecum of Walahfrid Strabo, Turnhout. 103 As a starting point, see Bisagni, J. (2014) ‘A new citation from a work of Columbanus in BNF Lat. 6400B,’ Peritia 24–25, 116–22. 99

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scripts showing unambiguous Breton features (Old Breton glosses). This project is in its infancy, but it will eventually shed very bright new light on eighth-century Breton culture. As a side product, this research draws attention to neglected texts crucial for the understanding of the scientific strand of the Carolingian educational reform movement.104 It needs to be remembered that Borst’s corpus breaks off at AD 818. It is far from complete up to this point, however, and for the remaining eight decades of the ninth century the manuscripts need to be mined systematically for cohesive texts. Two impressive cases to illustrate this are the encyclopediae preserved in Laon, Bibliothèque municipale, 422 and Vatican, Biblioteca Apostolica, Reg. lat. 123.105 Both appear to have been produced in the mid-ninth century in the Loire valley, and they are prime witnesses of how the Carolingian scientific endeavour progressed from the celebrated three encyclopedias of AD 793, 809/12, 818 published by Borst to these hardly known, but much more structured and more sophisticated texts. Certainly, a study of these and, no doubt, other comparable collections will shed much more light on the Carolingian invention of the scientific encyclopedia and its development in the slowly but surely disintegrating Empire. This also points to what I would consider the biggest desideratum in the study of the mid- to late ninth century: detailed studies of key (calendrical) scientific centres or Wissenschaftslandschaften, like St Gall and Fleury on the one hand, the Lake Constanz region and the Loire valley on the other. If the second half of the ninth century is massively understudied, this applies even more so to the tenth century, at least to its first seven decades. One of the most promising ways of approaching this century will be through the Liber de computo of Helperic, which was composed in AD 900/03, but then constantly updated right into the eleventh century and beyond. The transmission of this text and the manuscript contexts in which it appears (Helperic’s computus is preserved in over 80 manu104 Besides the two massive works discussed in the following, see also especially the neglected scientific encyclopedia in Cologne, Dombibliothek, 83-II, 15r–36v, which is currently also being studied by Bisagni. 105 On both, Bisagni has articles in preparation. For discussion of some of the texts and diagrams in those two manuscripts, see Obrist, B. (2002) ‘Les manuscrits du De cursu stellarum de Grégoire de Tours et le MS Laon, Bibliothèque municipale 422,’ Scriptorium 56, 335–44; Castiñeiras González, M.  A. (1998) ‘Diagramas y esquemas cosmográficos en dos misceláneas de cómputo y astronomía de la Abadía de Santa María de Ripoll (ss. XI–XII),’ Compostellanum 43, 593–646; Martínez Gázquez, J. and J. Gómez Pallarès (1992) ‘La Epistola de ciclo paschali del monje Oliba de Ripoll,’ Mittellateinisches Jahrbuch 27, 103–40.

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scripts), will shed substantial light on the computitistical tradition of the two centuries after its original composition. This fundamental text still awaits a proper introduction into scholarship by means of a modern critical edition, which hopefully will reverse the negligence of the past 40 years which saw no publication on this crucial textbook.106 One of the updates of Helperic’s text was produced by Abbo of Fleury in AD 978. More importantly, Abbo designed his own computus, a new edition of which will be published by Alfred Lohr shortly. Abbo’s computus, best preserved in Berlin, Staatsbibliothek, Phillipps 1833, signals a major change in the presentation of computistical material. When computus turned into a major subject in monastic education in the seventh century, it was almost exclusively text-based (with the exception of the Easter tables proper, of course). In the Carolingian era, some of the algorithms were pressed into table format, while diagrams helped to explain more complex concepts. Abbo’s computus, then, was almost exclusively based on tables and diagrams, the textual element was minimized. Whether Abbo can be credited with this revolution in the presentation of computistical material can only be determined by a systematic analysis of tenth-century manuscripts. Certainly, the tenth century has much more to offer than the current neglect of this period suggests, and Nadja Germann’s 2006 study of Abbo’s scientific mindset, together with the excellent volume on Abbo edited by Barbara Obrist two years earlier, are only a first step towards a better understanding of this crucial century.107 Abbo certainly set a trend by questioning the incarnation era. One of the reasons why the Easter controversy lasted well into Charlemagne’s reign was the fact that the Alexandrian reckoning in Dionysius’ translation was intrinsically connected to AD. AD marked one of the columns 106 Helperic’s Liber de computo was edited by Bernard Pez in 1721 (repr. in PL 137, 17–48), on the basis of one, late witness (of AD 1090!). A list of manuscripts can be found in Jullien, M.-H. (2010) Clavis scriptorum Latinorum medii aevi: auctores Galliae 735– 987, vol. 3, Turnhout, 421–29. Key studies are: Traube, L. (1893) ‘Computus Helperici,’ Neues Archiv der Gesellschaft für ältere deutsche Geschichtskunde 18, 73–105, repr. with additional notes in Traube (1909–20), iii 128–56; McGurk, P. (1974) ‘Computus Helperici: its transmission in England in the eleventh and twelfth century,’ Medium Aevum 43, 1–5. 107 Germann, N. (2010) De temporum ratione: quadrivium und Gotteserkenntnis am Beispiel Abbos von Fleury und Hermanns von Reichenau, Leiden. Obrist, B. (2004) Abbon de Fleury: philosophie, science et comput autour de l’an mil, Paris. See also the articles by Germann (‘A la recherche de la structure du temps: Abbon de Fleury et le comput’) and Obrist (‘Abbon de Fleury: cosmologie, comput, philosophie’) in Dufour-Malbezin, A. (2008) Abbon, un abbé de l’an mil, Turnhout. For an earlier study, see Engelen, E.-M. (1993) Zeit, Zahl und Bild: Studien zur Verbindung von Philosophie und Wissenschaft bei Abbo von Fleury, Berlin.

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of Dionysius’ Easter table, and this column provided its most contentious datum. Dionysius’ original table covered the years AD 532–626, but there was enough expertise in the seventh and eighth centuries not only to extend that table, but also to calculate its data back to the time of Christ. The Gospels suggest that Christ died in either the 31st or 34th year of his life.108 The Dionysiac reckoning provided lunar data for both years which could not be reconciled with the explicit information found in the Gospels. This made the Dionysiac reckoning theologically problematic at best. There are interesting seventh-century attempts to solve this obvious problem, and Bede tried to dismiss it by a blunt reference to authority. The question re-emerged, however, in Abbo’s days. Between the AD 990s and 1135, at least eight attempts at recalculating the incarnation era prove the seriousness of the issue. These have been thoroughly discussed by Peter Verbist in his 2003 Ph.D. thesis, which appeared in English translation (with financial help from the ArnoBorst-Foundation) in 2010 (following in the footsteps of van de Vyver, von den Brinken, and Wiesenbach, discussed above).109 Philipp Nothaft corrected Verbist’s account of Marianus Scottus’ recalculation in a seminal article of 2013, and he proved that Marianus’ ideas were still being discussed in the early 14th century.110 For the eighth and last of these recalculations of the incarnation era by Heimo of Bamberg, Verbist based his study on the later, longer of two versions of the text, and worked directly from the manuscript. As mentioned above, between Verbist’s Ph.D. of 2003 and the English translation of 2009, Hans Martin Weikmann published a critical edition of both versions in parallel, which will have to form the basis of all subsequent studies. In fact, the text discovered by Luciana Cuppo and published in the present volume, appears to represent a ninth attempt at recalculating the incarnation, rather than belonging to the seventh century as she argues. Abbo also set a trend in Anglo-Saxon England. Little is presently known about the Anglo-Saxon computistical tradition between Bede and the Benedictine reform movement, and this certainly would make For the wider context, see Nothaft, C. P. E. (2012) Dating the Passion: the life of Jesus and the emergence of scientific chronology (200–1600), Leiden. 109 Verbist, P. (2010) Duelling with the past: medieval authors and the problem of the Christian era (c. 990–1135), Turnhout. 110 Nothaft, C. P. E. (2013) ‘An eleventh-century chronologer at work: Marianus Scottus and the quest for the missing twenty-two years,’ Speculum 88, 457–82; Nothaft, C. P. E. (2012) ‘Nicholas Trevet and the chronology of the crucifixion,’ The Mediaeval Journal 2 (2012), 37–51. See also Baran-Kozłowski, W. (2009) Kronika świata, Mariana Szkota: studium źródłoznawcze, Poznań. 108

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for an excellent study for more enterprising students or scholars. Traditionally, most of what has been published on late Anglo-Saxon computus follows the practice inherent in English Studies of focusing on prominent authors, here two scholars influenced by Abbo, Ælfric and Byrhtferth. A new edition of Ælfric’s De temporibus anni was published by Martin Blake in 2009.111 In 2011, Ælfric was the theme of the XII Seminario avanzato in filologia germanica, and Giuseppe de Bonis prepared an article on Ælfric’s computus for the subsequent proceedings.112 Byrhtferth has been more prominently in the spotlight of recent scholarship. This is in part due to rekindled interest in the glosses to Bede’s computistica ascribed to his name, which John Contreni discussed in Galway in 2012.113 Even more attention has been given to Byrhtferth’s Enchiri­ dion, especially its diagrams. Philippa Semper in 2004 and Peter Baker in the 2005 Festschrift for Michael Lapidge reopened discussion,114 which was continued by Francesca Chiusaroli and Cristina Raffaghello in Italy.115 The last Galway conference in 2016 saw a strong panel on Byrhtferth with papers by Rebecca Stephenson and Sabine Rauch.116 Blake, M. (2009) Ælfric’s De temporibus anni, London. De Bonis, G. D. (2012) ‘Il De temporibus anni di Ælfric: una rielaborazione contenutistica e sintattica dell’ opera De temporibus di Beda,’ in V.  Dolcetti Corazza and R. Gendre, Lettura di Ælfric, Alessandria, 347–88. For Ælfric, see also Chardonnens, L. S. (2013) ‘Ælfric and the authorship of the Old English De diebus malis,’ in C. Gili­ berto and L. Teresi, Limits to learning: the transfer of encyclopaedic knowledge in the early Middle Ages, Leuven, 123–53. 113 Contreni, J.  J. (2011–12) ‘‘Old orthodoxies die hard’: Herwagen’s Bridferti Ramesiensis Glossae,’ Peritia 22–23, 15–52. Contreni’s recent scholarship on the subject is a reaction to earlier articles by Lapidge and Gorman: Lapidge, M. (2007) ‘Byrhtferth of Ramsey and the Glossae Bridferti in Bedam,’ Journal of Medieval Latin 17, 384–400; Gorman, M. (1996) ‘The glosses on Bede’s De temporum ratione attributed to Byrhtferth of Ramsey,’ Anglo-Saxon England 25, 209–32, repr. in M. Gorman, Biblical commentaries from the early Middle Ages, Firenze 2002, 175–98. 114 Semper, P. (2004) ‘Doctrine and diagrams: maintaining the order of the world in Byrhtferth’s Enchiridion,’ in P. R. Cavill, Christian tradition in Anglo-Saxon England: approaches to current scholarship and teaching, Woodbridge, 121–37. Baker, P. S. (2005) ‘More diagrams by Byrhtferth of Ramsey,’ in K. O’Brien O’Keeffe and A. Orchard, Latin learning and English lore, 2 vols, Toronto, ii 53–73. 115 Chiusaroli, F. (2005) ‘Costituzione e impiego del lessico tecnico nell’Enchiridion di Byrhtferth: l’ambito dell’astronomia,’ in D. Gottschall, Testi cosmografici, geografici e odeporici del medioevo germanico, Louvain-la-Neuve, 1–40; Raffaghello, C. (2009) ‘Il computo del tempo in ambiente anglosassone: l’Enchiridion de Byrhtferth di Ramsey,’ in L. Vezzosi, La letteratura tecnico-scientifica nel medioevo germanico: Fachliteratur e Gebrauchstexte, Allessandria, 213–36. 116 See also Stephenson, R. (2012) ‘Byrhtferth’s Enchiridion: the effectiveness of hermeneutic Latin,’ in E. Tyler, Conceptualizing multilingualism in England, 800–1250, 111 112

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More problematic is the question of how to approach the vast number of English computistical manuscripts of c.AD 950–1150. One route taken in recent years is through a series of texts that are classified as prognostics by modern scholars. Prognostics occur, more often than not, in computistical manuscripts because of the calendrical element in many of these texts. From the ninth to the eleventh centuries, prognostics appear to have developed into a genre in its own right. The lead here was taken by Roy Liuzza’s seminal article of 2001, which he followed up by a monograph on London, British Library, Cotton Tiberius A III in 2010.117 Sándor Chardonnens produced a detailed overview of the texts and manuscript tradition in 2007, which he in turn followed up with numerous articles.118 Marilina Cesario produced a series of articles on weather prognostics.119 This massive increase in interest in late Anglo-Saxon prognostics led to a panel on the subject at the 2012 Galway conference which brought the three aforementioned scholars into dialogue. Prognostics Turnhout, 121–44; Stephenson, R. (2016) ‘Saint who? Building monastic identity through computistical inquiry in Byrhtferth’s Vita  S. Ecgwini,’ in R.  Stephenson and E. Thornbury, Latinity and identity in Anglo-Saxon literature, Toronto, 118–37. 117 Liuzza, R. (2003) ‘Anglo-Saxon prognostics in context: a survey and handlist of manuscripts,’ Anglo-Saxon England 30, 181–230; Liuzza, R. (2010) Anglo-Saxon prognostics: studies and texts from London, British Library, MS Cotton Tiberius A.iii, Woodbridge. 118 Chardonnens, L. S. (2007) Anglo-Saxon prognostics, 900–1100: study and texts, Leiden. Of his subsequent articles, see, e.g., Chardonnens, L. S. (2007) ‘Context, language, date and origin of Anglo-Saxon prognostics,’ in R. H. Bremmer, Jr and K. Dekkers, Foundations of learning: the transfer of encyclopaedic knowledge in the early Middle Ages, Groningen, 317–40; Chardonnens, L. S. (2007) ‘London, British Library, Harley 3271: the composition and structure of an eleventh-century Anglo-Saxon miscellany,’ in P.  Lendinara, L.  Lazzari, and M.  A. D’Aronco, Form and content of instructions in Anglo-Saxon England in the light of contemporary manuscript evidence, Turnhout, 3–34; Chardonnens, L.  S. (2010) ‘Appropriating prognostics in late Anglo-Saxon England: a preliminary source study,’ in R. H. Bremmer, Jr and K. Dekker, Practice in learning: the transfer of encyclopaedic knowledge in the early Middle Ages, Leuven, 203–55; Chardonnens, L.  S. (2011) ‘Norm and practice of divination and prognostication in late Anglo-Saxon England,’ in L. Sturlese and K. Nauer, Mantik, Schicksal und Freiheit im Mittelalter, Wien, 51–64; Chardonnens, L. S. (2016) ‘Beyond long lines and column: experiments in the visual structure of knowledge in Anglo-Saxon manuscripts,’ R. H. Bremmer, Jr and K. Dekker, Fruits of learning: the transfer of encyclopaedic knowledge in the early Middle Ages, Leuven, 35–74. 119 See, e.g., Cesario, M. (2009) ‘La Revelatio Esdrae nella tradizione latina e anglosassone,’ in L. Vezzosi, La letteratura tecnico-sceintifica nel medioevo germanico: Fachliteratur e Gebrauchstexte, Allessandria, 57–84; Cesario, M. (2012) ‘Weather prognostics in Anglo-Saxon England,’ English Studies 93, 391–426; Cesario, M. (2015) ‘An English source for a Latin text? Wind prognostication in Oxford, Bodleian MSS Hatton 115 and Ashmole 354,’ Studies in Philology 112, 213–33.

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provide an interesting inroad into the English computistical tradition of the high Middle Ages, but a comprehensive focus on the computistica, especially those composed in Latin, still remains a major desideratum. Thorough studies of individual manuscripts as undertaken, e.g., by Faith Wallis will be one of the main avenues to open up this material to the wider scholarly community, and her website on Oxford, St. John’s College, 17 (launched in 2007) is an excellent starting point for anybody interested in English calendrical science of the high Middle Ages.120 The Benedictine reform movement brings us to the eleventh century, to Germany, and back to Arno Borst and Hermann of Reichenau. As outlined above, since 1975 Borst had toyed with the idea of writing a biography of Hermann of Reichenau, one of the most prominent intellectuals of that century. In order to do that, he started preparing first editions of Hermann’s key scientific texts. The project stalled in 1990, however, for various reasons, one of them being that more groundwork was needed on Hermann’s sources, which led to Schriften, Borst’s last publication. The basis for an appreciation of Hermann’s scientific writings had been laid, but, sadly, Borst was not given the time to finalize the project he had been working on for four decades. With Borst’s death in 2007, his microfilm copies went to the Monumenta Germaniae Historica in Munich, while his remaining papers went to the Universitätsbibliothek in Konstanz (his books were transferred to the Stiftsbibliothek in Zeitz). Among the Konstanz papers was the 1990 draft of his edition of Hermann’s scientifica, which came to the attention of the Monumenta Germaniae Historica. It was immediately understood that Borst’s edition of Hermann’s computistica was much further advanced than his work on the astrolabica. Therefore, it was agreed to publish one volume on Hermann’s computistica on the basis of Borst’s manuscript first, which will hopefully be followed by a thorough study of Hermann’s astrolabica.121 Hermann’s interest in the subject was sparked by Notker of St Gall (†AD 1022), whose Quatuor questionibus compoti Hermann considered inadequate. In order to understand the context, Borst decided to include Notker’s text in the volume.122 Notker was principally known for his http://digital.library.mcgill.ca/ms-17/index.htm. Borst, A. and I. Warntjes (2018) Hermann der Lahme: Schriften zur Zeitrechnung, Wiesbaden. See also the study by Germann (2006). 122 In this, Borst followed the pioneering study by Meier, G. (1887) ‘Die sieben freien Künste im Mittelalter 2,’ Jahresbericht über die Lehr- und Erziehungs-Anstalt des Benediktiner-Stiftes Maria-Einsiedeln der Studienjahre 1886/87, Einsiedeln, 3–36. 120 121

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writings in Old High German, which led to his epithet ‘the German’. Incidentally, he also translated his own computus from Latin into German, as a wonderful discovery by Norbert Kruse in the Fürstlich Quadt zu Wykradt und Isny’schen Archiv proves, published in 2003.123 One of the questions that Notker posed, on the basis of an eighth-century tract, was about the exact length of the mean synodic lunar month (the period from one new moon to the next). The mathematical exercise was a division of 6939 ¾ days by 235 lunar months (of the 19-year lunar cycle). Notker did not consider it worth the effort to follow the calculation through to the smallest fractions. Hermann, however, did just this (in his Epistola), and then created a new 19-year cycle on the basis of his precise value (Abbreviatio) and even tried to use it for the calculation of eclipses (Prognostica). Alongside Notker’s Questiones and Hermann’s three works, Borst also included commentaries on Hermann’s texts in this volume. In total, some 30 manuscripts from the eleventh and twelfth centuries are thoroughly analysed in this study, which therefore provides a good (but certainly not exhaustive) insight into the study of computus in high medieval southern Germany. Similar studies for other regions of Western Europe in the eleventh and twelfth centuries are still major desiderata. Borst’s ultimate goal of producing a biography of Hermann was finally achieved in a collective effort through a conference held in Weingarten in 2013. Its proceedings of 2016 show a strong focus on Hermann’s scientific (often mistaken as ‘quadrivial’) works.124 Besides a discussion of Hermann’s astrolabica (by David Juste) and his computistica (by myself ), the volume includes an introduction to Hermann’s abacus tracts (by Martin Hellmann) and his version of the Rithmomachia (by Menso Folkerts). It is a forceful reminder that computists, from the seventh to the eleventh century, were not scholars of a single discipline. What changed were the other areas they studied. In the seventh century, computists quite certainly were also trained grammarians and accomplished exegetes. From the tenth or eleventh century onwards, however, computists were actively embracing the new scientific knowledge entering the Latin West from the Arabic world; accordingly, they would also have called themselves experts on the astrolabe or abacists.125 The interrela123 Kruse, N. (2003) ‘Eine neue Schrift Notkers des Deutschen: der althochdeutsche Computus,’ Sprachwissenschaft 28, 123–55. 124 Heinzer, F. and T. Zotz (2016) Hermann der Lahme: Reichenauer Mönch und Universalgelehrter des 11. Jahrhunderts, Stuttgart. 125 See, e.g., the case of Abbo: Burnett, C. (2006) ‘Abbon de Fleury, abaci doctor,’ in B. Obrist, Abbon de Fleury: philosophie, science et computu autour de l’an mil, Paris, 129–39.

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tion of these subjects still remains to be studied in detail, for the seventh century as much as for the eleventh. Hermann’s ideas principally spread from his monastery of Reichenau in Lake Constance in two directions: north-eastwards into Bavaria and up the Rhine as far as Liège. One of the key computists influenced by Hermann was a certain Gerland, who worked in Lotharingia. His computus (of c.AD 1060–93) has been studied principally for his re-dating of the incarnation era, a chronological problem that Abbo of Fleury appears to have popularized.126 But his text contains many more important ideas, which can only be appreciated since Alfred Lohr’s editio princeps of 2013.127 Gerland applied Hermann’s exact value of the mean synodic lunar month more rigorously to the 19-year lunar cycle. The problem was that the 19-year cycle did not consist of an integer number of days (6939 ¾). This made the equal distribution of the fractions of the mean synodic lunar month difficult, as they could only be assigned to full calendar days, not their quarters. The solution was a 76-year period (4×19  years=4×6939  ¾  days=27,759  days). In one of the eleventhcentury redactions of Hermann’s computistica, his results were already applied to a 76-year period. Gerland followed this lead, and, like, Hermann, used a solar eclipse (of 29 September 1093) as the basis for his calculations. Lotharingia towards the end of the eleventh century and at the beginning of the twelfth appears to have been a hotbed of progressive computists. Shortly after Gerland, Walcher, also from Lotharingia, added more accuracy to Gerland’s model by using the lunar eclipse of 18  October 1092 as the basis of his calculations of all 940 conjunctions in 76 years (in a text now labelled De lunationibus by its recent editor). Unlike Gerland, who worked from an eclipse record rounded to the hour, Walcher was able to draw on his own astrolabe calculations to gain a precision of the event unparalleled in the Latin West. Alas, Walcher soon realized that his calculations did not match the observable realities of subsequent years, as solar and lunar eclipses occurred a few hours after Walcher’s estimated time. Walcher had the benefit of being able to draw on two research traditions. The ideas underlying his De lunationibus he surely acquired in his native Lotharingia. But he soon moved to England, where he became See above pp. 27–28. Lohr, A. (2013) Der Computus Gerlandi: Edition, Übersetzung und Erläuterung, Stuttgart. 126 127

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prior of Great Malvern. The West Midlands, at this time, developed into a centre of the early reception of Arabic science in Western Europe, personified by Abelard of Bath. Walcher came into dialogue with Petrus Alfonsi, a Jew trained in Arabic Iberia, who provided some insight into why Walcher’s calculations did not match the observable reality: the motions of sun and moon are not uniform, and the lunar node (the points when the moon’s orbit crosses the ecliptic) are not fixed, they have their own motion, in the opposite direction to the planets (De dracone). Walcher deserves the credit of heralding the potential of applying Arabic science to computistics. His pioneering work used to be known only to the initiated few among modern scholars, and it had never been studied either in detail or in full. It is only thanks to the masterful editio princeps published by Philipp Nothaft this year that the scholarly community can fully appreciate Walcher’s ingenious thought-world.128 Philipp Nothaft’s interest in the introduction of Arabic science into computus was partially triggered by the study of manuscripts transmitting Hermann’s work. Hermann paved the way for the division of computus into vulgaris or ecclesiasticus, and naturalis. Vulgaris was the traditional design for the ordering of the liturgical calendar, while na­ turalis attempted to find solid mathematical models for observable celestial phenomena. Hermann produced the first attempt at naturalis in the second part of his Abbreviatio and in his Prognostica. However, as Hermann’s theories did not work, this opened up the search for more accurate models. In the manuscripts, Hermann’s study of the computus naturalis was soon replaced, in south-eastern Germany, by discussions of the Arabic lunar calendar and its relation to the traditional system. A pioneering article in this respect is Nothaft’s ‘The application of Arabic science in twelfth-century computistics’ of 2013, which puts a spotlight on the diocese of Salzburg.129 In a second publication of 2015, Nothaft opens the same discussion for twelfth-century England by editing a hitherto unknown Collatio compoti Romani et Arabici.130

128 Nothaft, C. P. E. (2017) Walcher of Malvern, De lunationibus and De dracone: study, edition, translation, and commentary, Turnhout. 129 Nothaft, C.  P.  E. (2013) ‘The reception and application of Arabic science in twelfth-century computistics: new evidence from Bavaria,’ Journal for the History of Astronomy 44, 35–60. 130 Nothaft, C. P. E. (2015) ‘Roman vs. Arabic computistics in twelfth-century England: a newly discovered source (Collatio compoti Romani et Arabici),’ Early Science and Medicine 20, 187–208.

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The discussion of foreign lunar calendars as alternatives to the almost a millennium old Christian tradition reached its first height in the AD 1170s. At least four authors compiled in this decade substantial treatises on computus and the relation of the traditional Christian system to the newly introduced Jewish and Arabic lunar calendars (as well as additional astronomical information from newly translated Greek authorities). The first is Reinher of Paderborn in Westphalia, who vehemently advocated the supremacy of Jewish calculation (presumably influenced by intellectual developments in neighbouring Lotharingia). His Computus emendatus of AD 1170–71 was first transcribed in print by Walter van Wijk in 1951, and Werner Herold produced a new edition in 2011, which, however, falls short of modern critical standards.131 In 2015, Alfred Lohr produced a definitive text for Corpus Christianorum, and, admirably, this volume also contains editiones principes of the following two computists, thereby bringing to a successful fruition the pioneering (though sometimes misguided) studies by the Dublin-based scholar, Jennifer Moreton.132 If Reinher was the reformer, a certain Magister Cunestabulus, writing in AD 1175, presumably in Canterbury according to Lohr, was the traditionalist (and a new study of this text has just been published by Nothaft133). Cunestabulus was provoked by Gerland’s redating of the incarnation era, which he refuted extensively in his final chapter (39). In the three chapters before that (36–38), he cleverly employed Greek, Arabic, and Jewish knowledge to explain the difference between traditional computus and observable reality in the date of the equinoxes, solstices, and new moons. The rest of the work is more or less a textbook along conventional lines. The third author, Roger of 131 van Wijk, W. E. (1951) Le comput emendé de Reinherus de Paderborn (1171), Amsterdam; Herold, W. (2011) Reinher von Paderborn – Computus emendatus: die verbesserte Osterfestberechnung von 1171, Paderborn; see also Herold, W. (2005) ‘Der Computus emendatus des Reinher von Paderborn,’ Beiträge zur Geschichte des Bistums Regensburg 39, 39–47. 132 Lohr, A. (2015) Opera de computo saeculi duodecimi: Reinheri Paderbornensis Computus emendatus, Magistri Cunestabuli Computus, Rogeri Herefordensis Computus, Turnhout (CCCM 272). Moreton, J. (1995) ‘Before Grosseteste: Roger of Hereford and the calendar reform in twelfth-century England,’ Isis 86, 562–86; Moreton, J. (1999) ‘The Compotus of Constabularius (1175): a preliminary study,’ in J. Biard, Langage, sciences, philosophie au XIIème siècle, Paris, 61–82. For Moreton’s contribution to the study of medieval computus, see P. Nothaft, ‘On Jennifer Moreton’s edition of the Computus ecclesiasticus,’ https://durhamgrossetesteproject.files.wordpress.com/2015/10/nothaft_ preface-to-moreton_computus-ecclesiasticus_online-edition.pdf. 133 Nothaft, C. P. E. (2017) ‘A reluctant innovator: Graeco-Arabic astronomy in the Computus of Magister Cunestabulus (1175),’ Early Science and Medicine 22, 24–54.

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Here­ford, writing in AD 1176, was then the conciliator, outlining the details of the old system (books 1–3), Gerland’s new approach (here updated; book 4), and finally contrasting the computus vulgaris and naturalis with the Hebrew lunar calendar and Arabic astronomy (book 5). A fourth text of the AD 1170s, a computus by a certain Peter, has only very recently been (re-)introduced into scholarship by Philipp Nothaft and awaits detailed analysis.134 Twelfth-century computus therefore provides a wealth of information concerning the practical impact of the so-called Renaissance of the twelfth century.135 Even more so, it opens an interesting window into the cultural difficulties, the prejudices and biases of those responsible for the introduction of new knowledge that ran counter to established church doctrine. There has been a considerable turn over the past two decades within Medieval Studies towards comparative history between different cultures (especially between Western Europe and the Arab world) and towards what is often termed cultural transfer (principally of ideas). It must appear surprising that computus has not entered the discussion yet, as there is hardly any other field in which the appropriation of Greek and Arabic knowledge directly questions Christian authority. Computus has constantly been marginalized within Medieval Studies, for various reasons, but particularly because of its perceived complexity and an underestimation of its cultural impact. Sometimes, the more time-consuming engagement with complex topics yields the more fruitful results, and it can only be hoped that the current tendency of research being determined by arbitrary metrics, public impact, and projects designed to please funding bodies rather than simply pushing the boundaries of knowledge, will not create further obstacles in the future. Nothaft and Lohr have raised twelfth-century computistics, almost single-handedly, to a new level. Their research shows the massive potential of manuscript studies, which will certainly lead to new and important discoveries. On the other hand, it demonstrates the existence of exciting Wissenschaftslandschaften (regions of specialized learning) in south-eastern Germany, Lotharingia, and the West Midlands in England. In this, they have only started the academic discourse, and more systematic studies of all computistical manuscripts of these regions, and Nothaft (2017), 28; earlier Haskins (1924), 86. For the scientific strand of the Renaissance of the twelfth century, see especially the pioneering studies by Charles Homer Haskins: his detailed analyses collected in Studies in the history of mediaeval science (Cambridge 1924) and his sweeping classic The Renaissance of the twelfth century (Cambridge 1927). 134 135

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even more so of the so-far neglected France, will be very rewarding areas of research in the future. It certainly is crucially important to understand the role of computistics in the adoption of Arabic science. In the oneand-a-half centuries from Hermann to Roger of Hereford, computus became thoroughly divided into two separate strands: the traditional computus ecclesiasticus which was soon taught as the first basic scientific instructions in the newly emerging universities, while the computus naturalis had fulfilled its purpose by providing inroads into the critical examination of Arabic and Greek science which became the basis of the scientific discourse of subsequent centuries. C.AD 1200 therefore marks a major watershed in the study of the medieval computistical traditions, and I therefore end my survey here. I can do this with a good conscience, knowing that Philipp Nothaft’s seminal volume on the pre-history of the Gregorian calendar reform (Scandalous error) has been submitted to the publisher and will appear shortly (and is sure to replace Kaltenbrunner’s classic136). This will provide an excellent overview of the continued story of the calendrical endeavour from the late twelfth to the sixteenth centuries. It certainly is very heartening to see that computus has made it into the most recent surveys of medieval mathematics, which is a welcome present acknowledgement of its past importance.137 *** Among the six conferences on Computus so far held in Galway since 2006, the third, which forms the basis for the present volume, marked a notable watershed. The initial idea was to bring together all scholars worldwide who worked on the technical details of late antique and early medieval computus. The field was small enough, the time-range covered substantial, the source material overwhelming, so that the pattern established was to meet regularly to learn from each other as much as possible in a short period of time and to create a platform for communication. The third conference still followed this original spirit, with a strong focus on the scientific theory behind late antique and early medieval calendrical concepts, as this book demonstrates. One of the contributions to 136 Kaltenbrunner, F. (1876) ‘Die Vorgeschichte der Gregorianischen Kalenderreform,’ Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften, philosophischhistorische Classe 82, 289–414. 137 Hein, W. (2010) Die Mathematik im Mittelalter: von Abakus bis Zahlenkampfspiel, Darmstadt, 87–102; Katz, V. J. et al. (2016) Sourcebook in the mathematics of medieval Europe and North Africa, Princeton, 29–36.

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the third conference, Leofranc Holford-Streven’s edition of the Disputatio Chori et Praetextati, grew to monograph length in the process of preparation for print, so that it was decided to publish this seminal study as a separate book. The first five essays in this volume principally deal with Easter reckonings and the tables preserving the relevant data. Alden Mosshammer starts off with a discussion of the oldest computistical text that has survived, the famous Computus of AD 243. Mosshammer, who has been working on an edition of this important text, here explains its Roman contexts and the misrepresentation of its 112-year Easter cycle in the available editions. Jan Zuidhoek analyzes the Easter table in Anatolius Latinus’ De ratione paschali. He concludes that the data provided there suggest AD 271 as the base year of the table. Dan Mc Carthy examines the reference to St Patrick in Cummian’s letter of AD 632 and its context. In his view, the southern Irish as represented by Cummian adopted a Victorian 532-year Easter table with lunar limits 15–21 devised by Patrick, which provides additional evidence for the late chronology of the saint’s floruit. Luciana Cuppo continues the discussion of the introduction of Easter tables into the insular world in the seventh century. First she focuses on three manuscripts which she believes represent Victorian reactions to the gradual spread of the Dionysiac reckoning. The second part of her article argues that part of the famous codex Oxford, Bodleian Library, Digby 63 contains a Dionysiac dossier that was, in her opinion, probably sent to Anglo-Saxon England by Pope Vitalian (AD 672–76). From Britain to Iberia: Brigitte Englisch (in a more speculative article) assumes that the Christian communities in Spain followed an Easter reckoning that was based, at least in parts, on actual observations. The second half of this volume deals with texts of various genres and their computistical ideas. David Howlett discusses a didactic poem on computistics which he places in seventh-century Ireland. Marina Smyth examines the early medieval perception of one of the most though-provoking feature of the Julian calendar, the bissextile (leap-year) day. She surveys its understood length (three vs six hours per year) and the various theological, allegorical, etymological, and pseudo-scientific explanations of this calendrical device that circulated from seventh-century Ireland to eighth-century Francia. Philipp Nothaft reconstructs the ‘computistical chronology’ of Claudius of Turin, who calculated the Julian calendar date and weekday data for key biblical events. The most important one of these obviously was the passion of Christ, for which Claudius implied, according to Nothaft’s reconstruction (contra Borst),

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the impossible 21 March AD 34. Lisa Chen Obrist turns attention to Claudius’s contemporary Hrabanus Maurus, proving that in one strand of the manuscript transmission of his De rerum naturis he made use of Isidore of Seville’s De ecclesiasticis officiis. This volume therefore principally deals with the technical features of lunar and solar calendars designed in late antiquity and their application in the early Middle Ages. In the final article, Dáibhí Ó Cróinín rounds off the volume by turning to the history of scholarship on late antique and early medieval computus. The interests of 16th- and 17th-century scholars in the texts and tables discussed in the present volume lay in establishing biblical chronology and in understanding one of the most central debates within the early Church, the Easter controversy. James Ussher, working in the first half of the 17th century, was a key figure in this emerging scholarship, and Ó Cróinín unearthed in the Bodleian Library a list of late antique and early medieval texts related to the Easter question in chronological order, compiled by Ussher. This list demonstrates the rigour of 17th-century scholarship and serves as a reminder of the many texts written before AD 700 which still await a definite study. The third Galway conference was the last with such a strong focus on technical details. It was felt that, after establishing the technicalities, it was time to allow more room for the scientific, cultural, political, theological, and manuscript contexts of computus. To facilitate this and to attract more colleagues from neighbouring disciplines, special themes were developed for each of the subsequent conferences. The first of these, in 2012, was on early medieval prognostics, attracting scholars like David Juste, Roy Liuzza, Sándor Chardonnens, Faith Wallis, Marilina Cesario, and Richard Landes. The dialogue started there was very inspiring, and it certainly served one of the purposes of introducing these themes, namely creating a continued interest in the topics discussed for future conferences. Prognostics has since formed a central part of the Galway conference. In 2014, a platform was created to discuss the extremely impressive advances in high medieval computus, which I have highlighted in some detail above. In 2016, the impact of Borst’s Schriften on the advances in Carolingian computistics was reviewed ten years after the publication of his monumental work. For the first time, a call for papers was issued, which received an overwhelming response. In fact, the conference had to be extended from its traditional 1 ½ to 2 ½ days, which stretched the organizational limits of an academic culture that does not have a tradition of undergraduate and postgraduate Hilfskräfte; without the voluntary help of Galway MA students, the tremendous organiza-

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tional skills of Maura Walsh (Ó Cróinín), and Dáibhí’s exceptional abi­ lity of securing the basic funding necessary for such an event, this would not have been possible. It is hoped that the next Galway conference will have ‘computus and the vernacular’ as its special theme. The introduction of special themes also changes the direction of publication. The 2016 Carolingian theme was so popular that a separate volume on Carolingian computistics is envisaged for the near future. The proceedings of the 2012 and 2014 conferences will be published in one volume. In the future, to keep the publications as focused as possible, a volume on the special theme will be produced. At the same time, the Galway Conference will continue to invite papers on topics other than the special theme, to keep its character as a platform for the latest research in late antique and early medieval computistics; publication of these papers in key journals will be strongly encouraged to spread the awareness of the massive progress that has been made in this field. It is very much hoped that the present volume, the two previous ones, and those published in the future will push the boundaries of knowledge in this field, and will attract new recruits to the vibrant research culture of late antique and early medieval computus and its scientific, cultural, political, theological, and manuscript context.

Appendix: Alfred Cordoliani’s contribution to the study of computus138 ‘Les traités de comput ecclésiastique de 525 à 990,’ Positions des thèses de l’école des chartes (1942). ‘Études de comput I: notes sur le manuscrit latin 7418 de la Bibliothèque Nationale,’ Bibliothèque de l’école des chartes 103 (1942), 61–65. ‘Études de comput II: un texte espagnol de comput du VIIe (?) siècle,’ Bibliothèque de l’école des chartes 103 (1942), 65–68. ‘Les traités du comput du haut moyen âge (526–1003),’ Archivum latinitatis medii aevi 17 (1943), 51–72. ‘Une encyclopédie carolingienne de comput: les  Sententiae in laude compoti,’ Bibliothèque de l’école des chartes 104 (1943), 237–43. ‘Notes sur un auteur peu connu: Gerland de Besançon (avant 1100–après 1148),’ Revue du moyen âge latin 1 (1945), 411–19. ‘Les computistes insulaires et les écrits pseudo-Alexandrins,’ Bibliothèque de l’école des chartes 106 (1945–46), 5–34. This list was compiled by Philipp Nothaft and myself.

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‘Le comput de Gerland de Besançon,’ Revue du moyen âge latin 2 (1946), 309–13. ‘A propos de premier chapitre de De temporum ratione de Bède,’ Le moyen âge 54 (1948), 209–23. ‘Abbon de Fleury, Hériger de Lobbes et Gerland de Besançon sur l’ère de l’incarnation de Denys le Petit,’ Revue d’histoire ecclésiastique 44 (1949), 463–87. ‘La Logica de Gerland de Besançon,’ Revue du moyen âge latin 5 (1949), 43–47. ‘Un manuscrits de comput et d’astronomie des XIIe-XIVe siècles: le ms. 467 de l’Université de Glasgow,’ Scriptorium 3 (1949), 69–79. ‘A propos du manuscrit 467 de l’Université de Glasgow,’ Scriptorium 4 (1950), 115. ‘Les manuscrits de comput écclésiastique conservés dans les bibliothèques d’Aragon,’ Universidad Zaragoza 27 (1950), 592–616. ‘Los manuscritos de cómputo eclesiástico en las bibliotecas de Barcelona,’ Analecta sacra tarraconensia 23 (1950), 1–28. ‘Inventaire des manuscrits de comput ecclésiastique conservés dans les bibliothèques de Catalogne,’ Hispania sacra 4 (1951), 359–84. ‘Los textos y figuras de cómputo en los códices Emilianense y Vigiliano y el Tratado del cómputo de Rodríguez Campomanes,’ Revista bibliográfica y documental 5 (1951), 117–52. ‘Manuscrits de comput ecclésiastique de l’Escorial,’ La Ciuidad de Dios 163 (1951), 277–317. ‘Un manuscrits de comput ecclésiastique mal connu de la Bibliothèque Nationale de Madrid,’ Revista de archivos, bibliotecas y museos 57 (1951), 5–35. ‘Les manuscrits de comput ecclésiastique de la Bibliothèque Capitulaire de Tolède,’ Revista de archivos, bibliotecas y museos 58 (1952), 323–52. ‘Inventaire des manuscrits de comput ecclésiastique conservés dans les bibliothèques de Catalogne,’ Hispania sacra 5 (1952), 121–64. ‘La connaissance du comput ecclésiastique au moyen âge dans les abbayes de l’ancienne province de Normandie du VIIIe au XIIIe siècle,’ Bulletin philologique et historique du Comité des travaux historiques et scientifiques (1953–54), 359–76. ‘Manuscrits bernardins des bibliothèques des anciens monastères d’Utrecht,’ in Mélanges saint Bernard: XXIVe Congrès de l’Association bourguignonne des Sociétés savantes, Dijon, 1953 (Dijon, 1954), 399–407. ‘Les textes et figures de comput de l’Antiphonaire de León,’ Archivos Leonenses 8 (1954), 258–87. ‘Inventaire des manuscrits de comput ecclésiastique conservés dans les bibliothèques de Madrid,’ Hispania sacra 7 (1954), 111–43. ‘Les manuscrits de comput ecclésiastique des bibliothèques du Levant,’ Anales Universidad de Valencia 28 (1954–55), 2–22. ‘Les manuscrits de comput ecclésiastique de l’Abbaye de Saint Gall du VIIIe au XIIe siècle,’ Zeitschrift für schweizerische Kirchengeschichte 49 (1955), 161–200.

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‘L’évolution du comput ecclésiastique à Saint Gall du VIIe au XIe siècle,’ Zeitschrift für schweizerische Kirchengeschichte 49 (1955), 288–323. ‘Un autre manuscrit de comput ecclésiastique mal connu de la Bibliothèque Nationale de Madrid,’ Revista de archivos, bibliotecas y museos 61 (1955), 435–86. ‘Les manuscrits de comput ecclésiastique des bibliothèques de Madrid,’ Hispania sacra 8 (1955), 177–208. ‘Textes de comput espagnol du VIe siècle: encore le problème des traités de comput de Martín de Braga,’ Revista de archivos, bibliotecas y museos 62 (1956), 685–97. ‘Textos de cómputo espagñol del siglo VI: el Prologus Cyrilli,’ Hispania sacra 9 (1956), 127–39. ‘Les plus anciens manuscrits de comput ecclésiastique de la bibliothèque de Berne,’ Zeitschrift für schweizerische Kirchengeschichte 51 (1957), 101–12. ‘Un manuscrit de comput ecclésiastique du fonds de Jumièges,’ in Jumièges Congrès scientifique du XIIIe centenaire, Rouen, 10–12 juin 1954, 2 vols (Rouen, 1955), ii 691–702. ‘Les manuscrits de la bibliothèque de Berne provenant de l’abbaye de Fleury au XIe siècle le comput d’Abbon,’ Zeitschrift für schweizerische Kirchengeschichte 52 (1958), 135–50. ‘Un manuscrit de comput intéressant: Schaffhouse, Ministerialbibliothek, 61,’ Scriptorium 12 (1958), 247–53. ‘Textes de comput espangnol du VIIe siècle: le Computus Cottonianus,’ Hispania sacra 11 (1958), 125–36. ‘Le comput de Dicuil,’ Cahiers de civilisation médiévale 3 (1960), 325–37. ‘Contribution à la littérature du comput ecclésiastique au moyen âge,’ Studi medievali, Ser. 3, 1 (1960), 107–37. ‘Contribution à la littérature du comput ecclésiastique au moyen âge,’ Studi medievali, Ser. 3, 2 (1961), 169–208. ‘La table paschale de Périgueux,’ Cahiers de civilisation médiévale 4 (1961), 56–60. ‘Comput, chronologie, calendriers,’ in C.  Samaran, L’histoire et ses méthodes (Paris, 1961), 37–51. ‘Les manuscrits de comput des bibliothèque d’Utrecht,’ Scriptorium 15 (1961), 76–85. ‘Le computiste Hermann de Reichenau,’ Miscellanea storica ligure 3  (1963), 165–90. ‘L’activité computistique de Robert, évêque de Hereford,’ in P. Gallais and Y.J. Riou, Mélanges offerts à Réné Crozet, 2 vols (Poitiers, 1966), i 333–40. ‘Le comput ecclésiastique de l’abbaye du Mont-Cassin au Xe siècle,’ Anuario de estudios medievales 3 (1966), 65–89. ‘Les manuscrits de comput de l’abbaye du Mont-Saint-Michel,’ Sacris Erudiri 17 (1966), 55–65.

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ALDEN A. MOSSHAMMER

TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

Abstract The Pseudo-Cyprianic De pascha computus, composed in the year AD 243, is the earliest extant paschal computus (that attributed to Hippolytus being a list of dates, rather than a computistical text). This paper discusses the manuscripts and editions, summarizes the contents, and finally focuses on the reconstruction of the 16-year cycle and 112-year paschal period appended to the now lost Codex Remensis. Keywords Pseudo-Cyprian; Computus of AD  243; 112-year paschal cycle; Hippolytan cycle.

Introduction The Computus of AD 243 is the earliest extant Christian computistical text.1 It first came to the attention of scholars in the 1640s. While resident in London during the last fifteen years of his life, James Ussher discovered among the manuscripts in Robert Cotton’s library an anonymous text under the title Expositio Bissexti. Ussher mentions the work in his Chronologia Sacra, which was published posthumously from his manuscripts by Thomas Barlow in 1660. Ussher remarks that, according to a formula at the end, the text was written in the fifth year of Gordian,

1 The 112-year cycle preserved in an inscription at Rome and attributed to Hippolytus dates from AD 222, but this is a list of dates, not a computistical text. On that list, see Mosshammmer (2008), 116–25.

Late Antique Calendrical Thought and its Reception in the Early Middle Ages, ed.  by Immo Warntjes and Dáibhí Ó Cróinín, Turnhout, Brepols, 2017 (Studia Traditionis Theologiae, 26), pp. 43-70 © BREPOLSHPUBLISHERSDOI 10.1484/M.STT-EB.5.114733

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the consulship of Arianus and Papus, which is the year 243 of the Christian era.2 About the same time, the Jesuit scholar Jacques Sirmond discovered a similar text in a manuscript at Reims. That manuscript is now lost, probably destroyed in the fire of 1774.3 Sirmond sent a copy to Gilles Bouchier (alias Aegidius Bucherius) in Paris in August of 1648. According to Bouchier, the copy he received carried the name of Cyprian. He thought the attribution genuine and reported that the astronomer Gott­ fried Wendelin agreed.4

The editions John Fell, dean of Christ Church and bishop of Oxford, decided to include the tract in an appendix to his edition of the works of Cyprian, published at Oxford in 1682. In his preface, Fell defended Cyprianic authorship and identified the work with the Chronicam ualde utilem mentioned in an anonymous Life of Cyprian falsely attributed to Paul the Deacon, who was a contemporary of Charlemagne.5 Even if this ‘very useful chronicle’ is a reference to the Computus of AD 243, we would know only as much as can be inferred from the Reims manuscript—that the name of Cyprian had been attached to this work by the end of the ninth century. Fell entrusted the editorial work to the Oxford mathematician John Wallis. Wallis worked from Ussher’s copy of the Cotton manuscript, which he sent to Thomas Gale for collation with the original. Gale returned the text with some notes of his own. There are nevertheless a number of errors in the readings that Wallis attributes, either explicitly or by silence, to the manuscript.6 James Ussher in Barlow (1660), 29: In Paschali tractatu, quem ineditum habeo, anno quinto Gordiani, Ariano et Papo consulibus, id est, anno aerae Christianae 243. The form Ariano is based on the reading of the Cotton manuscript. The correct spelling is Arrianus, as in the Reims manuscript. 3 Mattei (2001), 35. 4 Bouchier (1655), 202 col. 1 (book 6, chapter 9, paragraph 1). I thank Leofranc Holford-Strevens for consulting a copy of Bouchier’s report for me in Oxford. 5 Fell (1700), 3. 6 For example, in Fell (1700), 211 (line 8 in the first full paragraph = p. 252.2 in Hartel’s CSEL edition, below n. 10), Wallis reads iniunxisse, with no note, whereas the reading of the Cotton manuscript is inluncxisse and the Vatican copy of the Reims manuscript (discussed below) has illuxisse. 2

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From Jean Mabillon Wallis obtained a copy of the text from the Reims manuscript. Wallis found that the text was the same as that contained in the Cotton manuscript, but in a fuller and more reliable form. The Reims text also included a set of tables illustrating the 16-year cycle and 112-year period that the author describes.7 In a discussion of the life and works of Cyprian published in 1797, Gottfried Lumper argued on stylistic grounds that the work could not possibly have come from Cyprian’s pen.8 Lumper also pointed out, as had others before him, that it is unlikely that Cyprian was a Christian as early as AD 243. Most scholars since that time have agreed, and the work is therefore often referred to as the Pseudo-Cyprianic De pascha computus. Wallis’s edition was reprinted in the Patrologia Latina in 1844.9 There are numerous typographical errors in that edition, and the editors also disturbed the format and arrangement of the appended tables. Wallis’s notes appear erroneously and inexplicably under the title Notae Henrici Dodwelli. Wilhelm von Hartel included the treatise among the dubia in the third volume of his edition of the works of Cyprian published in the Corpus Scriptorum Ecclesiasticorum Latinorum in 1871.10 Hartel was unable to find the Reims manuscript or any new witnesses to the text. He therefore asked his readers to be content with a redaction of Wallis’s edition, in which Hartel adopted a few readings rejected by Wallis.11 Unfortunately, Hartel seems to have used the edition from the PL reprint of 1844, instead of from the Oxford original or the Paris reprint. This conclusion follows from such erroneous readings as sexta decennitas feria quarta at p. 255, lines 14–15, in agreement with PL 4, 951, where Wallis, following both the Reims text and the logic of the description, reads sexta decennitas feria tertia. Hartel also reprinted and further disturbed the arrangement of the tables as they appear in the Patrologia Latina. Indeed, Hartel’s version of these tables is all but unintelligible.

This and the previous paragraph are derived from Wallis’s Monitio ad lectorem in Fell (1700), 208. 8 Lumper (1795), 375, repr. in PL 4, 816. 9 PL 4, 937–74. 10 CSEL 3,3, 248–71. 11 Wilhelm von Hartel in CSEL 3,3, lxii, lxv. 7

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The manuscripts Vatican, Biblioteca Apostolica, Reg. lat. 324 Of the two witnesses, the better text is that included in the now lost manuscript discovered at Reims. As remarked above, Wallis used a copy made for him by Mabillon. In 1896, Johann Ernst brought to the attention of scholars the existence of another seventeenth-century copy in Vatican, Biblioteca Apostolica, Reg. lat. 324.12 The manuscript consists of 25 leaves in two distinct parts.13 The first part includes the same Pseudo-Cyprianic works listed in a description of the Reims manuscript extant in Paris, Bibliothèque nationale de France, Lat. 11777, 244v, followed by the spurious Letter of Cornelius to Cyprian, apparently copied from a different (now lost) manuscript at Reims.14 The titles of De rebaptismate and De resurrectione are in the same hand as the text. The titles of De pascha and the Epistula Cornelii are in a different, but approximately contemporaneous hand. At the bottom of folio 20v, from the hand of the second writer, is a letter of Pope Alexander III to William of Sens, concerning the feast of St Thomas of Canterbury. This text too came from that second Codex Remensis. The last five pages of the manuscript are from a different hand entirely and on a different type of paper. On folios 21–25 is a copy, in French, of the Peace of Vervins between Henry III of France and Philipp II of Spain dated 2 May 1598. There are numerous corrections entered into the text of the three Pseudo-Cyprianic treatises.15 Some of these corrections seem to come immediately from the pen of the original copyist, making good his own errors. Other corrections come from the hand of the same writer who added the titles of De pascha and the Epistula Cornelii. They appear sometimes in the margin, sometimes supra lineam. That writer also supplies some omissions in the text and was therefore working by collation with the manuscript. The identity of these writers cannot be determined. They may have been the same person, possibly Sirmond or Mabillon, working at different times. The Vatican copy of De pascha is different from what Mabillon sent to Wallis, as several variant readings attest.16 14 15 16 12 13

Ernst (1896); Ernst (1898). Wilmart (1937–45), ii 224–25; see also Petitmengin (1974), 28–29. Petitmengin (1975), 126, 131. Mattei (2001), 35. For examples, see below on the text of the appended tables.

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London, British Library, Cotton Caligula A XV The only other direct witness to the text is London, British Library, Cotton Caligula A XV, 97v–105v. The manuscript consists of 153 leaves containing a miscellany of material, ranging in date from the eighth to the eleventh century.17 Folios 73r–117v contain a late eighth-century copy of a collection of computistical texts, including (fols 73r–80r) the so-called Cotton Computus of AD 689.18 On folio 107r there is a calendar giving the dates of new and full moons for every month of a year designated as AD 743 from the Incarnation, an eleventh indictional year, the first year of King Childeric, with Easter on 14 April, moon 15.19 This collection of material was therefore probably made somewhere in Gaul, during the reign of Childeric III (AD 743–51). The text in question begins in the middle of folio 97v, with the title Incipit Expositio Bissexti. That title is entirely inappropriate to the text. While the author mentions bissextile days and years, there is no discussion of the leap-year intercalation. The title perhaps originally belonged to the following text in the collection, which occupies folios  106r–v. That text is untitled, but begins certum est quod byssexti diem et aetatem lunae deputandam est in luna Feb.20 The Expositio Bissexti of Cotton Caligula A XV is a heavily redacted version of the text to which the lost Reims manuscript attests.21 The author of the original text argues (251.11) that the Sunday of Creation was 25 March and that the earliest permissible date for the paschal full moon is 17  March (252.19–22). The redactor of the Cotton copy changes both of those dates to 22 March. In the Reims copy, the author criticizes (251.11–15) certain predecessors for reckoning the first day of the new month between 15 March and 13 April. By that phrase, the computist actually means the full moon of the first month. The redactor substitutes the standard Alexandrian dates for the paschal new moon, as ranging from 8 March to 5 April.22 Where the Reims text refers (253.18–23) to 17 For descriptions, see Planta (1802), 45–46; CLA 2, 19 (no. 183); Gómez Pallarès (1986), 24–32. 18 See Gómez Pallarès (1986) and Warntjes (2011). 19 On the date, see Palmer (2011), 218–19, and Warntjes (2011), 174–76. 20 For a transcription, see Jones (1943), 372. 21 In what follows, citations are by page and line from Wilhelm von Hartel’s CSEL edition. 22 For these Alexandrian limits, see Dionysius Exiguus, Epistola ad Petronium (ed. by Krusch (1938), 65).

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the creation of the sun and the moon or its anniversary on 28 March, the redactor either substitutes the more standard 25 March or omits the datum.23 The Reims text says (256.18–20) that Jesus shared a paschal meal with his disciples on 8 April and suffered on 9 April. The redactor substitutes the traditional Roman dates of 24 and 25  March.24 Friday 9 April corresponds to AD 28 as the year of the Passion, which is consistent with the interval of 215 years from the Passion to the consulship of Arrianus and Papus in the dating formula at the end of the Cotton manuscript (216.18–20).25 The Reims manuscript changed the interval to 220 and moved the present year to a position in the 16-year cycle that corresponds to AD  248, instead of 243 (268.25). This is the only instance where the Cotton manuscript preserves a significantly better text. Sometimes the redactor omits whole sections of the text in order to suppress arguments with which he disagrees. Thus, in Chapter  4, where the author explains how the predecessors erroneously arrived at 15 March, the redactor omits the passage (251.12–13). In Chapter 7, the redactor omits a section (254.7–9,12–20) explaining how the Hebrews added an embolismic month to avoid a full moon earlier than 17 March. At the end of Chapter 18 and the beginning of 19, the redactor omits the computist’s unusual date for the Nativity on 28 March (266.8–11). In two places, where the redactor omits what he regards as an unorthodox passage, he interpolates into the text phrases borrowed from the last paragraph of the Disputatio Morini Alexandrini episcopi de ratione

For other texts with 22 March as the date of creation and 25 March for the sun and the moon, see the note on Munich Computus 44, ll. 18–36 (ed. by Warntjes (2010a), 146). 24 For 25  March 29 as the date of the Passion see Nothaft (2012), 38–56, who argues (p. 50) that the date in Tertullian, Aduersus Iudaeos 8.18 (ed. by Emil Kroyman in CCSL 2, 1363) is a later interpolation and (p.  51) that the tradition derives from Hippolytus. The date is attested also in the Chronograph of AD 354, ed. by Theodor Mommsen in MGH Auct. ant. 9, 57. For a convenient list of other texts attesting to this tradition see Warntjes (2010a), 151 (note on Munich Computus 44, ll. 55–80). 25 The traditional date for the Passion was the consulship of the two Gemini (C. Fufius and L. Rubellius), in the year AD 29. In some versions of the Roman consular list, that year corresponds to AD 28, instead of AD 29. The best-known example is the Cursus paschalis of Victorius, ed. by Krusch (1938), 16–52, which begins (p. 26) in the year of the Passion in the consulship of the two Gemini, with 1 January on a Thursday and 28 March on a Sunday—the calendrical data for the year AD 28. He finished his work, as he says in the prefatory letter (Victorius, Prologus 7 (ed. by Krusch (1938), 23)) in the consulship of Constantinus and Rufus, AD 457, which he numbers (p. 48) as the year 430 from the Passion. The reasons for the choice of the year 28 in the Computus of AD 243 are different. See below, near n. 50. 23

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TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

paschali, often referred to as Pseudo-Morinus.26 That last paragraph is not included in all of the manuscripts, but it does appear in the copy of Pseudo-Morinus preserved in the Cotton collection (fol. 83v). The relevant text is as follows: Post haec enim omnia breuiter dicam quod numquam factus est pascha apud Iudaeos ante XII K. Apr., quae luna nata in VIII id. Marti, XIIII est in XI [sic!] kal. Aprelium. Obserua igitur cursum lunarem iuxta regulam I Grecorum more Aegyptiorum et non secundum aepactas id est adiectiones lunares. ‘After this, I will briefly explain that Passover was never done among the Jews before 12 Kalends April (21 March), because the moon that is new on 8 Ides March (8 March) is the 14th on 11 Kalends April (22 March).27 Therefore observe the course of the moon according to the first rule of the Greeks in the manner of the Egyptians, and not according to the epacts, which are lunar additions.’

At the beginning of Chapter 6 in the Cottonian version of the Computus of AD 243, where the Reims text had Propter hoc ergo (252.13), the redactor substitutes Post haec enim omnia breuiter dicam quod. The redactor then omitted the section where the computist explains how the Hebrews arrived at 17 March as the date of the full moon in the first year after the Creation, and wrote instead: numquam fecerunt Pascha ante XII kal. Apr. quae luna nata in VIII id. Mart. XIIII est in XI kal. Apr. (‘They never did Passover before 12 Kalends April, 21 March, because the moon that is new on 8 Ides March, 8 March, is the 14th on 11 Kalends April, 22 March’). At the beginning of Chapter 7, where the computist says (253.25) that the Hebrews were divinely instructed circa cursum lunae, the redactor wrote: circa cursum lunarem iuxta regulam primam Graecorum more Aegyptiorum et non secundum aepactas (‘about the lunar course according to the first rule of the Greeks in the manner of the Egyptians and not according to the epacts’). When Wallis sent his copy of the Cottonian text to Gale for collation with the manuscript, Gale drew his attention to the parallel with Pseudo-Morinus in Chapter 6. Wallis included that report in his notes.28 See Graff (2011), 125–33. Correctly, 12 Kalends April, 21 March. For reconstruction of the text based on the witness of several manuscripts, see Graff (2011), 142. 28 Fell (1700), 211 n. 1. 26 27

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ALDEN A. MOSSHAMMER

Either Gale did not note or Wallis did not report the parallel in Chapter 7. Wallis included the phrase iuxta regulam primam Graecorum more Aegyptiorum et non secundum epactas in the text. The inclusion of this phrase in Wallis’s text, as in the Patrologia Latina reprint and in the edition of Hartel, has misled several scholars to state that the Computist of AD 243 rejected calculation by the epacts.29 If this phrase were genuine, it would be the earliest known reference to epacts of the moon in any Greek or Latin text.30 The Computist of AD 243 uses what looks like a system of epacts in his description of the 8-year cycle (253.18–23). He cannot therefore have rejected such a method. The redactor of the Cottonian version of the Computus of AD 243 may have belonged to the same seventh-century context from which what few references we have to the Disputatio Morini derive.31 It is unlikely that the person who collected these texts undertook this redaction himself. Origins The original author of this tract must remain unknown. The consular date in the year of Arrianus and Papus (AD 243) is secure and, as Lumper argued, excludes Cyprian as author, who is unlikely in any case to have written so barbarous a text.32 Since the year AD 243 does not correspond to a first year of the author’s cycle, it is likely to have been the date of composition.33 The author (216.18–20) calculated an interval of 215 years from that date to the Passion, which yields AD 28 as the year of the Passion. That agrees with the author’s calendar date on Friday 9 April (256.20). Jerome attributed a work De pascha to the Roman schismatic Novatian.34 Adolf von Harnack raised the possibility that the Computus of Schwartz (1905), 39; Fotheringham (1922), 53; Mosshammer (2008), 126. The Greek word is first attested in the later fourth century in Epiphanius’ description of the 8-year cycle (Panarion, ed. by Holl (1915–33), iii 246) and in Theon’s Lesser Commentary on the Handy Tables (ed.  by Tihon (1978), 256), where Theon explains how to calculate the epacts for any year of a 19-year cycle. The Latin word is not attested before Dionysius Exiguus, except in a fragmentary letter of Cyril to Leo (PL 54, 603), which is of doubtful authenticity and uncertain date. 31 Graff (2010), 125–33, and Warntjes (2011), 183, who suggests that this collection of texts comes from the circle of Willibrord. 32 See above, n. 8. 33 For the author’s cycle, see the discussion below under the heading Pinax. 34 Jerome, De viris illustribus 70 (ed. by Bernoulli (1895), 41). 29 30

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TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

AD 243 might be that work.35 August Strobel accepted that identification in a book published in 1977, but in the commentary on a translation of the text that he published in 1984 stated only that there are influences of Novatianist and Montanist theology in the tract.36 A work De pascha need not have been a computus. Novatian was a far better writer than the computist, and whether his literary activity can be dated to as early as AD 243 is doubtful.37 An African provenance for the work is suggested by the similarities between the Old Latin text of the Hebrew Bible used by the author and that used by Cyprian, in the few instances where comparison is possible.38 Apart from the manuscripts themselves, the only witnesses to the Computus of AD  243 are African. The Carthaginian Computist of AD 455 quotes from a brief portion of the text.39 Quintus Julius Hilarianus, author of a treatise De die paschae, subscribed on 5 March AD 397, seems to have known the Computus of AD  243. Hilarianus says that some people call the full moon the first day of the month, and this must be a reference to the peculiar usage of the Computist of AD 243.40 Contents The author advocates the use of a 16-year luni-solar cycle with a 112-year paschal period.41 Eusebius attributes to an author named Hippolytus a work De pascha with a ‘Canon’ of dates for a 16-year cycle ‘to the first year of Alexander Severus’, AD 222.42 Among the other works that Eusebius attributes to this Hippolytus are commentaries on the six days of creation and a book Against All Heresies. Whether this Hippolytus is the same person whom Eusebius had just mentioned as a prolific writer von Harnack (1893–1904), i 653. Strobel (1977), 174; Strobel (1984), 65–67. 37 On Novatian, see Papandrea (2008). 38 Monceaux (1901–23), i 121–22; ii 98. 39 Computus Carthaginiensis 1.6; 2.4–5 (ed. by Krusch (1880), 282–83, 287). 40 Quintus Julius Hilarianus, Expositum de die paschae et mensis 2 (ed. by Christopher Pfaff (Paris 1712), repr. in PL 13, 1107): alii in orbem plenam, qualis Dei jussu mundo orta fuerat, primam appellabant. 41 For discussion, see also de Rossi (1857), lxxxi–lxxxii; Mac Carthy (1901), xl– xliv; Schwartz (1905), 36–40; Ogg (1955). Krusch (1880), 189–92, provides a reconstruction of the 112-year list as it must have appeared in the Codex Remensis. He mentions the work in his narrative (esp. p. 20 n. 5), but there is no continuous discussion. 42 Eusebius, Historia ecclesiastica 6.22 (ed. by Schwartz and Mommsen (1903–09), ii 568–69). 35 36

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ALDEN A. MOSSHAMMER

and the bishop of an unknown, but apparently eastern city, or the same that the Chronograph of AD 354 says was exiled from Rome to Sardinia in AD 235, is a much-debated problem.43 Eusebius does not mention a 112-year list in connection with the 16year cycle of Hippolytus. About 1550, excavations at Rome unearthed just such a list. The headless statue of a figure seated on a chair was found near the church of San Lorenzo fuori le Mura. On one side of the chair there is inscribed in Greek characters a 16-year list of dates in the Roman calendar, with a letter from α to ζ indicating the weekday from 1 to 7 on which that date occurs. There follow six more columns with only the letters from α to ζ indicating the weekday for the full moon in each of six iterations, for a total of 112 years. Within that matrix there are short notes showing to what year in the 112-year period the several observations of Pascha mentioned in the Bible correspond. On the other side of the chair is a 112-year list of Easter Sundays in seven parallel columns of 16 years each. At the top of the first list is an inscription stating that the table begins in the first year of Alexander Severus, which was an embolismic year with the 14th day of the moon on Saturday, the Ides of April (13  April AD  222). The list of dates for Moon 14 repeats after eight years, so that this system is based on doubling an 8-year cycle.44 The earliest date for Moon 14 is 18 March, and the latest is 13 April.45 By inspection of the 112-year list of Sundays, one can see that what we know from later sources as the distinctively Roman rules are already in effect. Easter can be observed no earlier than the 16th day of the moon, no later than the 22nd day of the moon, with an outer limit of 21 April.46 43 Eusebius, Historia ecclesiastica 6.20 (Schwartz and Mommsen (1903–09), ii 566–67); Chronograph of 354 (MGH Auct. ant. 9, 72–75). For a brief summary of this ‘Hippolytan’ problem, see Mosshammer (2008), 118–21, and Burgess and Kulikowski (2013) 366–71, with full references to the earlier literature. The most comprehensive discussion is that of Brent (1995). 44 An 8-year cycle is based on the correspondence between eight Julian years (365.25×8=1922 days) and 99 lunar months, with 96 ordinary months alternating between 29 and 30 days (29.5×96=2832 days) and seven embolismic months of 30 days each (3×30=90). The cycle conveniently returns the same calendar date for the 14th moon every eight years, but with an error of about 1 ½ days with respect to the observable moon, three days after a 16-year cycle. 45 For further discussion, see de Rossi (1857), lxxix–lxxx; Mac Carthy (1901), xxxii–xl; Schwartz (1905), 29–35; Mosshammer (2008), 122–25. On the statue, see Guarducci (1977). Photographs of the inscription are among the plates in Brent (1995), at the front of the volume. The text is ed. by Scaliger (1595), 1–3; Fabricius (1716–18), i 37–41 (repr. in PG 10, 875–84). 46 For these ‘Latin’ rules, see Victorius, Prologus 4–5 (Krusch (1938), 19–21).

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TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

The name of Hippolytus does not appear on this statue, nor is the list of writings on the back of the chair the same as the list attributed to him by Eusebius. The 16-year cycle may or may not be the same list that Eusebius attributes to Hippolytus. Unless there was a very early corruption in the text, Eusebius said that the list of Hippolytus ended in the first year of Alexander, while that on the statue began in that year.47 Neither the Roman exemplar nor the Computus of AD 243 is likely to have been composed entirely independently of the 16-year cycle that Eusebius attributes to Hippolytus. The author’s cycle is not, however, simply a continuation of the Roman list that began in AD 222, nor of the Hippolytan list that either began or ended in that year. While the inscription at Rome began with a paschal full moon on 13  April, which was correct for the year AD  222, the Computus of AD 243 begins twenty years later with 1 April, correct for the year AD 242. It is thus a recalibrated version of the Hippolytan list, which in the Roman exemplar would produce 29 March in the year AD 242. It also has different lunar limits. The Roman list restricts the 14th day of the moon to the period between 18 March and 13 April, while the Computist of AD 243 uses the limits of 17 March and 12 April. The contents of the tract can be divided into three parts. In the first (250.1–253.23), the author argues from the facts of Creation that what he calls the first day of the new month must be 17 March. The Sunday of Creation was 25  March. The moon was created on the fourth day of the week. It was a full moon rising in its 15th day in the evening of 28 March. Therefore, in the following year, which is the first year when there was actually a 14th day of the moon in the first month, that moon corresponded to the daytime hours of 17 March. The author does not explain why that date should also be the earliest date for Passover. The reason presumably is that the first year of Creation cannot have been 47 Most scholars (see n. 43) have assumed that the list of Hippolytus began in the same year as the list on the statue. Recently, however, Burgess and Kulikowski (2013), 367–68, have challenged that consensus, pointing out that Eusebius says, in Historia ecclesiastica 6.22 (Schwartz and Mommsen (1903–09), ii 568), ἐπὶ τὸ πρῶτον ἔτος αὐτοκράτορος Ἀλεξάνδρου τοὺς χρόνους, which means ‘to the first year’, not ‘from the first year’. Burgess and Kulikowski do not refer to the ancient translations. Rufinus (Schwartz and Mommsen (1903–09), ii 569) translated intra primum Alexandri imperatoris annum concludit. The Armenian version (German translation by Preuschen (1902), 29) has, ‘setze er auch die Zeitrechnung auseinander und den Kanon von Zehn (sic!) Jahren und (zwar) mit den ersten zehn (sic!) Jahren des Kaisers Alexandros beendete er die ganze Berechnung’. Book 6 is missing in a lacuna in the Syriac text (ed. by Nestle (1901)). Perhaps the text originally read (or Eusebius meant) ἀπὸ τοῦ πρώτου ἔτους. Burgess and Kulikowski are not the first to notice this discrepancy. See Brent (1995), 308–09.

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ALDEN A. MOSSHAMMER

embolismic. Therefore, the 14th day of the first month in the following year must be 17 March, not 16 April. The author also claims that the 11 ¼-day differential between the lunar and solar year arises from the 11 ¼ hours during which the sun preceded the moon on that first day of their creation. The difference is 11 ¼ hours, not 12. The moon rose at the beginning of the night, he says (253.10–15), but not after sunset, because it had to appear on the same ‘day’ (i.e., during daylight). In the second part of the text (253.24–258.2), the author describes a 16-year cycle, with the year of the Exodus as its base-date, with a Passover moon on Monday, 12 April. Intervals are counted from the Exodus, which was in effect the year 0. The cycle itself begins the next year with moon 14 on 1 April. The author says that the Hebrews discovered the 8-year cycle and then doubled it because of the sequence of weekdays. He refers to the fact that after a 16-year interval, any given date in the Julian calendar will recede by one weekday. After seven repetitions, 112 years, 12 April returns to Monday. Unfortunately, because of the error in a 16year cycle, 12 April will not be moon 14, but moon 22.48 The third and longest portion of the text (258.3–269.5) offers a chronological summary of the years from the Exodus to the Passion. The author calculates the date of every Passover observance mentioned in the Bible. That which took place in the 18th year of Josiah (2 Kings 23:23), for example, was 970 years after the Exodus, on 12 Kalends April (21 March), weekday #2 (Monday).49 He enjoys finding typological associations between the week of Creation and events in biblical history. Thus (266.3–8), he dates the birth of Christ to 28 March and chooses the year 1548 from the Exodus, corresponding to 4 BC, because 28 March was a Wednesday in that year. To that he adds 31 years and dates the Passion to 1579, AD 28, choosing that year, rather than the traditional AD 29, because in his pinax the year 1579 has Moon 14 on a Thursday, to accommodate a Passover meal on that weekday.50 He also reflects on the mystical significance of the various numerals he generates, commenting especially (257.8–10, 268.10) on the 318 servants of Abraham (Genesis 14:14) as being represented in Greek by the numeral τιη, which consists Over the period of 112 years, an error of –22 days with respect to the observed moon will accrue, which is the equivalent of +8. 49 The author does not show the calculation, which is as follows: The interval 970 is divisible by 112 with a remainder of 74, which is divisible four times by 16 with a remainder of 10. The 10th year of a cycle always has Passover on 12 Kalends April. In the fifth column of the pinax, the 10th year has feria 2. 50 See Nothaft (2012), 52–54. 48

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TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

of tau, the symbol of the cross, plus iota-eta, the first two letters of the name Jesus (Ίησοῦς).51 The pinax and 112-year list of the Codex Remensis I am working on a new edition of the text for publication in the Corpus Christianorum Series Latina. Here I will focus on the set of tables included in the lost Reims manuscript, which implement the 16-year cycle and 112-year period that the author describes. A redactor has introduced a few changes. Figure 1 shows the 16-year cycle and Figure 2 the 112-year period as the author describes them in the text. In reconstructing what the author refers to as his pinax, I have followed Wallis’s lead and used the format in the Roman exemplar of the Hippolytan cycle, with seven parallel columns for the weekdays on which the paschal full moon occurs through the 112-year period.52 The 16-year cycle is actually a pair of 8-year cycles, and the list of Sundays repeats therefore after 56 years. The base-date is the year corresponding to the Exodus, while the 16-year cycle begins in the following year. The author says in Chapter 7 that the Hebrews first observed the Pasch on 12 April (254.2), in the next year on 1 April (254.10), and that the second year has moon 14 on 21 March (254.11). At the end of Chapter 18 (266.6) and again at the beginning of Chapter 23 (268.21), he says the year of the Exodus is the first line of the pinax (a primo pinacis uersu). We must therefore distinguish between the 17 lines of the pinax and the 16 years of the cycle. In constructing the 112-year list of Easter dates, I  have followed ­Krusch and Strobel in assuming that the author adopted the rule apparent in the Roman exemplar of the Hippolytan tables and explicit in descriptions of the later Roman 84-year cycle, that Easter Sunday may not precede the 16th day of the moon.53 My reconstruction in Figures 1 and 2 therefore does not differ from either of theirs.

51 Later the numeral became significant also as the traditional number of bishops assembled at the Council of Nicaea; so, for example, Dionysius Exiguus in the Epistola ad Petronium (ed. by Krusch (1938), 20). The earliest (c.AD 359) reference to the ‘318 fathers’ is from Hilary of Poitiers, De Synodis 86 (PL 10, 538), who also connects the number with the story of Abraham. See Aubineau (1966). 52 Fell (1700), 213. 53 Krusch (1880), 189–91; Strobel (1984), 57–59; for the rule, see Victorius of Aquitaine, Prologus 4 (Krusch (1938), 19).



Age of Moon on 28 March

30

11

22

4

15

26

7

19

30

Year

Exodus

I

II

III

IIII

V



VI

VII

VIII

Embolismic

Common Bissextile

Embolismic

Common

Common

Embolismic Bissextile

Common

Common

Type

Prid Id Apr 12 April

IX Kl Apr 24 March

Non Apr 5 April

XVI Kl Apr 17 March

V Kl Apr 28 March

VI Id Apr 8 April

XII Kl Apr 21 March

Kl Apr 1 April

Prid Id Apr 12 April

Passover (Luna XIV)

5

6

2

3

6

2

3

6

2

Weekday I

4

5

1

2

5

1

2

5

1

II

3

4

7

1

4

7

1

4

7

III

2

3

6

7

3

6

7

3

6

IIII

1

2

5

6

2

5

6

2

5

V

7

1

4

5

1

4

5

1

4

VI

6

7

3

4

7

3

4

7

3

VII

ALDEN A. MOSSHAMMER



11

22

4

15

26

7

19

30

VIIII

X

XI

XII

XIII

XIV

XV

XVI

Embolismic

Common Bissextile

Embolismic

Common

Common

Embolismic Bissextile

Common

Common

Type

Prid Id Apr 12 April

IX Kl Apr 24 March

Non Apr 5 April

XVI Kl Apr 17 March

V Kl Apr 28 March

VI Id Apr 8 April

XII Kl Apr 21 March

Kl Apr 1 April

Passover (Luna XIV)

1

2

5

6

2

5

6

2

Weekday I

Figure 1 The pinax and 16-year cycle as reconstructed from the author’s description.

Age of Moon on 28 March

Year

7

1

4

5

1

4

5

1

II

6

7

3

4

7

3

4

7

III

5

6

2

3

6

2

3

6

IIII

4

5

1

2

5

1

2

5

V

3

4

7

1

4

7

1

4

VI

2

3

6

7

3

6

7

3

VII

TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

23. VI Kl Apr 27 March

24. XVI Kl Mai 16 April

2. VII Kl Apr 26 March

3. XVIII Kl Mai 14 April

4. III Kl Apr 30 March

5. XI Kl Apr 22 March

6. III Id Apr 11 April

7. VII Kl Apr 26 March

XII Kl Apr 21 March

VI Id Apr 8 April

V Kl Apr 28 March

XVI Kl Apr 17 March



Non Apr 5 April

IX Kl Apr 24 March

Prid Id Apr 8. XVII Kl Mai 12 April 15 April

22. Prid Id Apr 12 April

21. X Kl Apr 23 March

40. XV Kl Mai 17 April

39. V Kl Apr 28 March

38. Id Apr 13 April

37. IX Kl Apr 24 March

36. Kl Apr 1 April

35. XVI Kl Mai 16 April

19. XVII Kl Mai 15 April

20. Prid Kl Apr 31 March

34. V Kl Apr 28 March

33. Non Apr 5 April

Cycle III

18. VI Kl Apr 27 March

17. Prid Non Apr 4 April

1. III Non Apr 3 April

Kl Apr 1 April

Cycle II

Easter Sunday, Cycle I

Luna XIV

56. XIV Kl Mai 18 April

55. IV Kl Apr 29 March

54. VII Id Apr 7 April

53. VIII Kl Apr 25 March

52. IV Non Apr 2 April

51. IV Id Apr 10 April

50. IV Kl Apr 29 March

49. VIII Id Apr 6 April

Cycle IIII

72. XIII Kl Mai 19 April

71. III Kl Apr 30 March

70. VI Id Apr 8 April

69. XIV Kl Apr 19 March

68. III Non Apr 3 April

67. III Id Apr 11 April

66. X Kl Apr 23 March

65. VII Id Apr 7 April

Cycle V

88. XII Kl Mai 20 April

87. Prid Kl Apr 31 March

86. V Id Apr 9 April

85. XIII Kl Apr 20 March

84. Prid Non Apr 4 April

83. Prid Id Apr 12 April

82. IX Kl Apr 24 March

81. VI Id Apr 8 April

Cycle VI

104. XVIII Kl Mai 14 April

103. Kl Apr 1 April

102. IV Id Apr 10 April

101. XII Kl Apr 21 March

100. Non Apr 5 April

99. Id Apr 13 April

98. VII Kl Apr 25 March

97. V Id Apr 9 April

Cycle VII

ALDEN A. MOSSHAMMER



9. VII Id Apr 7 April

10. X Kl Apr 23 March

11. III Id Apr 11 April

12. III Non Apr 3 April

13. XIV Kl Apr 19 March

14. VI Id Apr 8 April

15. III Kl Apr 30 March

16. XIII Kl Mai 19 April

Kl Apr 1 April

XII Kl Apr 21 March

VI Id Apr 8 April

V Kl Apr 28 March

XVI Kl Apr 17 March

Non Apr 5 April

IX Kl Apr 24 March

Prid Id Apr 12 April

32. XII Kl Mai 20 April

31. Prid Kl Apr 31 March

30. V Id Apr 9 April

29. XIII Kl Apr 20 March

28. Prid Non Apr 4 April

27. Prid Id Apr 12 April

26. IX Kl Apr 24 March

25. VI Id Apr 8 April

Cycle II

Figure 2  112-year list of Easter Sundays.

Easter Sunday, Cycle I

Luna XIV

48. XVIII Kl Mai 14 April

47. Kl Apr 1 April

46. IV Id Apr 10 April

45. XII Kl Apr 21 March

44. Non Apr 5 April

43. Id Apr 13 April

42. VIII Kl Apr 25 March

41. V Id Apr 9 April

Cycle III

64. XVII Kl Mai 15 April

63. VII Kl Apr 26 March

62. III Id Apr 11 April

61. XI Kl Apr 22 March

60. III Kl Apr 30 March

59. XVIII Kl Mai 14 April

58. VII Kl Apr 26 March

57. III Non Apr 3 April

Cycle IIII

80. XVI Kl Mai 16 April

79. VI Kl Apr 27 March

78. Prid Id Apr 12 April

77. X Kl Apr 23 March

76. Prid Kl Apr 31 March

75. XVII Kl Mai 15 April

74. VI Kl Apr 27 March

73. Prid Non Apr 4 April

Cycle V

96. XV Kl Mai 17 April

95. V Kl Apr 28 March

94. Id Apr 13 April

93. IX Kl Apr 24 March

92. Kl Apr 1 April

91. XVI Kl Mai 16 April

90. V Kl Apr 28 March

89. Non Apr 5 April

Cycle VI

112. XIV Kl Mai 18 April

111. IV Kl Apr 29 March

110. VII Id Apr 7 April

109. VIII Kl Apr 25 March

108. IV Non Apr 2 April

107. IV Id Apr 10 April

106. IV Kl Apr 29 March

105. VIII Id Apr 6 April

Cycle VII

TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

ALDEN A. MOSSHAMMER

This original arrangement was disturbed by a redactor, then distorted by the copyist of the lost Reims manuscript, slightly disarranged in the transcript that Mabillon sent to Wallis and in the copy at the Vatican, misunderstood by the editors of the reprint in the Patrologia Latina and finally rendered unintelligible in Hartel’s edition. The Vatican copy shows that Wallis faithfully reproduced the tables as they appeared in the copy of the Reims manuscript that Mabillon sent him. I use an image from Wallis’s edition to represent the basic arrangement, as it is in the public domain, more easily legible than Vatican, Biblioteca Apostolica, Reg. lat. 324, and with fewer errors. Plate 1 shows the tables as they appeared in the 1700 reprint of the 1682 edition.54 The tables in the manuscript began with columns on the left for the age of the moon and the weekday (the feria) on the Kalends of March. That information is useful, but in the text the author makes no reference to such data. It is probably a later addition.55 The pinax on the right, headed ‘Exodus’, has 16 lines, instead of 17, with the result that this table makes the 16-year cycle begin with 12 April in the year corresponding to the Exodus, instead of with 1 April in the following year. This is the work of a redactor. Either the redactor did not understand why there were 17 lines in the exemplar and omitted the last one or he made a deliberate decision to recalibrate the cycle with the Exodus as year 1. The redactor also entered a note at the eighth line that the year is numbered ∞DCCXVIII (1718).56 This is a scribal error for ∞DCCXCVIIII (1799), which is the numeral that appeared at the end of the narrative text in the Reims manuscript. There (268.18) the interval from the Exodus to the fifth year of Gordian, the consulship of Arrianus and Papus, is ∞DCCXCVIIII (1799), and the writer refers to the eighth line of the table. In the Cotton manuscript, the year is 1794, and is said to correspond to the third line of the first set of 16 years. That the Cotton manuscript is correct has already been noted above. The intervals given (268.20) from the Passion to the present year show The page also shows on the lower right Wallis’s reconstruction of the Hippolytan table. It is not the pinax of the Computus of AD 243, which Wallis reconstructs on p. 213. 55 Krusch (1880), 162, says: ‘Luna und ferie des 1. März finden wir übrigens schon in dem sogenannten Computus Cypriani notirt’; but the table that includes this column is clearly the work of a redactor and not necessarily a product of the third century. 56 The earlier form of the Roman numeral for 1000 was CIƆ, which evolved into a symbol resembling a figure-8 turned on its side. I have used the unicode symbol for the ‘infinity’ sign to represent it. See Capelli (1962), 44. 54



TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

that 1794 is the correct interval from the Exodus. The consulship of Arrianus and Papus was AD 243. The author dates the Passion to Friday 8 April in the year 1579. Friday 8 April is calendrically correct for the year AD 28. In the Cotton manuscript the interval from the Passion to the present year is 215. In the Reims copy the interval is 220. The Cotton copy refers to the third line of the table with the paschal full moon on Tuesday, 12 Kalends April (21 March), which is correct for the year AD 243. The Reims copy makes it the eighth line, with Passover on Friday, 9 Kalends April (24 March), correct for AD 248.57 The decision to include a table for the luna and feria of 1  March forced a change in the arrangement of the columns for the feriae in the pinax. There was not room for an additional six columns of feriae on the right side of the page. The list of feriae is arranged instead in four columns of 24 lines each and a fifth column of 11 lines, for a total of 107 lines, instead of six columns of 16 lines each. The fifth column appears on the left below the first column. To understand the ferial columns as they appeared in the Reims manuscript, the reader must compare Plate 1 with Figure 1. Consistent with his own 16-line version of the pinax, the redactor began with feria ii in the list of 14th moons and therefore with feria i in the first of the now displaced ferial columns. Although there are 107 lines, a horizontal space across the whole width, and another space within the second column, reduce the total number of entries to 102. There would have been 96 feriae included in the original six columns. A copyist erroneously repeated the last six entries at the end of the fifth column. There is only one mistake in Wallis’s version as compared with what appears in the Vatican manuscript. In the second column, after the horizontal space, the second numeral in Wallis’s copy was 5, instead of 6. The Vatican copy correctly has 6. There follows a list of calendar dates. They are all Sundays, as indicated by the word Domini or the abbreviation dom, for dies domini, the Lord’s Day. The entry for year eight is missing, so that the list has 111 entries, instead of 112. The list begins erroneously with XVI Kl. Mart. instead of XIV Kl. Mai.

Krusch (1880), 20 n. 5, thought that the change in the dating formula in the Reims manuscript derived from a copy or redaction made in the year AD 248. George Ogg (1955), 40, argued that a scribal error in the numerals led a copyist to make all of these changes. 57



ALDEN A. MOSSHAMMER

Plate 1  Fell (1700), 220–21.

At the end of the list is Wallis’s note, as follows:58 Sed non fui sollicitus de his vel ordinandis, vel restituendis: Quoniam quicquid sani hic contineatur, id omne haberi potest in Pinace quem (ad Auctoris mentem) ante restitueram et Notis inserueram, quam hanc conspexerim tabellam. Qua nihil aliud continetur, quam confusa materia, et male digesta, et depravata pro istiusmodi pinace: neque operae pretium est corrigere, vel ordinare, vel etiam divinare quo casu in hanc confusionem devenerit. Si cui id libet, per me licet. ‘I have not troubled to order or reconstruct these lists. Whatever here makes sense can be found in the pinax that I reconstructed according to the author’s intent and the notes I have inserted rather than in what I observed in this table. In it there is nothing but confused material, a poorly understood and corrupted version of that pinax. It would not Fell (1700), 221; the translation is mine.

58

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TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

be worth the effort to try to correct or rearrange it, much less to try to figure out how it came to its present condition. If that task appeals to anyone, he has my permission to try.’

George Ogg attempted to reconstruct the tables as he thought they must have appeared in the lost Reims manuscript.59 Unfortunately, Ogg worked from Hartel’s 1871 CSEL edition. As we saw previously, Hartel in turn used the reprint of Wallis’s edition published in the Patrologia Latina. The editors of the PL altered the format to fit the requirements of a page divided into two columns. Hartel seems not to have understood the format and rearranged its parts beyond recognition.60 Ogg correctly identified some of the elements of Hartel’s tables, but because of his failure to consult Wallis’s edition his attempt to explain how the tables came to be so disarranged is unnecessarily complex. The arrangement is less chaotic than Wallis realized. The Vatican copy has the same arrangement of four columns of feriae, with a fifth column appended to the first. In the Vatican copy, however, a page break occurs after line 17, instead of a horizontal space after line 16 (see Figure 4b, line  17). Where Wallis’s copy had the incorrect feria 5 (see Figure  4b, year 55), the Vatican has the correct numeral 6. Both manuscripts share five other mistakes compared with the original matrix, and those errors must therefore have already appeared in the lost Remensis (see Figure 4b, years 66, 89, 97, 104, 105). At the end of the list of feriae, Wallis’s copy had an extra six entries. The Vatican copy has three more extra entries, but a corrector has marked them for deletion. The list of Easter Sundays had few errors, and the order of the entries is less disturbed than it appeared to Wallis. The list of Sundays in the Vatican copy has a slightly different arrangement compared with Wallis’s copy. In particular, what was the fifth column in Wallis’s arrangement appears in the Vatican copy as if it were an extension of the third column. Figure 3 compares the list of Sundays as it appears in Wallis’s edition with that of the Vatican copy. The numerals represent the position of each date in a correctly reconstructed version (Figure 2). By numbering the list, one can easily see how the original arrangement came into its present format. Wallis’s version is almost correct. The year 8 is missing. The entries for years 11/34 and 12/33 have been reversed. The entries for years 79 to 83 and 108 to 112 have been shifted to the left. Ogg (1954). CSEL 3,3, 269–71.

59 60



ALDEN A. MOSSHAMMER

1 2 3 4 5 6 7 9 10 12

Wallis’s Copy 13 11 34 14 35 57 15 36 58 16 37 59 17 38 60 18 39 61 19 40 62 20 41 63 21 42 64 22 43 65 23 44 66 24 45 67 25 46 68 26 47 69 27 48 70 28 49 71 29 50 72 30 51 73 31 52 74 32 53 75 33 54 76 55 77 56 78 79 108 80 109 81 110 82 111 83 112

84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

1 2 3 4 5 6 7 9 10 11 35 36 37 38

Vat. Reg. lat. 324 12 39 13 40 14 41 15 42 16 43 17 44 18 45 19 46 20 47 21 48 22 49 20 50 21 51 22 52 23 53 24 54 25 55 26 56 27 79 28 80 29 81 30 82 31 83 32 84 33 85 34 86 57 87 58 88 59 89 60 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 108 109 110 111 112

Figure 3  Numbered list of Easter Sundays. Numerals in bold type highlight the displacements. For the dates associated with each numeral, see Figure 2.



TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

In the Vatican version, the year  8 is also missing. The years  11/34 correctly follows the years 10/33. The years 20 to 22 have been erroneously repeated. The years 35 to 38 and 57 to 60 appear at the bottom of the first two columns, instead of the top of columns three and four. The years 79 to 83 and 108 to 112 are shifted to the left as in Wallis’s copy, but what appeared as a fifth column in Wallis’s version is written as if it were an extension to the third column. Although the Vatican version is more seriously disarranged than what was in the transcript that Mabillon sent to Wallis, having the two copies of the Codex Remensis for comparison helps us to reconstruct what must have been the arrangement in that lost manuscript. Figures 4a–e show that reconstruction. Figure 4a shows the pinax and 4b the list of feriae. Figures 4c–e have the end of the list of feriae followed by the list of Sundays, written two columns to the page. A space before the last entry in the first two columns (Figure 4c, line 24) led Mabillon to treat them as the beginning of new columns (Figure 3, line 1), when he was transcribing the Reims manuscript for Wallis. A  space after the first four entries in columns three and four (Figure  4d) led the copyist of the Vatican manuscript to see those four entries as continuations of the first two columns (see Figure 3), rather than as new columns following in sequence after the second column. Both copyists (Mabillon and the unknown writer of Vatican, Biblioteca Apostolica, Reg. lat. 324) read the first five entries on the third page of Sundays (Figure 4e) as continuations of columns three and four, instead of four and five. The list of feriae was originally written 23 lines to the page. The list of Sundays was written 23 lines to the page, except that on the last page the copyist managed to fit 30 lines.



ALDEN A. MOSSHAMMER

Kl. Mart.

Luna et Feria

Exodus

Luna II

feria ii

Embolismus

Prid Id Apr

1 feria ii

Luna XIII

feria iii

Comm

Kl Apr

2 feria vi

Luna XXIII (recte XXIIII)

feria iiii

Comm

XII Kl Apr

3 feria iii

Luna VI

feria vi

Bissextus et Embolismus

4 vi

Luna XVII

feria vii

Comm

V Kl Apr

5 feria vi

Luna XXIIX

feria i

Comm

XIII Kl Apr (recte XVI Kl Apr)

6 feria iii

Luna IX

feria ii

Embolismus

Non Apr

7 feria ii

Luna XXI

feria iiii

Bissextus Comm

VIII Kl Apr (recte VIIII Kl Apr)

8 feria vi

∞dcxviii Luna II

feria v

Embolismus

Prid Id Apr

9 feria v

Luna XIII

feria vi

Comm

Kl Apr

10 feria ii

Luna XXVI (recte XXIIII)

feria vii

Comm

XII Kl Apr

11 feria vi

VI Idus Apr

12 feria v

Luna VI

feria vi Bissextus et (recte ii) Embolismus

Passio Luna XVII

feria iii

Comm

V Kl Apr

13 feria ii

Luna XXVIII

feria iv

Comm

XVI Kl Apr

14 feria vi

Luna VIIII

feria v

Embolismus

Non Apr

15 feria v

Luna XXI

feria vii

Bissextus Comm

VIIII Kl Apr

16 feria ii

Figure 4a  Reconstruction of first page of tables in the Codex Remensis Deperditus.



TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

1

17 feria i

40 feria iiii

62 feria iii

85 feria i

2

18 feria v

41 feria iii

63 feria ii

86 feria v

3

19 feria ii

42 feria vii

64 feria vi

87 feria iiii

4

20 feria i

43 feria iiii

65 feria v

88 feria i

5

21 feria v

(blank both copies)

66 feria iii (recte ii)

89 feria v (recte vii)

6

22 feria ii

44 feria iii

67 feria vi

90 feria iiii

7

23 feria i

45 feria vii

68 feria v

91 feria i

8

24 feria v

46 feria iiii

69 feria ii

92 feria vii

9

25 feria iiii

47 feria iii

70 feria vi

93 feria iiii

10 26 feria i

48 feria vii

71 feria v

94 feria i

11 27 feria v

49 feria vi

72 feria ii

95 feria vii

12 28 feria iiii

50 feria iii

73 feria i

96 feria iiii

13 29 feria i

51 feria vii

74 feria v

97 feria i (recte feria iii)

14 30 feria v

52 feria vi

75 feria ii

98 feria vii

15 31 feria iiii

53 feria iii

76 feria i

99 feria iiii

16 32 feria i

54 feria vii

77 feria v

100 feria iii

55 feria vi (feria v Wallis)

78 feria ii

101 feria vii

19 34 feria iv

56 feria iii

79 feria i

102 feria iiii

20 35 feria i

57 feria ii

80 feria v

103 feria iii

21 36 feria vii

58 feria vi

81 feria iiii

104 feria iiii (recte feria vii)

22 37 feria iv

59 feria iii

82 feria i

105 feria iiii (recte feria vi)

23 38 feria i

60 feria ii

83 feria v

106 feria iii

24 39 feria vii

61 feria vi

84 feria iiii

107 feria vii

17 Blank Space W 18 33 feria vii Page Break V

Figure 4b  Reconstruction of second page of tables in the Codex Remensis Deperditus.



ALDEN A. MOSSHAMMER

1

108 feria vi

12 III Id Apr Dom

2

109 feria iii

13 III Non Apr Dom

3

110 feria vii

14 XIIII Kl Apr Dom

4

111 feria vi

15 VI Id Apr Dom

5

112 feria iii

16 III Kl Apr Dom

6

feria vii

17 XIII Kl Mai Dom

7

feria vi

18 Prid Non Apr Dom

8

feria iii

19 VI Kl Apr Dom

9

feria vii

20 XVII Kl Mai Dom

10

feria vi

21 Prid Kl Apr Dom

11

feria iii

22 X Kl Apr Dom

12

23 Prid Id Apr Dom

13

24 VI Kl Apr Dom

14

1 XVI Kl Mart Dom (recte XIV Kl Mai)

25 XVI Kl Mai Dom

15

2 III Non Apr Dom

26 VI Id Apr Dom

16

3 VII Kl Apr Dom

27 VIIII Kl Apr Dom

17

4 XIIX Kl Mai Dom

28 Prid Id Apr Dom

18

5 III Kl Apr Dom

29 Prid Non Apr Dom

19

6 XI Kl Apr Dom

30 XIII Kl Apr Dom

20

7 III Id Apr Dom

31 V Id Apr Dom

21

9 XVII Kl Mai Dom

32 Prid Kl Apr Dom

22

10 XVII Id Apr Dom 33 XII Kl Mai Dom (recte VII Id Apr Dom)

23 24

11 X Kl Apr Dom

34 Non Apr Dom

Figure 4c  Reconstruction of third page of tables in the Codex Remensis Deperditus.



TOWARDS A NEW EDITION OF THE COMPUTUS OF AD 243

1

35 V Kl Apr Dom

57 XIII Kl Mai Dom

2

36 XVI Kl Mai Dom

58 III Non Apr Dom

3

37 Kl Dom (Kl Dom WV)

59 VII Kl Apr Dom (VII Id. Apr W)

4

38 VIIII Kl Apr Dom

60 XVII Kl Mai Dom (recte XVIII Kl Mai)

6

39 Id Apr Dom

61 III Kl Apr Dom

7

40 V Kl Apr Dom

62 XI Kl Apr Dom

8

41 XV Kl Mai Dom

63 III Id Apr Dom

9

42 V Id Apr Dom

64 VII Kl Apr Dom

5

10 43 VIII Kl Apr Dom

65 XVII Kl Apr Dom (recte XVII Kl Mai Dom)

11 44 Id Apr Dom

66 VII Id Apr Dom

12 45 Non Apr Dom

67 X Kl Apr Dom

13 46 XII Kl Apr Dom

68 III Id Apr Dom

14 47 IIII Id Apr Dom

69 III Non Apr Dom

15 48 Kl Apr Dom

70 XIIII Kl Apr Dom

16 49 XVIII Kl Apr Dom (recte XVIII Kl Mai)

71 VI Id Apr Dom

17 50 VIII Id Apr Dom

72 III Kl Apr Dom

18 51 IIII Kl Apr Dom

73 XIII Kl Mai Dom

19 52 IIII Id Apr Dom

74 Prid Non Apr Dom

20 53 IIII Non Apr Dom

75 VI Kl Apr Dom

21 54 VIII Kl Apr Dom

76 XVIII Kl Apr Dom (recte XVII Kl Mai Dom)

22 55 VII Id Apr Dom

77 Prid Kl Apr Dom

23 56 IIII Kl Apr Dom

78 X Kl Apr Dom

24 Figure 4d  Reconstruction of fourth page of tables in the Codex Remensis Deperditus.



ALDEN A. MOSSHAMMER

1 2 3 4 5 6

79 Prid Id Apr Dom 80 VI Kl Apr Dom 81 XVI Kl Apr Dom (recte XVI Kl Mai) 82 VI Id Apr Dom (V Id Apr W) 83 VIIII Kl Apr Dom

7

84 Prid Id Apr Dom

8

85 Prid Non Apr Dom

9

86 XIII Kl Apr Dom

108 IIII Id Apr Dom 109 IIII Non Apr Dom 110 VIII Kl Apr Dom 111 VII Id Apr Dom 112 IIII Kl Apr Dom

10 87 V Id Apr Dom 11 88 Prid Kl Apr Dom 12 89 XII Kl Apr Dom (recte XII Kl Mai) 13 90 Non Apr Dom 14 91 V Kl Apr Dom 15 92 XVI Kl Apr Dom (recte XVI Kl Mai) 16 93 Kl Apr Dom 17 94 VIIII Kl Apr Dom 18 95 Id Apr Dom 19 96 V Kl Apr Dom 20 97 XV Kl Mai Dom 21 98 V Id Apr Dom 22 99 VIII Kl Apr Dom 23 100 Id Apr Dom 24 101 Non Apr Dom 25 102 XII Kl Apr Dom 26 103 IIII Id Apr Dom 27 104 Kl Apr Dom 28 105 XVIII Kl Apr Dom (recte XVIII Kl Mai) 29 106 VIII Id Apr Dom 30 107 IIII Kl Apr Dom Figure 4e  Reconstruction of fifth page of tables in the Codex Remensis Deperditus.



JAN ZUIDHOEK

THE INITIAL YEAR OF DE RATIONE PASCHALI AND THE RELEVANCE OF ITS PASCHAL DATES

Abstract According to Mc Carthy and Breen, the Latin text De ratione paschali (DRP) is a translation of the original Greek paschal tract of Anatolius, bishop of Laodicea (†c.AD  282), containing his famous 19-year paschal cycle. The 19-year periodic sequence of epacts of this paschal cycle has a Metonic structure. This implies that there must have been a Metonic sequence of dates of paschal full moons (PFM) on the basis of which Anatolius constructed his paschal cycle. This particular Metonically structured predecessor of the classical Alexandrian cycle must have been constructed by Alexandrian computists somewhere between the years AD 250 and 270. Thus the sequence of dates of the Anatolian PFM marks the connection between the proto-Alexandrian cycle and the sequence of paschal dates of DRP. Subsequently, somewhere in the first quarter of the fourth century, the church of Alexandria replaced its Metonic sequence of dates of paschal full moons then in use, possibly the proto-Alexandrian cycle, with the classical Alexandrian cycle. It will be demonstrated that this was done by advancing the date of the March equinox by one day and the computistical lunar months by two days. Keywords Anatolius, Annianus, Bede, Theodor Zahn, Bruno Krusch; Christian era, Nisan, Jewish calendar, Julian calendar, Metonic sequence, Alexandrian calendar, classical Alexandrian PFM, proto-Alexandrian cycle, Six Millennium Catalog, lunisolar conjunction, Metonic adaptation, Passover, March equinox, synodic period, Anatolian PFM, Anatolian paschal Sunday, Anatolian calendar, Anatolian paschal data; Alexandria, Jerusalem, Nicaea; De ratione paschali.

Late Antique Calendrical Thought and its Reception in the Early Middle Ages, ed.  by Immo Warntjes and Dáibhí Ó Cróinín, Turnhout, Brepols, 2017 (Studia Traditionis Theologiae, 26), pp. 71-93 © BREPOLSHPUBLISHERSDOI 10.1484/M.STT-EB.5.114734

JAN ZUIDHOEK

Introduction According to Daniel Mc  Carthy and Aidan Breen, the Latin text De ratione paschali (DRP)1 is a fourth-century accurate translation of the original Greek paschal tract of Anatolius,2 a native of third-century Alexandria, who was a famous computist and bishop of Laodicea (on the coast of Syria), around the seventies of the third century.3 This implies that the paschal table that forms part of DRP is nothing less than Anatolius’ lost 19-year paschal cycle. This agrees with the view of Theodor Zahn that the Greek original of DRP could very well have been written around the year AD 270.4 This text was of great importance in the early Middle Ages to scholars of the Christian paschal calendar;5 however, it still remains unclear whether the paschal table of DRP was part of the original work. For Bruno Krusch, writing in the year 1880, the text was no more than a forgery.6 On the other hand, according to Mc Carthy and Breen, DRP represents a consistent whole,7 which was consulted by early medieval scholars of the Christian Pasch. The very core of DRP is formed by the 19-year periodic sequence of paschal dates, and it is by locating this sequence of dates in the Christian era that the relevance of these dates can be established. Paschal tables were used since the third century, most of them provided with a periodic sequence of dates of the paschal full moon (PFM) for consecutive calendar years. These periodic sequences of dates of PFM were originally intended as substitutes for dates of the observed fourteenth day of the Jewish month Nisan. It was not before the middle of the fourth century that the Jewish calendar began gradually to become fixed.8 1 Mc Carthy and Breen (2003), 25. I would like to express my gratitude to both Dáibhí Ó Cróinín and Daniel Mc Carthy for their encouragement and support in developing my ideas about the development of the Alexandrian computus in the second half of the third century and the first quarter of the fourth; to Dáibhí for his constructive editorial patience; to Dan for the continuously inspiring e-mail correspondence on these ideas since July 2009. 2 Mc Carthy and Breen (2003), 142. 3 Mc Carthy and Breen (2003), 18. 4 Zahn (1884), 196. 5 Mc Carthy and Breen (2003), 24. 6 Krusch (1880), 311. 7 Mc Carthy and Breen (2003), 142. 8 Mosshammer (2008), 88.

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THE INITIAL YEAR OF DE RATIONE PASCHALI

In this article various periodic sequences of dates of PFM falling between 21 March and 20 April (inclusive) of 19 consecutive Julian calendar years will be considered. In these sequences each following date can be obtained by advancing its immediate predecessor by 10, 11, or 12 days modulo 30 days, such that, over each period of 19 years, the total number of advanced days amounts to 210 (e.g., 4×10+10×11+5×12=210 days). Among such 19-year periodic sequences of dates, particularly those in which each following date can be obtained by advancing its immediate predecessor by 11 days modulo 30 days, or, once every 19 years, by 12 days modulo 30 days (i.e. 18×11+1×12=210 days), are of especial interest, because they reflect in the most natural way the phenomenon of the 19-year lunar cycle. That is, a time interval of 19 solar years consists on average of nearly as many days as 235  synodic lunar months. Taking the solar year to be the tropical year consisting of 365.2422 days,9 and the synodic lunar month to consist of 29.53059  days,10 we find 19×365.2422≈6939.602  days and 235×29.53059≈6939.689  days respectively. Thus in about 6940 days the sun and moon return to the same relative positions. This astronomical synchronism was known as early as the fifth century before Christ in Mesopotamia, as well as in Greece, where the Athenian astronomer Meton rediscovered it. Hence, such sequences of dates, as well as their particular structure, are called Metonic. Metonic sequences of dates can be separated into two types: the first type is characterized by 11 ordinary advances of 11 days, 1 saltus advance of 12 days, and 7 ordinary regressions of 19 days; the second type is characterized by 12 ordinary advances of 11 days, 6 ordinary regressions of 19 days, and 1 saltus regression of 18 days. For example, it is easy to verify that four out of the five 19-year periodic sequences of dates of PFM in Figure 6, being Metonic, are Metonic sequences of dates of the first type. We note that two of them have not 21 March but 23 March as their earliest permissible date, and, as a consequence, not 18 April but 20 April as their latest permissible date. The historically most important example of a Metonic sequence of dates is the classical Alexandrian cycle, which is of the first type. Its initial year is considered to be the year AD 285, the first year of consulship of the Emperor Diocletian,11 and its saltus is inserted between the years AD 303 and 304 modulo 19, i.e., at the end of its nineteenth Smart (1958), 146. Smart (1958), 169. 11 Mosshammer (2008), 26–27. 9

10



JAN ZUIDHOEK

year AD 303 modulo 19.12 This Metonic sequence of Alexandrian calendar dates of the classical Alexandrian PFM, which was constructed by Alexandrian computists somewhere in the first quarter of the fourth century, say around the year AD 310,13 forms the backbone of the classical Alexandrian paschal cycle.14 That is the great 532-year Easter cycle,15 which was invented, and drawn up, by the Alexandrian monk Annianus around the year AD 400.16 Its 532-year structure, unknown to Dionysius Exiguus, was reconstructed by insular computists (most famously Bede) in the late seventh-, early eighth-century.17 The initial year of Annianus’ Alexandrian calendar version is AD 285;18 and the initial year of Bede’s Julian calendar version is AD 532.19 Although constructed before the council of Nicaea in AD 325, the classical Alexandrian cycle cannot have been the very earliest example of a Metonic sequence of dates of PFM, because the Metonic sequence of epacts of the first type is a structural feature of the framework of DRP.20 Consequently it must also have formed a structural feature of the Greek original of DRP, and therefore must have originally derived from a thirdcentury proto-Alexandrian cycle, i.e., the Metonic sequence of dates of the proto-Alexandrian PFM, from which Anatolius started in order to construct his 19-year paschal cycle. Because 19-year periodic sequences of dates of PFM were not constructed before the middle of the third century,21 it must have been around the year AD 260 that this particular Metonic sequence of dates of PFM was constructed by computists of the church of Alexandria, among them possibly Anatolius before his episcopal consecration.22 From the dates of this proto-Alexandrian PFM it was relatively straighforward to calculate the corresponding dates of the

14 15 16 17 18 19 20 21 22 12 13

Mosshammer (2008), 149. Declercq (2000), 66. Neugebauer (1979), 59–63. Wallis (1999), 392–404. Declercq (2000), 30. Declercq (2000), 156–58. Neugebauer (1979), 59; Wallis (1999), 392–404. Wallis (1999), 392. Mc Carthy and Breen (2003), 68. Lejbowicz (2006), 45. Declercq (2000), 65–66.

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THE INITIAL YEAR OF DE RATIONE PASCHALI

paschal Sunday, namely by applying the third-century Alexandrian principle ‘paschal Sunday is the first Sunday after the paschal full moon’.23 Unfortunately, no ancient or medieval manuscript has survived containing a third-century Metonic sequence of dates of PFM; in fact, we know next to nothing with certainty about the proto-Alexandrian cycle. However, by using NASA’s Six Millennium Catalog of Phases of the Moon24 we can calculate Metonically structured good approximations to it. Thus, with the help of the Six Millenium Catalog we can endeavour to reconstruct the Julian calendar equivalent of the proto-Alexandrian cycle. These Metonically structured approximations, with the 19-year periodic sequence of dates of the Anatolian PFM, will enable us to precisely locate the sequence of paschal dates of DRP in the Christian era. To summarize: in this article I wish to work out how the following three fundamental 19-year periodic sequences of Julian calendar dates of PFM relate to each other: 1. The proto-Alexandrian cycle, i.e., the unknown Metonic sequence of dates of the PFM constructed by Alexandrian computists around the year AD 260. 2. The sequence of dates of the PFM constructed by Anatolius around the year AD 270, deduced from the sequence of paschal dates of DRP. 3. The classical Alexandrian cycle, i.e., the important Metonic sequence of dates of the classical Alexandrian PFM constructed by Alexandrian computists around the year AD 310.

Constructing Metonically structured approximations to the proto-Alexandrian cycle To reconstruct the proto-Alexandrian cycle, it is necessary to first construct the best possible Metonically structured approximations to the spring equinoctal full moons (luna 14) as observed from Alexandria. To do this, we need to imagine the way in which, around the year AD 260, the third-century Alexandrian computists endeavoured to construct their first Metonic sequence of dates of PFM. They must first have experimented with various sequences of successive ‘dates of the fourteenth Lejbowicz (2006), 48. Espenak (2007).

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day of Nisan’, remembering that, at that time, dates of the fourteenth day of Nisan were generally not exactly calculable. To be more precise, various sequences of successive dates had to be substituted for the observed fourteenth day of Nisan. I take these dates as the fourteenth day of an abstract computistical month, Nisan*, to be defined hereafter in this section. To obtain these dates of the fourteenth day of the computistical month Nisan*, of course via their dates of the first day of the computistical month Nisan*, the computists in question needed tables of then relatively recent dates and times of lunisolar conjunction; such tables they could have derived from the schematic lunar calendar as adapted to the Alexandrian calendar by Alexandria’s Jewish community of the time.25 These dates of new moons must have belonged to a sufficient time interval, encompassing at least twice 19 calendar years. Let us take the interval I* consisting of the period of years AD 220 to 260. It is likely that, in order to obtain their dates of the fourteenth day of their computistical month Nisan*, these Alexandrian computists made use of the old rule concerning the beginning of Nisan, i.e. the rule that the first day of Nisan usually began with the second sunset in Jerusalem after the new moon of Nisan, each individual new moon of Nisan actually being the lunisolar conjunction just preceding Nisan. The rule that Nisan usually began with the second sunset in Jerusalem after lunisolar conjunction is an obvious consequence of the Babylonian criterion for first new-moon visibility, namely the rule that around the beginning of spring every new moon will be visible for the first time, weather permitting, at the beginning of the evening, between 24  and 48  hours after lunisolar conjunction.26 Applying their rule concerning the beginning of Nisan means that, for each lunisolar conjunction which heralded the beginning of the month Nisan, the date of the first day of this month Nisan can be estimated quite accurately by simply adding 2 or 3 days to the local Jerusalem date (from midnight to midnight) of this lunisolar conjunction, depending on whether the local Jerusalem time of this lunisolar conjunction fell before or after 18:00, respectively. However, in order to reconstruct the proto-Alexandrian PFM cycle, we have not only to take into account the Babylonian 24 hours rule, but also the Jewish religious principle that Pesach had to be celebrated as early as possible in spring, which meant since the mid-third century that the fourteenth day of Nisan had to fall Neugebauer (1979), 8. Bruin (1977), 333.

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on or immediately after the spring equinox.27 It is because of this Jewish religious principle that, in the second half of the third century and in the earlier fourth century, Alexandrian computists reckoned with the March equinox. At least from about the year AD 253, their date for the March equinox was the Ptolemaic date of 22 March.28 This implies that the earliest possible date of the proto-Alexandrian PFM must have been either 22 or 23 March. Incidentally, in the second half of the third century, the astronomical date of the March equinox was, in reality, either 21 March or 20 March. We may safely assume that it is only since about the year AD 303 that the church of Alexandria considered 21 March to be the date of the March equinox.29 Reconstructing the proto-Alexandrian cycle, as well as reconstructing the classical Alexandrian cycle, appears to be possible because of the fact that, at least during the years AD 253 and 325, Alexandrian computists respected the rule of the equinox. On the other hand, estimating the dates of the fourteenth day of the real Jewish month Nisan during these same years is a precarious undertaking, because, at the time, their Jewish contemporaries in actual practice ignored the rule of the equinox, and in consequence began their month Nisan, and celebrated their paschal feast, in fact a month too early many a time.30 Normally, in order to obtain estimated dates of the fourteenth day of Nisan, we would apply the rule concerning the beginning of Nisan to the local Jerusalem dates and times of lunisolar conjunction. However, in order to reconstruct the proto-Alexandrian cycle, we must apply an adapted version of the rule concerning the beginning of Nisan, in which the computistical month Nisan* differs from Nisan in that, firstly, its fourteenth day must obey the rule of the March equinox falling on 22 March; secondly, its first day must begin with the second sunset in Alexandria after its new moon. In practice, this means that from each individual local Alexandrian date and time of new moon of the computistical month Nisan* belonging to I*, an approximate (from midnight to midnight) date of the daylight part of the first (from sunset to sunset) day of Nisan* can be obtained by simply adding 2 or 3 days to the date in question, depending on whether the time in question falls before or after 18:00, respectively. 29 30 27 28

Stern (2001), 50. Lejbowicz (2006), 48. Lejbowicz (2006), 48. Stern (2001), 66–67.

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The local Alexandrian dates and times of the new moon of the computistical month Nisan* that we need for the determination of the approximate dates of the first day of Nisan* can be obtained from the table of local Greenwich dates and times of lunisolar conjunction in the NASA Six Millennium Catalog. Local Greenwich dates and times can be converted to local Alexandria dates and times by adding 1:59, this being the time difference between the geographical longitudes of Greenwich and Alexandria. After having thus obtained approximate dates of all first days and then of all fourteenth days of the computistical month Nisan* belonging to the time interval I*, we are in a position to construct the best possible Metonically structured approximation(s) to the protoAlexandrian cycle. To do this, Alexandrian third-century computists must have been able to distinguish good Metonically structured approximations to sequences of their dates of the fourteenth day of Nisan from poor ones, and able to determine which good ones were the very best. In order to be able to appropriately constrain the Julian calendar position of the computistical month Nisan* over the time interval I*, we must establish suitable lower and upper limit dates, with a difference of about 29.5  days, between which limits to analyze local Alexandria times of lunisolar conjunction. These must determine a first day of the month Nisan*, such as to guarantee that the corresponding dates of the fourteenth day of Nisan* will be not earlier than 22  March, and not later than 20 April (a period of 30 days, inclusively). This is achieved by taking our lower limit as 6 March 18:00, and our upper-limit as 5 April 6:00, because adding 3+13 days to 6 March gives 22 March, and adding 2+13 days to 5 April gives 20 April. The structure of Figure 1 suggests the way in which third-century Alexandrian computists must have succeeded in constructing their Metonic sequence of dates of the proto-Alexandrian PFM. Hence the column Conjunction* shows the estimated local Alexandria times of new moon of the computistical month Nisan*, each one obtained by adding 1:59 to the corresponding local Greenwich times of lunisolar conjunction as reckoned by the NASA Six Millennium Catalog.31 These conjunctions are chosen so that the corresponding local Alexandria times all fall between 6 March 18:00 and 5 April 6:00. Column 1 Nisan* shows our estimated dates of the first day of the computistical month of Nisan*, each of them obtained from the corresponding time in Conjunction* by applying the Nisan* version of the rule concerning the beginning of Nisan. Column Espenak (2007).

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Year 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259

Conjunction* 11 March 17:01 30 March 14:57 19 March 19:58 7 March 20:06 26 March 13:13 15 March 19:16 3 April 17:35 23 March 9:06 13 March 1:49 1 April 2:02 21 March 12:21 9 March 15:21 28 March 7:53 17 March 9:27 5 April 4:55 24 March 16:42 14 March 8:54 2 April 10:08 23 March 0:47 11 March 8:57 30 March 3:00 19 March 3:12 8 March 6:12 26 March 2:50 15 March 16:09 3 April 16:58 24 March 9:45 12 March 22:50 31 March 19:38 20 March 22:26 9 March 22:37 27 March 16:41 17 March 1:18 5 April 0:27 25 March 16:54 14 March 8:59 2 April 8:22 22 March 16:13 11 March 17:29

1 Nisan* 13 March 1 April 22 March 10 March 28 March 18 March 5 April 25 March 15 March 3 April 23 March 11 March 30 March 19 March 7 April 26 March 16 March 4 April 25 March 13 March 1 April 21 March 10 March 28 March 17 March 5 April 26 March 15 March 3 April 23 March 12 March 29 March 19 March 7 April 27 March 16 March 4 April 24 March 13 March

14 Nisan* 26 March 14 April 4 April 23 March 10 April 31 March 18 April 7 April 28 March 16 April 5 April 24 March 12 April 1 April 20 April 8 April 29 March 17 April 7 April 26 March 14 April 3 April 23 March 10 April 30 March 18 April 8 April 28 March 16 April 5 April 25 March 11 April 1 April 20 April 9 April 29 March 17 April 6 April 26 March

Adv – −19 +10 +12 −18 +10 −18 +11 +10 −19 +11 +12 −19 +11 −19 +12 +10 −19 +10 +12 −19 +11 +11 −18 +11 −19 +10 +11 −19 +11 +11 −17 +10 −19 +11 +11 −19 +11 +11

ProtoAlexPFM* 26 March 14 April 3 April 23 March 11 April 31 March 19 April 8 April 28 March 16 April 5 April 24/25 March 12 April 1 April 20 April 9 April 29 March 17 April 6 April 26 March 14 April 3 April 23 March 11 April 31 March 19 April 8 April 28 March 16 April 5 April 24/25 March 12 April 1 April 20 April 9 April 29 March 17 April 6 April 26 March

Figure 1  Best Metonic approximations to the proto-Alexandrian PFM cycle All years are AD and all dates Julian calendar dates; the column Conjunction* refers to the estimated local Alexandrian times of the new moon of the computistical month Nisan* as obtained from the NASA Six Millenium Catalog; 1 Nisan* refers to the estimated dates of the first day of Nisan* on the basis of Conjunction*; 14 Nisan* refers to the estimated dates of the fourteenth day of the computistical month Nisan*; Adv refers to the advance in days from the preceeding 14 Nisan*; ProtoAlexPFM* contains the two equally best Metonically structured approximations to the sequence of dates of 14 Nisan*.

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14 Nisan* shows our estimated dates of the fourteenth day of the computistical month Nisan*, each of them obtained by adding 13 days to the corresponding date in column 1 Nisan*. Note that the sequence of dates in column 14 Nisan* is not Metonic, not even 19-year periodic. It is by systematically examining the advance or regression in 14 Nisan* from year to year given in the column Adv of Figure 1 that we obtain the best two Metonically structured approximations to that sequence of dates, as given in protoAlexPFM* of Figure 1. The first of these two sequences, which includes 24 March at AD 232 and 251 and which I will refer to as MSA1, is a Metonic sequence of the first type and has its saltus advance between the years AD 231–32 and 250–51. The second sequence, which includes 25  March at AD  232 and 251 and which I will refer to as MSA2, is of the second type and has its saltus regression between the years AD 232–33 and 251–52. Next we turn our attention to the column Adv of Figure  1 where positive values show the advances in 14 Nisan* ranging from +10 to +12 days, and negative values show the regressions in 14 Nisan* ranging from −17 to −19 days. Examination of these shows that in most instances a non-standard Metonic advance of +10 or +12, or regression of −17 or −18, is followed within two or three years by another, balancing, advance or regression. For example, at AD 223–24 the advances of +10 and +12 may both be adjusted to the standard advance of +11 by advancing the date of 14 Nisan* at AD 223 by one day. We may concisely list all such instances as follows: Years 223–24 225–26 227–29 236–37 239–40 244–47 252–53

Adv +10; +12 −18; +10 −18; +11; +10 +12; +10 +10; +12 −18; +11; −19; +10 −17; +10

14 Nisan* Adjustment Advance 223 one day Retard 225 one day Retard 227–28 one day Retard 236 one day Advance 239 one day Retard 244–46 one day Retard 252 one day

Resultant Adv +11; +11 −19; +11 −19; +11; +11 +11; +11 +11; +11 −19; +11; −19; +11 −18; +11

These ten one-day adjustments leave only AD 232 with its advance of +12, and AD 252 with its regression of −18, as the only non-standard values, and consequently these identify the two possible locations for the saltus. If we advance 25 March AD 251 by one day we obtain MSA1, or if we retard 24 March AD 232 by one day we obtain MSA2.

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Each of these choices to locate the saltus implies just one further adjustment to the 14 Nisan* sequence, and so they both represent equally good Metonic approximations, and consequently both dates are given under ProtoAlexPFM*. Thus the minimum number of adjustments required to obtain a Metonic sequence from 14 Nisan* is eleven, and these are shown in ProtoAlexPFM*. This results in two possible best Metonic sequences, MSA1 with date 24 March, and MSA2 with date 25 March. In more mathematical terms: in order to obtain the best Metonically structured approximation(s) to the sequence of dates in column 14Nisan*, divide the period AD 222–59 (because the dates of 14Nisan* are the same in AD 221 and 259) into two parts of 19 years each, AD 222– 40 and 241–59. Compare the dates of 14Nisan* of the first 19 years with the second. The maximum of consecutive years in which the two 19year periods agree is 3, and this happens only once, in AD 229–31 vs AD 248–50 (28 March, 16 April, 5 April). This three-year sequence is Metonically structured, with a regression of 19 days between 28 March and 16 April, and an advance of 11 days between 16 April and 5 April. Because of the already existing Metonic structure of the two advances (–19 and +11) between these three years, it is best to insert the saltus in any of the remaining 17 advances of the years AD 231–48 (and its 19-year equivalents). This leads to 17 possible Metonically structured approximations, depending on the place of the saltus, defined as MSAx, with the saltus inserted at the end of the year AD (230+x)modulo19 (e.g., MSA9 refers to the saltus being inserted at the end of the year AD 239modulo19). For each integer x with 0