The Tables of 1322 by John of Ligneres: An Edition With Commentary (Alfonsine Astronomy, 2) 9782503596099, 2503596096

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The Tables of 1322 by John of Lignères

Alfonsine Astronomy Studies and Sources

Series Editors: Matthieu Husson (Observatoire de Paris, Paris, France) José Chabás (Universitat Pompeu Fabra, Barcelona, Spain) Richard Kremer (Dartmouth College, USA) Editorial Board: Charles Burnett (Warburg Institute, London, United Kingdom) Karine Chemla (CNRS, SPHERE, France) Bernard R. Goldstein (Pittsburgh University, USA) Alena Hadravová (Academy of Sciences of the Czech Republic) Danielle Jacquart (EPHE, France) Marie-Madeleine Saby (Université Grenoble Alpes, France) Julio Samsó (University of Barcelona, Spain) Glen Van Brummelen (Quest University, Canada)

This collection is created under the auspices of the European Research Council project ALFA: Shaping a European scientific scene, Alfonsine astronomy, CoG 723085.

The Tables of 1322 by John of Lignères An Edition with Commentary

Edited by José Chabás and Marie-Madeleine Saby

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© 2022, Brepols Publishers n.v., Turnhout, Belgium This is an open access publication made available under a cc by-nc 4.0 International License: https://creativecommons.org/licenses/by-nc/4.0/. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, for commercial purposes, without the prior permission of the publisher, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. D/2022/0095/195 ISBN 978-2-503-59609-9 e-ISBN 978-2-503-59610-5 DOI 10.1484/M.ALFA-EB.5.125035 Printed in the E.U. on acid-free paper

Table of Contents Preface John of Lignères: Iberian Astronomy settles in Paris 1. John of Lignères’ works 2. Works attributed to John of Lignères 3. The set of tables 4. An edition 5. Commentaries to the tables

Edition of the Tables with Comments 1. Sine 2. Shadow 3. Solar declination 4. Ascensional difference 5. Right ascension 6. Oblique ascension 7. Equation of time 8. Planetary latitudes 9. Lunar latitude 10. Daily unequal motion of the planets 11. Retrogradation of the planets 12. Planetary stations 13. Planetary phases 14. Mean syzygies for collected years 15. Mean syzygies for expanded years 16. Mean syzygies for months in a year 17. Mean motion in elongation 18. Corrections of the hourly lunar motion 19. Equations and hourly velocities of the Sun and the Moon 20. Velocities of the Sun and the Moon in a minute of a day 21. Velocities of the Sun and the Moon at intervals of 6º 22. Parallax 23. Proportions for correcting lunar parallax 24. Solar eclipses with argument of lunar latitude as argument 25. Solar eclipses with lunar latitude as argument 26. Lunar eclipses with argument of lunar latitude as argument 27. Lunar eclipses with lunar latitude as argument 28. Eclipsed parts of the solar and lunar discs 29. Finding lunar latitude from the argument of latitude

7 9 10 19 24 28 38

42 50 54 58 62 66 72 76 80 84 88 90 92 94 96 98 102 106 108 114 120 122 126 128 132 134 138 142 144

30. Corrections 31. Tabula reflexionis tenebrarum 32. Proportions at intervals of 2º

146 148 150

List of manuscripts

153

Bibliography

157

Preface

Medieval astronomers solved a great majority of the problems they faced using specific astronomical tables. These were often assembled into collections, which can thus be considered as veritable toolboxes for addressing most of the issues with which astronomical practitioners were confronted. Among Arabic astronomers, it was customary to accompany such sets of tables with a text explaining their use. These ensembles were known by the Arabic term zij. The most influential zijes, first in al-Andalus and later in Latin Europe, were those compiled by al-Khwārizmī (first half of the ninth century) in the version by Maslama and by al-Battānī (d. 929). Both zijes are available to modern scholarship thanks to the editions compiled by H. Suter (1914) and C. A. Nallino (1903­–07), respectively, based on a very limited number of manuscripts. The two zijes merged into the Toledan Tables, which were originally written in Arabic by a group of Andalusian astronomers working in Toledo during the middle of the eleventh century. They are only extant in Latin, in a great number of manuscripts. In 2002, F. S. Pedersen published a splendid edition of the Toledan Tables based on more than one hundred manuscripts, which has proven to be very useful in identifying the tables used by subsequent astronomers and the actual manuscripts they employed. As explained below,  John of Lignères followed this well-established tradition in compiling zijes and, in the early 1320s, he composed an extensive set of astronomical tables aimed at solving the most common problems in mathematical astronomy and wrote the associated canons. This was the first major set to be compiled in Latin. It seems worth recalling that another such zij was compiled in Toledo during the second half of the thirteenth century with an associated text in Castilian, and it is now known as the Castilian Alfonsine Tables. However, the tables themselves have not been preserved or not yet found. This set evolved into what is now known as the Parisian Alfonsine Tables, which did not reach their standard form until well into the fourteenth century. This version was first printed in 1483 (Venice: E. Ratdolt, ed.) and was reprinted many times, but it has never been edited. The present monograph provides an edition of the Tables of 1322 by John of Lignères. It is based on a dozen manuscripts selected from more than forty containing the set, or parts of it. This volume is divided in two parts. In the first, we set the background for the Tables of 1322, focusing on the author, John of Lignères, his multiple works (Section 1), and those attributed to his pen (Section 2). We then turn to the tables composing the set (Section 3), and we present the criteria used to select the manuscripts of the edition, offering succinct descriptions of each of them (Section 4). The first part concludes with a general explanation of the commentaries to the tables (Section 5). The second part consists of the transcription

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and edition of the tables themselves, each of which has been collated with five manuscripts. A specific commentary is given for each table. Finally, we would like to thank Bernard R. Goldstein (University of Pittsburgh) and two editors of this ALFA series, Matthieu Husson (Observatoire de Paris) and Richard L. Kremer (Dartmouth College), for their useful comments during the preparatory stages of this work. We are also grateful to Natasha Saint-Geniès for carefully reviewing the text and to the various librarians for facilitating our task. This work was written in the framework of the ERC project ALFA: Shaping a European Scientific Scene (CoG 723085, 2017–2022). We are happy to highlight that this monograph will be published in 2022, on the 700th anniversary of the compilation of the Tables of 1322. José Chabás and Marie-Madeleine Saby Barcelona and Grenoble, December 2021

José Ch a bás a nd Ma rie- Ma delein e Sa by

John of Lignères: Iberian Astronomy settles in Paris

John of Lignères was a leading figure in the astronomical milieu in Paris in the early fourteenth century and, together with other astronomers by the name of John ( John Vimond, John of Murs, and John of Saxony), he recast the Castilian Alfonsine Tables compiled in Toledo under the patronage of King Alfonso of Castile and León (reigned: 1252–84) into what was to become the Parisian Alfonsine Tables.1 Regarding the biography of John of Lignères, not much can be added to the information given about fifty years ago, in 1973, by Poulle: ‘Originally from the diocese of Amiens, where any of several communes could account for his name, John of Lignères lived in Paris from about 1320 to 1335’.2 However, more can be said about his scholarly activity, since several of his astronomical and mathematical works, including two major sets of astronomical tables and various texts, have been studied in recent years. The object of the present monograph is his first set, the Tables of 1322, associated with two canons usually called Cuiuslibet and Priores owing to their respective opening words.3 John of Lignères was a successful and prolific author, whose astronomical tables and texts are frequently found in miscellaneous astronomical manuscripts of the fourteenth and fifteenth centuries. His reputation quickly gained traction and as early as 1327 his name was mentioned by his disciple, John of Saxony, who wrote canons to the Parisian Alfonsine Tables beginning Tempus est mensura. This text and this set of tables would become the most widely diffused astronomical material for more than two centuries. The Tables of 1322 consist of a series of tables addressing some of the most common problems faced by practitioners of astronomy in the early fourteenth century.4 As is shown below, this set by John of Lignères is heir to the Islamic astronomical tradition developed on the Iberian Peninsula and, in particular, to the Toledan Tables composed in Toledo by



1 José Chabás and Bernard R. Goldstein Bernard R., The Alfonsine Tables of Toledo, Archimedes, New Studies in the History and Philosophy of Science and Technology, 8 (Dordrecht-Boston-London: Kluwer Academic Publishers, 2003). 2 Emmanuel Poulle, ‘John of Lignères’, in Dictionary of Scientific Biography, ed. by Charles Gillispie, 16 vols (New York: Charles Scribner’s Sons, 1973–1980, republished 2001), 7, pp. 122–128; Marie-Madeleine Saby, ‘Les canons de Jean de Lignères sur les tables astronomiques de 1321’ (Unpublished thesis: École Nationale des Chartes, Paris, 1987). A summary appeared as ‘Les canons de Jean de Lignères sur les tables astronomiques de 1321’, École Nationale des Chartes: Positions des thèses, pp. 183–190. 3 These two texts were edited, but not published, by Marie-Madeleine Saby, 1987. A monograph on them is expected to appear in this book series. 4 José Chabás and Marie-Madeleine Saby, ‘Editing the Tables of 1322 by John of Lignères’, in Richard L. Kremer, Matthieu Husson, and José Chabás (eds), Alfonsine Astronomy: The Written Record (Turnhout: Brepols, 2022), 243–255.

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a group of Muslim astronomers in the second half of the eleventh century. On the other hand, many of the tables examined here were integrated in what was later called the Parisian Alfonsine Tables. For this reason, we first identify all tables composing this set by John of Lignères, edit them, and then write commentaries for each of them. 1. John of Lignères’ works Canons associated with the Tables of 1322

John of Lignères composed two lengthy canons related to his astronomical tables of 1322 and dealing with most of the problems faced by astronomers at the time. The first features the incipit Cuiuslibet arcus propositi sinum rectum, and it is structured in forty-four chapters, with a total length of about 9,400 words. The second, beginning Priores astrologi motus corporum celesti, is more than twice as long, for it has about 21,000 words, and it is presented in forty-six chapters. The two canons address different topics, and they are often found separately in codices. Even when found together in the same codex, their chapters have distinctive numberings.5 Nevertheless, the two texts were considered by John of Lignères to form a unified treatise, which is corroborated by the various cross-references appearing in them; for example, Chapter 1 of the Cuiuslibet refers to Chapter 9 of the Priores, and, vice versa, Chapter 35 of the Priores mentions Chapters 30 and 36 of the Cuiuslibet. The canons beginning Cuiuslibet arcus propositi sinum deal primarily with the daily rotation of the celestial vault, the primun mobile, as it was usually called at the time, and addresses trigonometrical problems related to it. The first chapter defines the sine of an arc and refers to a table of sines in which the argument is given at intervals of half a degree and the norm is set at 60, hence this is no doubt the table usually found heading up the Tables of 1322. After the first six chapters with instructions on how to deal with what we now call trigonometric functions, Chapters 7, 8, and 9 cover astronomical instruments, such as a quadrant and a parallactical rule, following the pattern established by Ptolemy in Almagest, V and in the zij of al-Battānī.6 These three long chapters were omitted in various manuscripts. The ensuing chapters refer to the solar declination and give the value used for the obliquity of the ecliptic (23;33,30º), the shadow cast by a gnomon of twelve units, right ascension, oblique ascension; and instructions on how to derive or use the solar altitude, geographical latitude, daily arc, length of daylight, rising times, houses, and stellar coordinates. While some of these topics are closely related to tables found in John of Lignères’ set of 1322, others are not addressed in this or any other table known to have been compiled by him.





5 In several manuscripts, only the Cuiuslibet is found, e.g. Paris, Bibliothèque nationale de France [BnF], lat. 7378A, 46r–52r (fourteenth century) and lat. 7290, 66r–75v (fifteenth century). In contrast, the Priores is found separately in Rome, Biblioteca Casanatense, 643, 105r–108v (fourteenth century), and BnF, lat. 7295A, 174r–180v (fifteenth century), among other manuscripts. 6 For Ptolemy, see Gerald J. Toomer, Ptolemy’s Almagest (New York: Springer Verlag, 1984), pp. 217–220, 244–247; for al-Battānī, see Carolo Alfonso Nallino, Al-Battānī sive Albatenii Opus Astronomicum, 3 vols (Milan: U. Hoepli, 1899–1907), pp. 138–144, Chapter LVII.

john of lign èr es: iberia n a stronomy settles in  pa ris

The issues addressed in the primum mobile were revisited shortly after by a former student of John of Lignères, John of Saxony, who in about 1335 wrote a text entitled Expositiones canonum primi mobilis per magistrum Johannem de Lineriis a magistro Johanne de Saxonia, beginning Quia plures astrologorum, providing full explanations and examples for the forty-four chapters of the Cuiuslibet.7 Priores astrologi is the incipit of the second text, in forty-six chapters, mainly devoted to the motion of the planets and the luminaries. In some manuscripts, the numbering and the order of the chapters differ, and in others, mostly in fifteenth-century codices, some chapters have even been omitted.8 The first chapters deal with the sexagesimal counting of the days, and references to several tables are given. These tables were not integrated into the Tables of 1322, but they found their way into the editio princeps of the Parisian Alfonsine Tables. Then follow several chapters on the mean motions of the celestial bodies, and it is worth emphasizing that the tables mentioned display sub-tables for collected years, expanded years, months, and days. We are also told that the year begins in January and that signs of 30º are used. A critical piece of information stated explicitly is that the radices for the use of these tables are set for the city of Paris at noon previous to the first day of January (Chapters 5 and 6). As was the case before, John of Lignères did not integrate the tables for mean motion mentioned here into the Tables of 1322. Rather, they appear in a later set of his, the Tabule magne (dated 1325). Chapter 10 opens with a crucial statement on Alfonsine theory of precession/trepidation: the motion of the apogees has two components, one due to the motion of access and recess, and another due to the continuous motion of the eighth sphere. The table for the equation of the eighth sphere mentioned here was not integrated in the Tables of 1322 either. We then find chapters on the equations of the luminaries and the planets, direct and retrograde motions, solar declination, and lunar and planetary latitudes, requiring the use of tables usually found in previous sets of tables, such as the Toledan Tables, but not always integrated into the Tables of 1322. Chapter 24 on the latitudes of the planets is one of few to explicitly state numerical values, and the maximum values for northern and southern latitudes are given for the five planets. However, although equal or similar in some cases, these numbers are not always in agreement with those in the corresponding table of his set, which was taken either from the Toledan Tables or from the zij of al-Battānī. Mean and true syzygies, parallax, and eclipses are the subjects of Chapters 25–40. The length of a mean ‘lunation’, which is the synodic month, is given in Chapter 27 as the difference between the thirty-one days in a month and 11;15,57h. The result, 29d 12;44,3h, is indeed the Alfonsine value embedded in the tables for syzygies in the Tables of 1322 (see Table 16). It is noteworthy to find two worked examples to explain the computation of the solar diameter (Chapter 36) and the time of a lunar eclipse (Chapter 38) and two chapters to explain how to draw figures for the solar and lunar eclipses, following a practice



7 For an edition of John of Saxony’s Quia plures astrologorum, see Saby 1987, based on two manuscripts: BnF, lat. 7281, 222r–232r and Erfurt, CA 2º 386, 26r–32r. 8 See, for instance, BnF, lat. 7329 and Basel, F II 7.

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already established by the Toledan Tables.9 All the tables mentioned in this section are found among the Tables of 1322. The last chapters concern the visibility of the planets, the unequal motion of the planets, a star catalogue based on that of Ptolemy, and the revolution of the years. While the first two subjects have their corresponding tables among the set for 1322, the other two do not. In 1327, John of Saxony wrote new chapters on the same topics in a text beginning Tempus est mensura, which became the standard canons associated with the Parisian Alfonsine Tables.10 Two conclusions can already be drawn at this point. The canons Cuiuslibet and Priores cover most of the problems faced by contemporary astronomers. These canons follow the schemes set in the zij of al-Battānī and those in the Toledan Tables, or even in the Castilian Alfonsine Tables, among others. However, in contrast to previous zijes, in the set compiled by John of Lignères not all of these problems were addressed by means of tables or have a corresponding table, and thus the correspondence between the two canons and the tables associated with them is not close. John of Lignères was not prone to mentioning his ‘auctoritates’ or his sources. One finds that in the canons Cuiuslibet and Priores, Ptolemy is not mentioned at all. This is a surprising feature because almost all medieval astronomical texts mention his name, and more so those embracing such a wide range of topics, as is the case here. Moreover, Alfonso is only mentioned once, in passing, when referring to a catalogue of stars that the astronomers in the service of King Alfonso adapted from Ptolemy’s catalogue. One would expect an early Alfonsine astronomer such as John of Lignères to refer repeatedly to Alfonso, as John of Murs did, for example. There are only three authorities mentioned. All the references appear in the Priores (none in the Cuiuslibet), and they are concentrated in two chapters: Azarquiel is mentioned once in Chapter 35, an otherwise unknown Abraham Benbthegar or Benbihegen is given two references in the same chapter,11 and al-Battānī is mentioned six times (once in Chapter 34 and five times in Chapter 35).

9 For the Toledan Tables, see Fritz S. Pedersen, The Toledan tables: a review of the manuscripts and the textual versions with an edition (Copenhagen: C.A. Reitzel, 2002), pp. 379, 464–467, 470–472. 10 For an edition of this text with a commentary and translation into French, see Emmanuel Poulle, Les tables alphonsines avec les canons de Jean de Saxe (Paris: Ed. du C.N.R.S.,1984). It is worth noting that the Tempus est mensura by John of Saxony integrates much previous textual material mostly taken from John of Lignères’ Priores, sometimes verbatim or very close to it. This is especially so in the chapters often found after the Tempus est mensura, which have been called the membra adiuncta. 11 The name Abraham Benbthegar seemed to be totally unknown to the copyists of most of the manuscripts, who all spelt the name very differently: Habraham Benbthegar (Erfurt, CA 2° 377), Abraham Benthegor alias Benhigen (Cracow, BJ 551), Abraham Benthegar (Catania, 85), Abraham Benhihegen (Erfurt, CA 4° 366), Abraham ben Bahegen (Paris, BnF lat.7281). No one with this name, or anything similar, has been identified, but the name of Abraham ben Waqār, a translator at the service of King Alfonso, has been evoked ( José Chabás and Bernard R. Goldstein 2003, p. 282). According to John of Lignères, this astronomer composed canons ‘that have not yet been translated from Hebrew, although it seems to me from what I have heard that they are the best of all, with the exception of those of al-Battānī’: Concordat similiter in isto secundo Habraham Benbthegar in canonibus suis qui de hebreo nondum sunt translati, licet ut michi videtur, per ea que audivi sint meliores omnibus aliis excepto Albategni (Marie-Madeleine Saby 1987, p. 237).

john of lign èr es: iberia n a stronomy settles in  pa ris The Tabule magne and their canons

This second set was compiled shortly after the Tables of 1322 and was dedicated to Robert the Lombard in 1325,12 together with two treatises on astronomical instruments, the saphea and the equatorium. The canons associated with the Tabule magne begin Multiplicis philosophie variis radiis. The text, not hitherto edited, mentions some outstanding astronomers, none of whom were contemporaries, or even Alfonso himself.13 As was the case in his Tables of 1322, John of Lignères uses signs of 30º and computes his geographically dependent tables for the meridian of Paris. The tables in this set can be grouped into six categories.14 (1) The table for the positions of the apogees of the Sun and the planets displays entries from 1320 to 1520 at twenty-year intervals, as well as their motions per year. This table is frequently found in manuscripts separately from the rest of tables in this set. (2) The tables for mean syzygies give entries for the first syzygy of years from 1321 to 1609 at intervals of twenty-four years. They are computed for the meridian of Paris with the years beginning in January; this was a systematic practice followed by John of Lignères in his Tables of 1322 on which he depended for his new tables. (3) The mean motions of the Sun, the Moon, and the planets are presented in separate tables, in collected and expanded years, and at twenty-year intervals, up to 2,000 years. (4) The table for the solar equation, with a maximum value of 2;10º at arguments 92º–94º is the same as that used by John of Lignères in his previous set for 1322. (5) The true longitude of the Moon is presented in a double argument table of 1800 entries. The entries represent the increment in longitude of the Moon to be added to its mean longitude at the preceding mean conjunction. Similar tables for the same purpose had previously been compiled by John Vimond and John of Murs. (6) The planetary equations are also displayed as double argument tables, where the arguments are the mean centre and mean anomaly, both at intervals of 6º. These compact double argument tables are probably the most outstanding feature of the Tabule magne, and of John of Lignères’ production as a table-maker. This format, however, was not new in Latin astronomy, but John of Lignères was the first to use it systematically for the equations of the planets. The canons also explain the use of a division table for finding the time from mean to true syzygy for each integer of the hourly velocity in elongation from 27 to 33 minutes per hour. For a list of twenty manuscripts containing the tables (whether totally or partially) and/or the canons, we invite the reader to consult J. Chabás, Computational Astronomy, pp. 205–206.

12 Roberto (di) Bardi (1290–1349), a member of a wealthy family in Florence, Tuscany, studied theology at the University of Paris. Although he is almost exclusively referred to as Dean of Glasgow (1318), according to the Dizionario Biografico degli Italiani (online) he was later Dean of Verdun (1323), Dean of Notre Dame de Paris (1335), and Chancellor at the University of Paris. In astrology, he is known as the author of a treatise in defence of astrological interrogations beginning Quesitum fuit utrum per interrogationes astronomicas, extant in two manuscripts: one in Brussels (Bibliothèque Royale, 926–40, 215r–221v) and the other in the Vatican (Vat. lat. 4275, 29r–34v). 13 Matthieu Husson, ‘Les domaines d’application des mathématiques dans la première moitié du quatorzième siècle’ (Unpublished doctoral thesis, Ecole Pratique des Hautes Etudes, 2007). 14 For a detailed description of the Tabule magne, see José Chabás, Computational Astronomy in the Middle Ages (Madrid: Consejo Superior de Investigaciones Científicas, 2019), pp. 199–206.

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Although no edition of the Tabule magne exists for the moment, this overview allows us to have a clear idea of its contents. It focuses on two topics: the computation of planetary positions by means of tables for the apogees, mean motions, and equations, and the prediction of the time of eclipses using the tables for syzygies. The tables associated with the first topic nicely complement those of the Tables of 1322, where this subject is not addressed. As for syzygies and the computation of eclipses, the three tables for mean syzygies of the Tables of 1322 were reused in the Tabule magne. Canons to the Tables of Alfonso

John of Lignères also wrote a text in sixteen chapters to accompany the Tables of Alfonso, as they are called in some manuscripts in which this text is found: Canones Tabularum Alfoncii, ordinati per magistrum Ioannem de Lineriis. The text begins Quia ad inveniendum loca planetarum and it contains about 4,600 words.15 According to Poulle, these canons by John of Lignères were written between 1322, the date he composed his set of tables under examination here, and 1327, when John of Saxony wrote his own canons to the Alfonsine Tables beginning Tempus est mensura.16 We note that John of Saxony, a disciple of John of Lignères, wrote longer and more complete canons shortly after his master. The Quia ad inveniendum was thus the first text compiled in Paris to explain the use of the set associated with King Alfonso. The first three chapters are devoted to the conversion of dates in different calendars. In particular, there is an explicit reference to some ‘Tables of the Arabs’ and the beginning of their era. Various tables are mentioned providing the number of days in collected and expanded years and months (whether in a leap year or not), as well as a table for the difference in the number of days between different eras to convert a date from one to another. Already in the introduction preceding the three chapters on the conversion of dates it is made clear that the entries in the tables are given in sexagesimal notation. These two elements, the sexagesimal counting of days and the variety of tables for different eras, are indeed found in the Castilian Alfonsine Tables, as explained in detail in the first chapters of these canons.17 Chapter 4 of the canons by John of Lignères is devoted to the determination of the day of the week in different calendars, and a table for the day of the week at the beginning of the different eras is explicitly mentioned (Tabula radicum notarum anni). A table with the same title is found in the editio princeps of the Parisian Alfonsine Tables (Ratdolt 1483, c7r). It displays entries for ten eras, from the Flood to the era of Alfonso, also including the eras of Nabonassar, Philippus, Alexander, the Incarnation, and the Hijra. This is a table that would seem to have been of little use in Paris at the beginning of the fourteenth century. It no doubt originated in a much more multicultural environment, and indeed

15 See Alena Hadravová and Petr Hadrava, ‘John of Lignères, Quia ad inveniendum loca planetarum: edition and translation’, in Richard L. Kremer, Matthieu Husson, and José Chabás (eds), Alfonsine Astronomy: The Written Record (Turnhout: Brepols, 2022), 257–302. 16 Emmanuel Poulle, Dictionary of Scientific Biography, p. 124. 17 For the edition of the Castilian Alfonsine Tables, see José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo, pp. 20–35, 143–151.

john of lign èr es: iberia n a stronomy settles in  pa ris

the canons to the Castilian Alfonsine Tables have two chapters on the day of the week associated with all these calendars.18 The following three chapters deal with the conversion of hours and minutes into fractions of a day and vice versa, clearly indicating the determination to extend the use of base 60 for counting days to submultiples of a day. Chapter 8 is concerned with the mean motion of the planets. This long chapter offers a description of a table for mean motion, with one column for the argument, from one to sixty, and ‘at least eight other columns’, together with a lengthy explanation on how to enter the table for each of the components of a time given in sexagesimal form. We are also told that the resulting signs ‘have the value of two [zodiacal] signs’, that is, 60º. This description corresponds to a compact and single table for each object, consisting of sixty successive multiples of a basic parameter for each case. It differs from the description given in the canons to the Castilian Alfonsine Tables, based on the use of sub-tables for collected and expanded years, months, days, and fractions of a day.19 The text mentions tables for the mean motions of the Sun, the planets, the lunar node, the anomalies of the Moon, Venus, and Mercury, the argument of lunar latitude, the eighth sphere, the apogees, and the elongation, in that order. The canons to the Castilian Alfonsine Tables do not specify the quantities appearing in the mean motion tables.20 Significantly, the tables for the radices of all these quantities are valid for the meridian of Toledo. Mean syzygies are the subject of Chapter 9. The tables for conjunctions and oppositions described have entries for four elements: the time of the event, the mean motion of the Sun and the Moon, the mean lunar anomaly, and the mean argument of lunar latitude. The same four quantities are used in the corresponding table described in the canons to the Castilian Alfonsine Tables.21 Another critical piece of information is given here: the year begins in January. Chapter 10 explains how to find the longitude of the planetary apogees, and it refers specifically to two tables for the mean motion and equation of the eighth sphere. No sign of this model with two components for trepidation is found in the extant canons to the Castilian Alfonsine Tables, although in his Expositio (1321) John of Murs ascribed it to the Tables of Alfonso. Apparently, the source of this claim made by John of Murs, and here echoed by John of Lignères, differed in some respects from the Castilian canons.22 The determination of the true positions of the two luminaries, the lunar node, and the five planets is presented in Chapters 12–15. Tables for the equation of the Sun, the Moon, and the five planets are invoked, and rules for their use are given. The procedures for determining the true positions agree with those explained in the canons to the Alfonsine Tables except for the true position of the Moon. It is worth noting that in Chapter 15, after explaining how to determine the true positions of Venus and Mercury, John of Lignères indicates that ‘in this way, anyone can compile an almanac’, with the following frequencies

18 19 20 21 22

See José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo, Chapters 6 and 7, pp. 25–28, 146–147. See José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo, Chapter 15, pp. 36–37, 151­–152. See José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo, Chapter 13, pp. 35–36. See José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo, Chapter 30, pp. 56–58, 188–189. For a detailed explanation, see José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo, pp. 256–266.

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for the entries: daily for the Sun and the Moon, every five days for the inferior planets, and every ten days for the superior planets. And indeed, these are the frequencies he used in his own almanac for the planets (see below). Finally, Chapter 16 concerns the possibility of an eclipse. For both kinds of eclipses, the range is between –12º and +12º of the argument of lunar latitude. This very crude estimate for the limits of an eclipse is the only specific numerical data given in the text. We note that there is no mention of any table for computing eclipses or determining their circumstances. These are not the only tables that are missing. The text Quia ad inveniendum focuses on general topics regarding Alfonsine astronomy but does not deal with such topics as solar declination, the lunar and planetary latitudes, equation of time, retrogradation and stations of the planets, parallax, and velocities of the luminaries and the planets. In contrast, these topics are found in major sets such as the Toledan Tables or those mentioned in the Castilian Alfonsine Tables, as well in the set of Tables of 1322 by John of Lignères. It is thus at the very least odd that in his text on the Tables of Alfonso he did not even mention any of these other tables. We are led to believe that perhaps this text is an unfinished work or, if we also take into account the simplicity of the explanations and the overall absence of numerical data, maybe it was intended as an introductory course on astronomy, to teach students the basic rules for using astronomical tables. Theory of the planets

The Theorica planetarum by John of Lignères, dated 1335 in Paris, BnF, lat. 7281, begins Spera concentrica vel circulus dicitur. This treatise is found on ff. 165r–172r in this splendid anthology of texts written by early Alfonsine astronomers in Paris: John of Murs, John of Lignères, John of Saxony, and John of Genoa, among others. We note, however, that after f. 165, numbering restarts at 164. In this case, the incipit is Prima pars continet descriptionem et numerum. This interesting treatise has not yet been the subject of an edition or a scholarly study. Kalendarium

Madrid, Biblioteca Nacional, 9288, 10v–14v, contains the unique copy of a hitherto unknown work by John of Lignères. It consists of a short text headed Canon supra kalendarium magistri Johannes de Lineriis (10v) and a table (11r–14v) for all mean conjunctions in the period from 1321 to 1396. The time of the conjunctions is given in months, days, hours, and minutes, beginning in January 1321, and the days begin at noon. All entries were computed for Paris. The characteristics and the dates of this 76-year period point to the conclusion that the table was indeed computed by John of Lignères or derived from tables compiled by him, building on those by John of Murs by applying a correction of 0;48h to account for the time difference between Toledo and Paris.23

23 José Chabás, ‘New texts and tables attributed to John of Lignères: context and analysis’ in Richard L. Kremer, Matthieu Husson, and José Chabás (eds), Alfonsine Astronomy: The Written Record (Turnhout: Brepols, 2022), 303–316.

john of lign èr es: iberia n a stronomy settles in  pa ris Almanac for the planets

As anticipated in Chapter 15 of his Quia ad inveniendum, John of Lignères compiled an almanac, that is, a collection of tables giving the true positions of the five planets at intervals of a few days (ten for Saturn, Jupiter, and Mars, and five for Venus and Mercury) for periods of return to their initial position (59y for Saturn, 83y for Jupiter, 79y for Mars, 8y for Venus, and 46y for Mercury).24 The entries, rounded to the nearest degree, were computed for the meridian of Paris using Alfonsine parameters. The pattern in the almanac closely follows that in the Almanac of Azarquiel and the Almanac of Jacob ben Makhir, compiled around 1300 and based on the Toledan Tables.25 We have located John’s almanac in five manuscripts: Munich, Universitätsbibliothek, F 593, 12r–21r; Paris, Bibliothèque nationale de France, Mélanges Colbert 60, 26r–32r (tables); Philadelphia, Free Library, Lewis E.3, 3r–10r (tables, incomplete), 10r (canons), late fourteenth century; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 446, 219v (canons), 220r–227v (tables), early fifteenth century; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1446, 36r–47v (tables), late fourteenth / fifteenth century.

The tables go along with a short text that differs slightly in the manuscripts examined. However, they provide the same basic information: the epoch year is 1341, starting in January, and the entries are displayed in zodiacal signs of 30º. As far as we can determine, this was the first time Alfonsine parameters were used to compile an almanac. This was innovative, but in terms of precision it was a regression with respect to previous almanacs. As was the case for the canons to the Tables of Alfonso, John of Lignères’ disciple, John of Saxony, went further and computed a larger and more precise version. Following his master, John of Saxony computed for Paris and used signs of 30º. In contrast, he substantially modified the pattern and also included true positions of the Sun and the Moon, paving the way to the compilation of ephemerides. Astronomical instruments

John of Lignères wrote treatises on two instruments, the saphea and the equatorium, which he dedicated to Robert, Dean of Glasgow in 1325, together with the Tabule magne.26

24 See José Chabás and Bernard R. Goldstein, ‘The Master and the Disciple: The Almanac of John of Lignères and the Ephemerides of John of Saxony’, Journal for the History of Astronomy, 50 (Cambridge: Science history pub., 2019a), pp. 82–96. 25 For the Almanac of Azarquiel, see José M. Millás, Estudios sobre Azarquiel (Madrid–Granada: Editorial Maestre, 1943–1950); for the Almanac of Jacob ben Makhir, see Chabás and Goldstein, ‘The Almanac of Jacob ben Makhir’ in Matthieu Husson, Clemency Montelle and Benno van Dalen (eds), Editing and Analysing Numerical Tables: Towards a Digital Information System for the History of Astral Sciences (Turnhout: Brepols), pp. 53–68. 26 Emmanuel Poulle, Équatoires et horlogerie planétaire du XIIe au XVIe siècle (Geneva: Droz. Paris: H. Champion, 1980); see esp. p. 213.

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The saphea is an astronomical instrument based on an analogous universal instrument, al-ṣafīḥa, developed by Azarquiel.27 The text of this late-eleventh-century Andalusian astronomer was translated from Arabic into Castilian and Latin by the astronomers at the court of King Alfonso in Toledo in the second half of the thirteenth century and into Hebrew by Jacob ben Makhir. The text by John of Lignères’ saphea is preserved in four manuscripts: Erfurt, Universitätsbibliothek, CA 4º 355, 73r–81v; Erfurt, Universitätsbibliothek, CA 4º 366, 40r–49r; Madrid, Biblioteca Nacional, 9288, 100r–105v; Paris, Bibliothèque nationale de France, lat. 7295, 2r–14r.

In the Madrid manuscript, John of Lignères’ text, spanning thirty-four chapters, begins Descriptiones que sunt in facie instrumenti notificate. Limbus seu circulus exterior and ends Expliciunt canons magistri Iohannis de Lineriis supra quoddam instrumentum mirabile, cuius anima cum Christo in eternum possideat sempiterna. Amen. John of Lignères authored two treatises on the equatorium. One is an adaptation of an instrument by Campanus de Novara for computing the positions of the planets. It is associated with a text sometimes bearing the title Abbreviatio Campani de Novaria equatorii, and it begins Quia nobilissima scientia astronomie and ends ad cetera sunt tabule.28 It should be noted that the same incipit appears in a different treatise, which is associated with the instrument called the Oxford equatorium, ending cum instrumentis prius dictis. John of Lignères’ text is extant in at least the following manuscripts: Brussels, Bibliothèque Royale, 10117–26, 142v–146v; Cracow, Biblioteka Jagiellońska, 555, 21r–24r; Cracow, Biblioteka Jagiellońska, 557, 11v–120v; Oxford, Bodleian Library, Digby 168, 65v–66r; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1375 8v–10v.

In his second treatise, the main goal is to determine the planetary equations.29 It consists of two parts: one for the construction of the instrument and another for its usage. The first part begins Fiat primo regula and the second Primo linea recta. The treatise is found in the following manuscripts: Oxford, Bodleian Library, Digby 228, 53v–54v; Vatican, Biblioteca Apostolica Vaticana, Urb. lat. 1399, 16r–21r. 27 Roser Puig, Los tratados de uso y construcción de la azafea de Azarquiel (Madrid: Instituto Hispano-Árabe de Cultura, 1987). 28 For an edition, see Derek. J. Price, The Equatorie of the Planetis (Cambridge: The University Press, 1955), pp. 188–196. 29 Matthieu Husson, ‘Compositio equatorium planetarum: construire un instrument de calcul astronomique au 14e siècle, le second équatoire de Jean de Lignières’, in Danièle Jacquart, Catherine Verna, Joël Chandelier, and Nicolas Weill-Parot (eds), Théorie et pratique: une intersection pertinente, hommage à Guy Beaujouan (Presses Universitaires de Vincennes, 2014).

john of lign èr es: iberia n a stronomy settles in  pa ris Arithmetic

Although not specifically addressing astronomical issues, the text by John of Lignères called Algorismus de minutiis or Algorismus minutiarum deals with sexagesimal fractions, which are frequently used in astronomy. This work saw great success, and it is extant in at least 30 manuscripts. The usual incipit is Modum representationis minutiarum vulgarium et phisicarum, but it varies slightly from one manuscript to another.30 This work went into print in Padua in 1483 by Matthaeus Cerdonis, where it was bound together with a text, also on arithmetic, by the Paduan astronomer Prosdocimo de’ Beldomandi, Algorithmus (1r–18v). Prosdocimo’s text precedes John of Lignères’ (19r–27v), beginning Representationis minutiarum vulgarium et phisicarum.31 2. Works attributed to John of Lignères As can be seen from the list of his works, John of Lignères was a prolific author addressing a great variety of topics on mathematical astronomy, ranging from the compilation of tables to the composition of texts regarding theory and instruments, as well as canons to tables. We also note that his work is not concerned with astrological matters. If we are to judge from the number of copies of his Tables for 1322 and the Tabule magne preserved in manuscripts, he was an authoritative and prestigious voice in the field of astronomy. Clearly, his sets of tables were especially appreciated. Therefore, it comes as no surprise that his name was used as auctoritas in order to prove the reliability and authority of a text or a table to add value to it.32 It is thus not uncommon to find texts in miscellaneous manuscripts, or in manuscript catalogues, that are attributed to him either by a copyist, a manuscript owner, or even a modern cataloguer. We have found several examples, which are described below, and we are convinced that more remain to be uncovered. 1. De significationibus planetarum in singulis domibus

Basel, Universitätsbibliothek, F II 10, is a thick fifteenth-century composite manuscript mostly devoted to astrology and medicine. In particular, it contains a short text attributed to John of Lignères on 163rb–164vb, with the title, Canon magistri Johannis de Lineriis de significationibus planetarum in singulis domibus. The incipit reads Saturnus cum fuerit in ascendente significat mortem, and the explicit is Haec sunt significationes capitis et caude draconis in 12 domibus planetarum si fuerit in bono significat bonum. Si in malo malum etc. This short text deals with the attributes of the planets, and no reference to tables for the celestial houses or computation of the cusps is found.33 Apparently, it is a version of Chapter 5 of De significationibus septem planetarum,

30 See Lynn Thorndike and Pearl Kibre, A Catalogue of Incipits of Mediaeval Scientific Writings in Latin (London: Mediaeval Academy of America, 1963), 878. For a critical edition, see Hubertus L. L. Busard, Het rekenen met breuken in de middleleeuwen, in het bijzonder bii Johannes de Lineriis (Brussels: Vlaamse Academie, 1968). 31 ISTC ib00299000: for a digital copy, see https://archive.org/details/3999345/page/n. 39/mode/2up. 32 José Chabás, ‘New texts and tables attributed to John of Lignères’. 33 A similar work beginning De significationibus planetarum, also attributed to John of Lignères, is preserved in Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1188.

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an astrological work attributed to Messahala, and sometimes to Jirjis or Gergis and even to Alcabitius.34 In any case, this text does not seem to have been composed by John of Lignères. 2. Short canons and tables for syzygies

Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1390, 86r–92r and Pal. lat. 1445, 219r–221v contain two copies of the same canons and tables.35 In both cases the incipit is Volens invenire medios motus planetarum, and the explicits refer to John of Lignères: Explicit canon Iohannis de Lineriis in tabulas sequentes (Pal. lat. 1390) and Explicit canon Iohannis de Lineriis in tabulis precedentibus brevis et utile valde (Pal. lat. 1445). The canons deal with mean motions, mean conjunctions, and apogees of the planets. The entries in the corresponding tables are computed for Toledo, not Paris, and the years begin in March, not January. These two features are not characteristic of John of Lignères’ other works. If we add that the interval of twenty-eight years in the list of conjunctions differs from the one commonly used by John of Lignères and the span of the list (1385–1556), we are led to the conclusion that he had nothing to do with the composition of this particular work, despite the mention of his name in the explicit. 3. Tables for the motions of all planets

Erfurt, Universitätsbibliothek, CA 2º 388, 1r–35r is a fifteenth-century manuscript containing a set of tables with the general heading Incipiunt tabule Iohannis de Lineriis de loci motuum omnium planetarum.36 One would expect to find the Tables of 1322, but this is not the case. Alternatively, we find tables for the mean motions of the planets and the two luminaries as well as tables for their equations. For each celestial body, the table for the equations follows those for the mean motions. The tables for the equations are indeed those in John of Lignères’ Tabulae magne, and the tables for the mean motions are presented as twelve tables; one for each month of the year, where the entries are the accumulated daily mean motions and mean anomalies. The entries are easily obtained by successive additions of the corresponding parameter for one day. A list of radices is also supplied. As far as we know, there are no grounds to conclude that these tables of accumulated mean motions were compiled by John of Lignères: they could have been compiled by almost anyone. In any case, the mean motions tables compiled by John of Lignères in his other works display a very different pattern. 4. De aspectibus

In his descriptive catalogue of the manuscripts preserved at the Amplonian Library at Erfurt, Schum attributed a treatise on the aspects extant in Universitätsbibliothek, CA 2º 34 Francis J. Carmody, Arabic Astronomical and Astrological Sciences in Latin Translation, a Critical Bibliography (Berkeley: University of California Press, 1956), pp. 29–30. 35 José Chabás, ‘New texts and tables attributed to John of Lignères’. We have also found a copy of this text in Frankfurt, Stadt- und Universitätsbibliothek, Barth. 134, 158v–162v. 36 The general heading was written by the Flemish scholar John of Wasia (d. 1395). For a description of the manuscript, see Wilhelm Schum, Beschreibendes Verzechnis der Amplonianischen Handschriften-Sammlung zu Erfurt (Berlin: Weidman, 1887), p. 273.

john of lign èr es: iberia n a stronomy settles in  pa ris

395, 40r–43v, beginning Tempus quarti aspectus solis et lune invenire, to John of Lignères. We are also told that the end of it reads facies secundum doctrinam magistri Iohannis de Lineriis, a quo scienciam mean habeo.37 Contrary to Schum’s suggestion, this corresponds to a work by John of Saxony. The above incipit is the beginning of Chapter 24 of John of Saxony’s canons, Tempus est mensura (Ratdolt 1483, b2r; Poulle 1984, p. 90), and the ending reproduces the last words of the chapter on lunar eclipses in the so-called membra adiuncta (Ratdolt 1483, b7v). It is worth noting that these additional chapters on eclipses, often found just after the 27 chapters traditionally included in the Tempus est mensura, are assigned to John of Saxony in Brussels, Bibliothèque Royale, 1022–47, 39v, and they were ordinati parisius per magistrum Iohannes de Saxonia anno domini 1330. This suggests that the membra adiuncta on eclipses were written by John of Saxony shortly after the date appearing in his Tempus est mensura ( July 1327). To sum up, the text extant in Erfurt, Universitätsbibliothek, CA 2º 395, 40r–43v was not authored by John of Lignères. 5. Star list

Bernkastel-Kues, Cusanusstiftsbibliothek, 211, 22r–v, displays a list of sixty stars under the title Huiusmodi stelle verificate sunt per magistrum Johannem de Lineriis anno domino 1240 (sic) prima die januarii anno imperfecto (see Figure 1). The explicit repeats this information, but it refers (correctly) to 1340. For each star we are given its name, longitude, latitude, magnitude, and an indication of the hemisphere in which it is located. The list only contains stars of first and second magnitudes. The first star listed belongs to the Ursa Minor constellation: Meridiana duarum que sunt in latere sequenti and the associated entries are 4s 6;11º (longitude), 72;50º (northern latitude), and 2 (magnitude). In Ptolemy’s catalogue, the longitude of this star is given as Cnc 17;10º. Therefore, the increase in longitude since Ptolemy’s catalogue is 19;1º. The same value is obtained for the rest of the stars in this list. Now, such an increase due to precession does not correspond to the year 1340, but to the year 1434, and this is indeed a date mentioned at the end of the list. Therefore, the entries in this list do not correspond to John of Lignères’ time. Moreover, the same stars, in the same order, are found in a list where their longitudes are increased by 18;0º compared to Ptolemy’s, a value of precession corresponding to 1340, and the longitude of the star Meridiana duarum is 4s 5;10º. The author of this star list is Swabian astronomer Heinrich Selder (fl. 1370’s), and it is extant in a few manuscripts.38 We conclude that someone in the first half of the fifteenth century adapted a star list for 1340 (that of Selder) to his own time by adding 1;1º to the longitudes of the stars, and, ignorant of the author’s name, added that of the well-known astronomer John of Lignères, to whom he attributed authorship. This is a new example of the widespread fame of the Parisian astronomer.

37 This text is also extant in Cracow, BJ 715, 67v–68r, and Rostock, UB, math.-phys. 1, 76r. 38 On Selder, see C. Philipp E. Nothaft, ‘Vanitas vanitatum et super omnia vanitas: The Astronomer Heinrich Selder and A Newly Discovered Fourteenth-Century Critique of Astrology’, Erudition and the Republic of Letters, 1 (Leiden-Boston: Brill, 2016), pp. 261–304.

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Figure 1. Star list in Bernkastel-Kues, 211, 22v.

john of lign èr es: iberia n a stronomy settles in  pa ris 6. Instruments

At least two instruments are attributed to John of Lignères. In his Recueil des plus celebres astrologues, Simon de Phares assigns an instrument called ‘directoire’ to John of Lignères, which is explained in a text beginning Accipe tabulam planam rotundam, and he added that this occurred ‘l’an mil IIIcXX’.39 The second is an armillary instrument described in a text beginning Trianguli equilateri ex tribus quartis preserved in Vatican, Biblioteca Apostolica, Urb. lat. 1399, 2r–15r. On f. 1v, the first item in the index of the codex reads, In hoc codice continentur instrumentum armillare Iohannis de Lineriis. 7. Canon primi mobilis

Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1412, 30r–35r, contains a text beginning Nunc autem de celestium diversitate. According to the catalogue of Palatine manuscripts at the Vatican Library (Schuba, p. 191), it is identified as a ‘Canon primi mobilis super tabulas’ and attributed to John of Lignères. Furthermore, the explicit of the text, on f. 35r, reads as follows: Explicit canon primi mobilis super tabulas Parisius 17ª die julii (…) anno domini 1454, ordinati per magistrum Johannem de Lineriis. The attribution of this text to John of Lignères is plausible, for it is followed by two other canons by him, the Cuiuslibet (35v–46v) and the Priores (ff. 46v–71v), and it is preceded by the canons to the Parisian Alfonsine Tables by John of Saxony (ff. 10r–24v) and canons on eclipses (ff. 25r–29v), which are usually considered as additions to the Alfonsine set (see Ratdolt 1483, b4r–b7v), here explicitly ascribed to John of Saxony. Moreover, Thorndike and Kibre (col. 961) list the text under the given incipit as a work by John of Lignères. The words Nunc autem de celestium diversitate actually correspond to the beginning of a sentence in the most popular version (Cb) of the Canons to the Toledan Tables. A comparison of the two texts reveals that the work attributed to John of Lignères in this manuscript corresponds to canons 51b–126 of the Toledan Tables (see Pedersen 2002, pp. 402–430). It is worth noting that these canons, mainly on trigonometry, had their own independent life, for they are preserved as such in at least one other manuscript: Zürich, Zentralbibliothek, II.88, 275r–298v. The difficulty in identifying the text attributed to John of Lignères derives from the fact that the first sentence, Nunc autem de celestium diversitate, is not the beginning of any paragraph or chapter, and that the text to which it belongs and that precedes it is not extant in the manuscript. In any case, the text here called ‘canon primi mobilis’ was not written by John of Lignères, contrary to what was claimed by the fifteenth-century scribe and by modern scholars.

39 Jean-Patrice Boudet, Le recueil des plus célèbres astrologues de Simon de Phares (Paris: Editions pour la société de l’histoire de France, 1997), pp. 469–470.

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3. The set of tables A set of astronomical tables is often preserved in miscellaneous manuscripts under the general title Incipiunt tabule magistri Iohannis de Lineriis. This set, which we have called the Tables of 1322 by John of Lignères, is preserved in many manuscripts frequently following copies of the Parisian Alfonsine Tables, and it is restricted to tables for chronology, radices, mean motions, and equations of the Sun, the Moon, and the planets. We find this set, whether complete or not, in 46 manuscripts -listed below- and we are convinced that this list will continue to grow as more codices are examined. *Basel, Universitätsbibliothek, F II 7, 62r–77v (fifteenth century); Bernkastel-Kues, Cusanusstiftsbibliothek, 210, 117v–123v; *Bernkastel-Kues, Cusanusstiftsbibliothek, 212, 74r–93r (fifteenth century); Bernkastel-Kues, Cusanusstiftsbibliothek, 213, 45r–60v; Bonn, Universitäts- und Landesbibliothek, S 498, 40v–60v; *Cologne, Historisches Archiv der Stadt, W* 178, 1r–18r (fifteenth century); Cracow, Biblioteka Jagiellońska, 459, 29r–37r; Cracow, Biblioteka Jagiellońska, 546, 23r–29v; Cracow, Biblioteka Jagiellońska, 547, 48v–55r; Cracow, Biblioteka Jagiellońska, 549, 23r–29v; Cracow, Biblioteka Jagiellońska, 550, 24v–52r; *Cracow, Biblioteka Jagiellońska, 551, 74v–90r, 94v–95r (fourteenth century); Cracow, Biblioteka Jagiellońska, 553, 147r–165v; Cracow, Biblioteka Jagiellońska, 602, 60r–75v; Cracow, Biblioteka Jagiellońska, 610, 317r–338r; Cracow, Biblioteka Jagiellońska, 613, 15r–22r, 59r–62v, 118r–137r; Cracow, Biblioteka Jagiellońska, 618, 18v, 23v–24r, 47r–64r; *Erfurt, Universitätsbibliothek, CA 2º 377, 41v–46v (fourteenth century); Erfurt, Universitätsbibliothek, CA 2º 384, 26r–45v; *Florence, Biblioteca Medicea Laurenziana, San Marco 185, 102r–117v (fourteenth century); Leipzig, Universitätsbibliothek, 1484, 43r–58v; *London, British Library, Egerton 889, 31r–52v (fifteenth century); Lüneburg, Ratsbücherei, Miscell. D 2°11, 26r–45r; Lüneburg, Ratsbücherei, Miscell. D 2°13, 36r–57v; *Madrid, Biblioteca Nacional, 10002, 23r–47r (fifteenth century); Moscow, Russian State Library, F 68 N 450, 36r–57r; Munich, Bayerische Staatsbibliothek, Clm 5640, 117r–132v; *Oxford, Bodleian Library, Can. Misc. 27, 78v–104r (fifteenth century); Oxford, Bodleian Library, Can. Misc. 499, 124r–144r; *Paris, Bibliothèque nationale de France, lat. 7282, 113r–128v (fifteenth century); Paris, Bibliothèque nationale de France, lat. 7285, 62r–v, 76r–83v, 115v–116r; *Paris, Bibliothèque nationale de France, lat. 7286C, 24v–28v, 48r–52r, 53v–55r, 56r (fourteenth century); *Paris, Bibliothèque nationale de France, lat. 7295A, 155r–171r (fifteenth century); Philadelphia, University of Pennsylvania, LJS 174, 27r–38r, 40v–42r; Prague, National Library, X A 23, 22v–40v;

john of lign èr es: iberia n a stronomy settles in  pa ris

Rome, Biblioteca Casanatense, MS 653, 34v–37r, 40r–64r; Rome, Osservatorio astronomico, III C 14, 15or–159v; Rostock, Universitätsbibliothek, math.-phys.1, 135r–144v; Vatican, Biblioteca Apostolica Vaticana, Ottob. 1826, 130r–140r; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1367, 27v–40v, 61r–62r, 70v–77r, 90r–v; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1373, 97v–108r, 123r; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1374, 26r–46v; Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1376, 34v–49r, 51v–56r; *Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1412, 35v–46v, 95r–101v, 109r–114r, 117r–120r (fifteenth century); Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1413, 46r–51r, 55r–58r; Venice, Biblioteca Nazionale Marciana, lat. VI, 29 (2526), 51r–56r, 60r–65v, 75r–76v; Wolfenbüttel, Herzog August Bibliothek, 36.21 Aug. 2º (2401), 304r–311r; Zürich, Stadtbibliothek, MS 361, 244r–273r.

To identify the components of the set, we used the thirteen manuscripts marked here with an asterisk, covering both the fourteenth and the fifteenth centuries and extant in a variety of European libraries. The result is a set of thirty-two tables.40 Not all manuscripts contain all the tables in the set. In Figure 2 we list the tables extant in each of the manuscripts closely examined in the edition of the Tables of 1322 by John of Lignères. A major feature is that none of them contains all thirty-two tables. In three of them (MSS Cracow 551, Egerton 889, and Oxford 27), only one table is missing; in all cases, the missing tables are different. We also note that in eleven manuscripts a maximum of three tables are missing, indicating that this set was considered by their users as an entity from which very few tables could be detached. If we now consider which tables are systematically found in the various manuscripts examined here, we find a total of twelve, and there are another thirteen tables that are missing only from one or two other manuscripts. The table that appears least frequently in the manuscripts under consideration, in just eight of them to be precise, is Table 20, a table for the velocities of the luminaries. This is probably due to the fact that John of Lignères’ set contains two other tables for the same purpose. Not all manuscripts have an incipit and an explicit for this set, but those that do are useful guides to identifying the tables belonging to it. The tables are usually arranged in the same order and provide a stable and coherent sequence of tables addressing the basic problems in mathematical astronomy faced by medieval astronomers. The set usually begins with a table representing the sine function at intervals of half a degree of the argument. Not all manuscripts have the same tables, and it is common to find additional tables not featured in most of the manuscripts, sometimes making the definition of the set a complicated issue. The thirty-two tables on which the edition of John of Lignères’ Tables of 1322 is based appear in nearly all manuscripts we have closely reviewed. The tables can be grouped in various categories: (i) Trigonometry and spherical astronomy,

40 José Chabás, Computational Astronomy in the Middle Ages 2019, pp. 175–198; José Chabás and Marie-Madeleine Saby, ‘Editing the Tables of 1322 by John of Lignères’.

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Tables 1–6; (ii) Equation of time, Table 7; (iii) Lunar and planetary latitudes, Tables 8 and 9; (iv) Planetary velocities, Table 10; (v) Stations, retrogradations, and planetary phases, Tables 11–13; (vi) Mean syzygies, Tables 14–16; (vii) Mean motion in elongation, Table 17; (viii) Velocities, Tables 18–21; (ix) Parallax, Tables 22–23; and (x) Eclipses, Tables 24–32. We note that there are no tables for chronology and conversion of dates, precession, mean motions (except one single table for elongation), equations (except for the Sun and the Moon, both integrated in Table 19), fixed stars, geographical coordinates, or astrology. In contrast to the set of Parisian Alfonsine Tables, John of Lignères’ Tables of 1322 do not reproduce the model of Arabic zijes. In these thirty-two tables, we have counted more than 18,200 sexagesimal numbers, including the entries and the arguments. This means that the Tables of 1322 compose a major but small set, as compared to the 51,000 sexagesimal numbers in the standard version of the Parisian Alfonsine Tables (Venice: Ratdolt, 1483) or the 315,000 numbers in the Tables for the Planets by Giovanni Bianchini (Venice: Bevilaqua, 1495).41 As deduced from the commentaries to the individual tables, a major conclusion is that most of the tables in this set by John of Lignères were not compiled by him. As was the case for most medieval astronomers, he reproduced tables according to the Ptolemaic tradition available in the Iberian Peninsula, in particular the Toledan Tables. Nevertheless, for his own set, John of Lignères computed a few tables by adapting already existing tables to the meridian of Paris and maintaining their format. In many respects, the Tables of 1322 can be considered as an intermediary set from the Toledan Tables to the Parisian Alfonsine Tables, in the sense that John of Lignères borrowed a great number of astronomical tables from his predecessors of the Iberian Peninsula that in turn were later transferred to the standard version of the Parisian Alfonsine Tables, thus providing a new example of continuity in the transmission of knowledge. 4. An edition Criteria for the choice of manuscripts

Two major criteria have been used to select a few manuscripts from the many containing the Tables of 1322 by John of Lignères for this edition: the completeness of the set and the quality of the individual copies. Even though we have not made any attempt to list all extant manuscripts in which this set is totally or partially extant, the numerous manuscripts known to us (46 at the time of writing) show some variability in their composition. As noted above (see Figure 2), there are more than a dozen copies with almost all thirty-two tables in the set, all following roughly the same order. The second criterion concerns the quality of the individual copies. Quality is indeed a vast topic, and it embraces several aspects. One is readability. Some manuscripts are 41 José Chabás and Bernard R. Goldstein, The Astronomical Tables of Giovanni Bianchini (Leiden-Boston, 2009), p. 11.

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poorly preserved, and thus sometimes difficult, or plainly impossible, to read. In others the copyist’s handwriting poses a problem and is subject to interpretation, leaving the reading uncertain. It is also frequent to find ‘corrected’ entries, resulting in a stain or an illegible number, although fortunately it is often the case that corrections were displayed as marginal annotations. Another aspect is the presentation of the table. A neat and well-spaced layout is far more readable than a compact series of numbers, and probably contains fewer scribal errors too. We note, for example, that in Erfurt, Universitätbibliothek, CA 2° 377, the copyist managed to include eleven sub-tables (from six tables) on a single folio (f. 45r). Other aspects associated with the quality of each individual copy, such as paratexts, annotations, corrections, and the presence of an incipit and an explicit, have also been taken into account. The milieu, mainly in terms of time and place, has also played a role, albeit secondary, in the selection of manuscripts for the edition. Furthermore, special attention has been given to fourteenth-century manuscripts, which are considered to carry versions of this set that more closely resemble the original one. Among the selected manuscripts are Erfurt, Universitätbibliothek, CA 2° 377, which is the earliest extant compilation of the tables (c. 1323–24), and Paris, BnF, lat. 7286C (second half of the fourteenth century). Nevertheless, we note that the oldest copies of the Tables of 1322, or those written close to the source, are not necessarily the best copies to use for such an edition.42 Description of the manuscripts

For the edition of the Tables of 1322 by John of Lignères, we used ten of the manuscripts listed above. As a base manuscript we mainly relied on the Basel manuscript, and, whenever this was not possible, we referred to the Cracow (BJ 551) and the London (Egerton 889) manuscripts. For each table, we compared the base manuscript with four others, notably manuscripts Erfurt, UB, CA 2º 377; Paris, BnF, 7286C; and Vatican, Pal. lat. 1374. Occasionally, four other manuscripts were used: Cologne, Historisches Archiv der Stadt, W* 178; Oxford, Bodleian Library, Can. Misc. 27; Vatican, Pal. lat. 1412; and Vatican, Ottob. lat. 1826. Succinct descriptions follow. 1. Base manuscripts Basel, Universitätsbibliothek, F II 7, 62r–77v (fifteenth century, MS B)

This manuscript is dated 1432, and it was written by a single copyist, Henricus Amici, a doctor of medicine in Montpellier and city physician of Basel.43 The volume was bequeathed to

42 Additional information can be found in José Chabás and Marie-Madeleine Saby, ‘Editing the Tables of 1322 by John of Lignères’. 43 For a description of the codex, see Albert Bruckner et al., Katalog der datierten Handschriften in der Schweiz in lateinischer Schrift vom Anfang des Mittelalters bis 1550. Die Handschriften der Bibliotheken von Aarau, Appenzell und Basel, 6 vols (Dietikon-Zürich: Urs Graf, 1977–1991), I, p. 174, n. 483.

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the Carthusian monastery of Basel upon Amici’s death in 1451.44 The handwriting is neat and clear, and the manuscript is well organized and heavily annotated in the margins. It contains a selection of the most relevant astronomical texts and tables compiled in Paris in the fourteenth century. It opens with a copy of John of Saxony’s twenty-seven canons to the Parisian Alfonsine Tables, beginning Tempus est mensura, although the first chapter is not numbered. The text is followed by an incomplete set of Parisian Alfonsine Tables, where the mean motion tables display radices for Paris, Montpellier, and Liège. On ff. 36r–37r, one finds the table and canons by Nicholaus de Heybech of Erfurt for computing the time from mean to true syzygies.45 Then follow the two texts by John of Lignères associated with his Tables of 1322: a complete copy of the canons for the primum mobile, beginning Cuiuslibet arcus propositi (38r–46r) and thirty canons out of forty-six in Priores astrologi (ff. 46v–57v). We note that the manuscript displays the three figures on instruments in the first canons and the three for eclipses in the second. The Tables of 1322 fill in ff. 62r–77v. There are twenty-nine tables out of thirty-two and many of their entries have been correctly emended by another hand. Among them are several latitude-dependent tables for oblique ascensions for latitudes 45;22º (sic), 48º, 50º, and 51º, and parallaxes for 45;24º and 48º. For the edition of these tables, we have only transcribed those for latitude 48º, corresponding to Paris. As is the case for many other texts and tables in this manuscript, the Tables of 1322 are bound by an incipit and an explicit (see Figures 3 and 4). Immediately after, there are several tables of various origins, beginning with four tables belonging to ephemerides compiled by John of Saxony, including tables for the daily positions of the Sun for 1336–39, as well as a table for a solar correction needed to extend its validity from 1340 to 1456.46 On f. 81r, we find a list of sixty cities (beginning with Jerusalem and ending with Oxford) in which only four have a mark at the left of their names: Toledo, Paris, Montpellier, and Liège. The table for the semidiameters and the hourly velocities of the luminaries by John of Genoa is found on f. 81v, followed by three tables for the mean motions in elongation for the superior planets. The manuscript closes with a text by John of Murs on the computation of sines, beginning Omnes sinus recti incipiunt and including an explanatory figure inside of which we find the following text: Figura inveniendi sinus kardagarum et aliarum circuli porciones demonstratione ordinata per magistrum Johannem de Muris bene mori modo et faciliori quo potest tradi. It is indeed not frequent to find works by some of the most significant early Parisian Alfonsine scholars such as John of Murs, John of Lignères, John of Genoa, John of Saxony, and Nicholaus de Heybech in a short manuscript (only 85 folios), suggesting that the copyist Amici held a deep interest in mathematical astronomy.

44 ‘Liber Cartusianum Basiliensium datus per bone memorie magistrum Henricum Amicum artium et medicine doctorem phisicum civitatis Basiliensis’ (f. 1r). 45 José Chabás and Bernard R. Goldstein, ‘Nicolaus de Heybech and His Table for Finding True Syzygy’, Historia Mathematica: International Journal of Mathematics, 19 (Elzevier, 1992) pp. 265–289. 46 See José Chabás and Bernard R. Goldstein, ‘The Master and the Disciple: The Almanac of John of Lignères and the Ephemerides of John of Saxony’, p. 91.

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Figure 3. Basel, UB, F II 7, 62r.

Figure 4. Basel, UB, F II 7, 77v.

Cracow, Biblioteka Jagiellońska, 551, 74r–95r (late fourteenth century, MS C)

This codex of 130 folios was probably by and large copied in Prague in 1388.47 The date is mentioned several times in the manuscript (see, in particular, f. 127r, where 1388 is mentioned twice). About thirty years later, in 1420, the manuscript was in Cracow, owned by a master of the University of Cracow, Johannes de Środa (1493), before passing on to a bishop of the city, who donated it to the University’s college of theologians. As is often the case for manuscripts mainly containing material on Alfonsine astronomy, the first works are a set of Parisian Alfonsine Tables and their canons by John of Saxony beginning Tempus est mensura. The radices given in the tables for the mean motions were computed for Prague, Paris, Magdeburg, Erfurt, Vienna, and Cracow. We also note that the last two chapters deal with solar and lunar eclipses, ending with secundum doctrinam magistri Johannis de Lineriis expressam. These are two canons considered as additions to the 1483 edition to the Parisian Alfonsine Tables. Next, we find several short texts on the astrolabe and the quadrant as well as a Theorica planetarum attributed to Gerard of Cremona. John of Lignères’ work on arithmetic, Algorismus de minutiis, is found on ff. 52v–56r, followed by the canons Cuiuslibet in forty-four chapters (ff. 58r–63r) and Priores in thirty-eight 47 For a detailed description of its contents, see Maria Kowalczyk et al., Catalogus codicum manuscriptorum medii aevi latinorum qui in Bibliotheca Jagellonica Cracoviae asservantur (Wrocław: Institutum Ossolinianum: Officina Editoria Academiae Scientiarum Polonae, 1984), vol. III, pp. 333–345.

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chapters out of forty-six (ff. 63r–70v). The Tables of 1322 are displayed on ff. 74v–90v, 94v–95r, without any incipit nor explicit. Only one table is missing out of the thirty-two composing the set: that for the velocities of the Sun and the Moon in minutes of a day. In this manuscript, rather, one finds a copy of John of Genoa’s table for the semidiameters and velocities of the two luminaries (f. 74r) and a copy of John of Montfort’s table for solar and lunar velocities (ff. 90v–92r). Then follows a heterogeneous mixture of astronomical tables and texts, some of them focusing on astrology, in particular medical astrology. Among them are a table for the duration of pregnancy, a table for the astrological houses, John Vimond’s list of 225 stars, two other short star lists, a list of geographical places, tables for the projection of rays, planetary dignities, terms, radices for Cracow for the year 1420, solar declination, mean motions of the planets, revolution of the years, fragments of the Oxford Tables, a table for the daily positions of the Sun in a year with a short text and a reference to John of Murs,48 a list of solar and lunar eclipses for 1342–86, and an annotated medico-astrological treatise by Guillelmus Anglicus, De urina non visa. This manuscript has been extensively used for comparisons and as the base manuscript for the edition of two tables (17 and 18). London, British Library, Egerton 889, 31r–52v (fifteenth century, MS L)

The astronomer-astrologer John Holbrook (d. 1437), Master of Peterhouse, Cambridge, was the owner of this manuscript, now in London, and the copyist of part of it. Holbrook compiled a set of tables for the meridian of Cambridge (latitude 52;19º), extant on ff. 134r–151v, with parameters for the mean motions of the celestial objects slightly different from those in the standard Parisian Alfonsine Tables, and preceded by a short chapter on f. 133v.49 Among the material assembled by Holbrook in this manuscript, we find John of Lignères’ canons Cuiuslibet and Priores (incomplete), which were numbered continuously up to 69. These texts follow his set of Tables of 1322 (ff. 31r–52v), of which the first is for the sine function (as is most often the case), and its heading mentions ‘Johannes de Lineriis’. There is no closing reference at the end of the set. There are several tables added that do not belong to this set but are often found together with it: a short table for the duration of pregnancy, a table for the possibility of eclipses, and a long table for planetary latitudes (bipartialis and quadripartialis). The manuscript abounds in Alfonsine material. There is also an incomplete copy of the Parisian Alfonsine Tables where, for the mean motions of the luminaries and the planets, we are given radices for Toledo, Oxford, London, Colchester, Paris, and Cambridge, in that order. The canons written by John of Saxony, beginning Tempus est mensura, ensue. It is worth noting that after the chapter on conjunctions of the planets and the stars, which in 48 A note on f. 107r indicates that it was written by John of Murs in 1334 in Paris. However, according to José Chabás and Bernard R. Goldstein, A survey of European astronomical tables in the late middle ages (Leiden-Boston: Brill, 2012), p. 83, this is a version of a table by this author computed for the year 1321 and the meridian of Toledo, and it belongs to his Tables for 1321. 49 C. Philipp E. Nothaft, ’John Holbroke, the Tables of Cambridge, and the ‘true length of the year’: a forgotten episode in fifteenth-century astronomy’, Archive for History of Exact Sciences, 72 (Berlin: Springer ScienceBusiness Media, 2018), pp. 63–88.

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the editio princeps (Ratdolt 1483) marks the end of John of Saxony’s canons, the chapters on solar and lunar eclipses, traditionally integrated in the so-called membra adiuncta, follow with no interruption. Oxford is well represented in this manuscript. There are tables for the houses compiled by John Walter, which are preceded by a text explaining their use, beginning Volenti [igitur] operari, and a copy of the Tables of William Rede for the latitude of Oxford (51;50º), preceded by its canons, with the incipit Volentibus prognosticare…50 In addition to these sets of tables with their canons, there are also several treatises on sines and chords and, at the very end of the manuscript, other tables for Cambridge dated 1433 (f. 157r). This manuscript has only been used as the base manuscript for the edition of Table 20. 2. Other manuscripts for comparison Erfurt, Universitätsbibliothek, CA 2º 377, 41v–46v (fourteenth century, MS E)

This manuscript in the Amplonian collection at the Erfurt University library is a composite unit with three parts bound together. The first and third parts were written in the fifteenth century and the central part in the fourteenth. A table of contents was inserted at the beginning of the manuscript, and a detailed description is found in the library’s catalogue; a codicological study, in particular about the hands of the manuscript, is also available.51 The central part of this manuscript can be considered as a landmark of Alfonsine astronomy. The only known copy of John Vimond’s Planicelium is extant on ff. 21r–22r.52 In the upper margin of f. 21r, one can just about read a note, in another hand (probably John of Wasia,53 d. 1395): Compositio astrolabii…magistri Johannis de Parvo Sancto… This name attributed to John Vimond is totally unknown. The planicelium is an astronomical instrument briefly described in ten chapters ending with a reference to Ptolemy. On f. 21r, two dates can also be read in the right-hand column: 1324 and 1326. This text is now under study by Samuel Gessner and Marie-Madeleine Saby. We then find John of Lignères’ canons for the primum mobile, beginning Cuiuslibet arcus propositi (ff. 22r–26v), with forty-four chapters, followed by the forty-six chapters of the Priores astrologi (ff. 26v–35r). The explicit of this second text informs the reader that in 1323 John of Saxony, under the name John of Danecowe, copied the canons written by his master, John of Lignères, only one year before.54 In the Cuiuslibet, figures of astronomical instruments or demonstrations are drawn in the margins (ff. 22r–v, 23r–v), sometimes very 50 José Chabás, Computational Astronomy in the Middle Ages, pp. 207–213. 51 Wilhelm Schum, Beschreibendes Verzechnis der Amplonianischen Handschriften-Sammlung zu Erfurt, (Berlin, 1887), p. 262–264. See also Matthieu Husson and Marie-Madeleine Saby, ‘Le manuscrit d’Erfurt CA 2° 377 et l’astronomie alphonsine’, in Les Miscellanées scientifiques au Moyen Age, Micrologus 27, 2019, pp. 205–233. 52 Lynn Thorndike and Pearl Kibre, 1050 and physical inspection by Marie-Madeleine Saby. Incipit: Planicelium vero componitur ex eis que sunt diversorum operum secundum quod apparet…-… et si sit in parte super orizontem illa erit meridionalis. Explicit tractatus Johannis Vimundi de floribus illorum que per instrumenta spere signorum possunt inveniri ad faciendum judicia astronomica secundum intentionem summi Ptholomei Feludiensis. 53 Adriaan Pattin, ‘À propos de Joannes de Wasia’, Bulletin de philosophie médiévale, 20, 1978, p. 74. 54 F. 35r: Expliciunt canones tabularum magistri ordinati per magistrum Johannem Pychardum de Lineriis, et completi Parisius, anno ab incarnatione Christi, filli dei, 1322. Scripte Parisius per manum Johannis de Danecowe, anno Domini M CCC XXIII, in die cathedra Petri. Deo gratias.

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clumsily, and numerous notes are visible in the margins, often consisting of additions to the text. A hypothesis already proposed by Saby suggests that all these notes written by the same cursive hand that reviewed and annotated John of Lignères’ text a few months after the copy were possibly made by John of Saxony himself.55 The same cursive hand added four works by John of Murs: The Arbor Boetii (ff. 35v–36r), composed in 1324;56 the Mare fusile (f. 36r–v), a text dealing with metrological aspects;57 the Prognosticationes, a short astrological text uniquely preserved in this manuscript; and the Tabula tabularum (ff. 37r–38r), a work concerning sexagesimal notation, composed in 1322.58 Dealing again with mathematical matters, on ff. 38v–41r there is a copy of John of Lignères’ Algorismus de minutiis,59 which includes a table for multiplication displayed on three pages in the wrong order. This collection of mathematical works could have been quite useful for a young astronomer such as John of Saxony when confronted with astronomical tables. The Tables of 1322 are found on ff. 41v–46v, and they represent the longest item in the central and oldest part of the manuscript. Altogether, there are twenty-seven tables out of thirty-two without an incipit or explicit. The set is quite compact, for the copyist tried as much as possible to use the available space; for example, he managed to fit eleven tables and sub-tables on f. 45r. He also mixed the tables of his set with others that did not belong to it: a Tabula longitudinis horarum, a table for computing true syzygies, and tables for the revolutions of the months and revolution of the years.60 The central and oldest part of this manuscript collects works by the four major early Alfonsine astronomers, John Vimond, John of Murs, John of Lignères, and John of Saxony as the copyist of it. This is perhaps an autograph manuscript. Paris, Bibliothèque nationale de France, lat. 7286C, 24v–28v, 48r–52r, 53v–55r, and 56r (fourteenth century, MS P)

This is an exceptional fourteenth-century manuscript for it contains the first appearance of Alfonsine parameters in Parisian astronomy.61 These parameters concern the mean

55 Marie-Madeleine Saby, 1987 p. 61. 56 The Erfurt manuscript provides the earliest copy of this text, also found in Paris, BnF. lat. 16621, 62v–64r and other fifteenth-century manuscripts. 57 Marie-Madeleine Saby, ‘Mathématique et métrologie parisienne au début du XIVème siècle: le calcul du volume de la mer d’airain, de Jean de Murs’, Archives d’histoire doctrinale et littéraire du moyen-âge, 58 (Paris: J. Vrin, 1991), pp. 197–213. 58 Matthieu Husson, ‘La tabula tabularum de Jean de Murs: nombres et opérations arithmétiques en astronomie au début du XIVème siècle’, Cahiers de recherches médiévales et humanistes, 27 (Auxerre: Centre d’études médiévales d’Orléans, 2014), pp. 96–122. 59 The text was published in Venice in 1540 by Ioannes Antonius de Vulpinis de Castrogiafredo. See also Hubertus. L. L. Busard Het rekenen met breuken in de middleleeuwen, in het bijzonder bii Johannes de Lineriis. (Brussels: Vlaamse Academie, 1968). 60 See José Chabás 2019, Computational Astronomy in the Middle Ages, p. 194 and José Chabás and Bernard R. Goldstein, A Survey, 2012, pp. 221–223. 61 According to Donatella Nebbiai, La Bibliothèque de l’Abbaye de Saint-Denis en France (Paris: Editions du Centre National de la Recherche Scientifique, 1985), p. 218, this manuscript was probably copied in Paris, and in the fourteenth century it belonged to the Abbaye of Saint-Denis. The contents of the manuscript suggest it could have been produced in the mid- fourteenth century.

john of lign èr es: iberia n a stronomy settles in  pa ris

Figure 5. Paris, BnF, lat. 7286C, 23v.

motions and the equations of the celestial bodies and are implicitly or explicitly found in a set of tables compiled by John Vimond (1320), and they are uniquely preserved in this manuscript (ff. 1r–8v). Several tables with radices computed for 1 January, 1321 for the meridian of Paris come next: they are part of John of Lignères’ Tabule magne. We also find the two canons Cuiuslibet arcus (ff. 10r–23r) and Priores astrologi (ff. 23v–41r), surrounding the Tables of 1322, but the text on any particular folio does not correspond to the featured tables. This is yet another uncommon feature of the manuscript. After the Tabule magne there are twelve tables, one for each month of the year, displaying the accumulated daily mean motions of the luminaries and the planets (ff. 11v–23r). Then follow the Tables of 1322, from which only three are missing. Mixed among them are other tables for the equations of the Sun, the Moon, and the planets. In the case of the solar equation, the maximum value of the entries is 2;10º, which is indeed the standard Alfonsine value, already embedded in John Vimond’s table. The table for the lunar equations (ff. 30v–32r) is certainly peculiar, because the equation of anomaly is presented in two different columns: equatio argumenti and equatio argumenti Alfonsi. And indeed, the entries in both columns differ and reach maximum values of 5;0,59º and 4;56º, respectively. We note that for Ptolemy, al-Battānī, and the Toledan Tables, the maximum equation of anomaly is 5;1º. In contrast, 4;56º is the maximum value used by Vimond, and in many zijes developed in the Iberian Peninsula as part of the Indian tradition, and it became the standard parameter for the equation of anomaly used by Alfonsine astronomers. To our knowledge, no previous table offers this double possibility, and one may consider that the

35

36

j os é c h a b ás an d mar ie-madelein e sa by

table was compiled at an early time, when this parameter had not yet been fully adopted, indicating that this is a table in transition from pre-Alfonsine to Alfonsine astronomy. Equations for the five planets follow those of the Sun and the Moon. All of them share their characteristic parameters with the Toledan Tables and the zij of al-Battānī. The parameters for the equations of centre of Jupiter and Venus are undoubtedly given as 5;15º and 1;59º, respectively; these are the ones found in the Toledan Tables, differing from 5;57º and 2;10º as used in Alfonsine astronomy, at least from the time of Vimond. Drawing conclusions about the authorship of these equation tables seems problematic, even though almost all tables in this manuscript were compiled by John Vimond and John of Lignères in the early 1320s in Paris. Yet, at the end of the manuscript, two tables by other authors are found: John of Genoa’s table for the velocities and the semidiameters of the Sun and the Moon (f. 56v) and Peter of Dacia’s lunar table (f. 57r), together with tables for the astrological dignities and for purposes of computus. On ff. 57v–58r, there is a triangular multiplication table ranging from 60 to 1 at the head of the table and from 1 to 60 along the sides.62 Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1374, 26r–46v (fifteenth century, MS V)

This manuscript consists mainly of astronomical tables, all in the Alfonsine framework.63 It opens with an incomplete set of Parisian Alfonsine Tables, where the tables for the mean motions include radices for Prague, Erfurt, and Paris, as well as other cities in Central Europe. In the explicit (f. 24v) we are told that this set of Alfonsine Tables was copied in Prague in 1407 by Reinhard von Nuremberg, a Bachelor of Arts, and a precise date is given: Saint Ludmila’s day. The Tables of 1322 by John of Lignères come later and, as is the case of Basel, UB, F II 7, an incipit and an explicit frame the set. We note that the first table in the set is not the table of sines, as is usually the case, but the table for the mean motion in elongation. Furthermore, in general, the usual order of the tables is not followed. On f. 46v there is an explicit, with about the same information as above including the name of the copyist, the year, and the place as given above. A date is also provided: anno domini 1407º undecimo Kalendas octobris. As Saint Ludmila’s day falls on 16 September, this means that Reinhard spent about a week copying tables onto a couple of dozen pages. As is often the case, after the Tables of 1322 there is a table for the semidiameters and velocities of the two luminaries compiled by John of Genoa. It is followed by a long and heterogenous series of tables including tables for planetary apogees, projections of the rays, various sets of tables for the houses, a calendar with accumulated daily mean motions, sets of radices for Prague for 1396 and 1400, tables of proportions, velocity tables, and unfinished tables, among other miscellaneous tables. On ff. 87v–102v, we see a unique and interesting item: a double argument table of the complete lunar equation. It is a large table, consisting of 10,800 entries, compiled by Nicolaus Mülhus, a wool merchant and layman in Zittau, now in Saxony, accompanied by 62 Fritz S. Pedersen 2002, p. 157; Marie-Madeleine Saby, 1987, pp. 516–520. 63 Ludwig Schuba, Die Quadriviums-Handschriften der Codices Palatini Latini in der Vatikanischen Bibliothek (Wiesbaden: Ludwig Reichert Verlag, 1992), pp. 86–88.

john of lign èr es: iberia n a stronomy settles in  pa ris

a short text.64 Although no date is assigned to the table, it is likely that it was computed in the early fifteenth century, at a time when new actors, such as this merchant and layman, appear in mathematical astronomy. A final item remains to be mentioned: a long table displaying the time and the positions of the Sun and the Moon for all mean conjunctions from 1400 to 1432. There is no indication of where this table is valid, but Prague is first candidate. For the years 1433–1519, we are given the times of mean conjunctions and only a few entries for the positions of the luminaries. 3. Other manuscripts Cologne, Historisches Archiv der Stadt, W* 178, 1r–18r (fourteenth century, MS K)

This short manuscript almost exclusively contains fourteenth-century Alfonsine astronomy. The copy of the Tables of 1322 by John of Lignères is almost complete, for only three tables are missing. There is also an incomplete copy of his Tabule magne, followed by the table for the velocities and semidiameters of the Sun and the Moon compiled by John of Genoa and tables for syzygies by Nicholaus de Heybech. In contrast, the tables for planetary latitudes (bipartiales and quadripartiales) are pre-Alfonsine. Oxford, Bodleian Library, Can. Misc. 27, 78v–104r (fifteenth century, MS Ox)

This complex codex from the fifteenth century opens with Jacob ben David Bonjorn’s canons and tables for 1361. These are followed by a set of Tabulae resolutae that Nicholas Polonius adapted for Salamanca while holding the chair of astronomy at the University of that city, no later than 1464: see Chabás 1998. The canons associated with this set, beginning Quoniam tabularum Alfonsii laboriosa difficultas, are also preserved in this manuscript.65 Between both works by Polonius there is a disordered copy of John of Lignères’ Tables of 1322. A partial copy of the canons Priores begins on f. 130r, but the first eighteen chapters, among others, are missing. A complete copy of the Cuiuslibet begins on f. 138r. Vatican, Biblioteca Apostolica Vaticana, Pal. lat. 1412, 95r–101v, 109r–114v, 117r–120r (fifteenth century, MS W)

This manuscript was copied in the middle of the fifteenth century. It opens up with Gerard of Cremona’s Theorica planetarum. The canons Tempus est mensura by John of Saxony follow, and additional chapters on eclipses are included, as well as John of Lignères’ canons Cuiuslibet and Priores, and a set of Parisian Alfonsine Tables. A rather comprehensive version of the Tables of 1322 begins on f. 95r, with only five tables missing. Moreover, it is mixed with other tables, including John of Genoa’s tables for velocities and semidiameters.

64 José Chabás and Bernard R. Goldstein, ‘The medieval Moon in a matrix: double argument tables for lunar motions’, Archive for History of Exact Sciences 73 (2019b) pp. 335–359. 65 José Chabás and Bernard R. Goldstein, ‘Nicolaus de Heybech and His Table for Finding True Syzygy’, Historia Mathematica: International Journal of Mathematics, 19 (Elzevier, 1992) pp. 265–289.

37

38

j os é c h a b ás an d mar ie-madelein e sa by Vatican, Biblioteca Apostolica Vaticana, Ottob. lat 1826, 127r–128r, 130r–140r (fourteenth century, MS X)

This codex consists of astronomical texts and tables (dating from the fourteenth and fifteenth centuries). It includes canons to the Toledan Tables, John of Lignères’ canons to the Alfonsine Tables (Quia ad inveniendum) and canons for the primum mobile (Cuiuslibet arcus), as well as a set of Parisian Alfonsine Tables, among other Alfonsine material.66 The explicit of the Cuiuslibet (f. 61v) reads Expliciunt canones supra tabulas de primo mobile quas compilavit Johannes de Lineriis ex dictis Albategni Parisius. One wonders why this unusual reference to al-Battānī was included. 5. Commentaries to the tables For each table, we have included a commentary with specific information and remarks. In addition to the title in the base manuscript, we list the titles found in four other manuscripts used for collating the table, as well as the titles of all latitude-dependent tables, such as those displaying the oblique ascension (Table 6) and parallax (Table 22), in the various manuscripts. Then follows a succinct description of the table, together with references to its antecedents in previous sets. We also indicate the chapter(s) of John of Lignères’ canons in which a table is mentioned, noting that there is no one-to-one correspondence between text and tables. Additionally, not all tables are mentioned in the canons and not all tables referred to in the canons belong to this set. In a few cases, a recomputation of the table is provided, thus allowing for a comparison of the entries between text and computation. In order to facilitate the interpretation of the tables and their comments, it was decided to display face to face, as much as possible, the comments on the even pages and the corresponding tables on the odd pages. We have also compared the entries in each of the tables with those in four other manuscripts, as mentioned above. Most of the time, the variant readings indicate the correct reading for a specific entry and thus make it possible to identify scribal errors in the manuscript used as a basis. In certain cases, other conclusions can be drawn. We made no attempt to derive any stemma of the manuscripts from the variant readings. In order to facilitate the identification of the entries, we have numbered the columns in a table with the letters a, b, c, …, that we have placed at the bottom of the table. In some cases, we have also added an extra column, at far left and outside the table, with the numbers 1, 2, 3, … to label the rows in a table. In the Tables of 1322, we counted 18,264 sexagesimal numbers, including those of the entries and the arguments. Manuscript B has served as base manuscript for 16,172 such numbers, whereas MSS L and C were used as base manuscripts for 1,440 and 652 sexagesimal numbers, respectively. The four manuscripts we compared with MS B, whenever possible, are C, E, P, and V. The error rate in a particular manuscript can be defined as the number of differences between a particular manuscript and the base manuscript divided by the number of entries examined. The error rates that can be deduced from these comparisons 66 A description is found in Fritz S. Pedersen, 2002, p. 177.

john of lign èr es: iberia n a stronomy settles in  pa ris

give an idea as to what extent a particular scribe made errors in copying numbers, or better, on how much the result of his work did not agree with the table he was copying. No substantial differences appear between the main manuscripts used: 2.56% for MS E, 2.86% for MSS C and V, and 3.14% for MS P. These percentages amount to a mean rate of about 2.8%. We did not compute error rates for other manuscripts, because of the comparative small sizes of the population involved. However, it is possible to compare this mean rate of errors with that scored by other copyists of tables during approximately the same period, who reproduced sets of tables with a similar volume of sexagesimal numbers. The Tables by Jacob ben David Bonjorn, with epoch 1361 and preserved in many manuscripts in Hebrew, Latin, Greek, and Catalan, display about 13,000 sexagesimal numbers. In nine Latin and Catalan manuscripts surveyed, the mean rate of errors is lower, 1.15%, with one fifteenth-century scribe –whose name is known, Ausiàs Sancho– scoring only 0.64%.67 The study of the variants found in the entries in the selected manuscripts has proven to be useful not only in identifying faulty entries but also in providing hints for further research. For example, comparing with entries in other manuscripts containing the Toledan Tables and extant in Paris at the time, in particular BnF, lat. 16655, makes it possible to draw some hypotheses about the materials actually used by John of Lignères. Two general ideas emerge from the study of the Tables of 1322 by John of Lignères. As is the case for all sets of tables in the Middle Ages, this is a toolbox in which practitioners of astronomy could find ways to address and solve most of the astronomical problems they had to face in their profession. As most table-makers in that period did, John of Lignères adapted location-dependent tables in his set to the coordinates of his city, Paris, thus sharing the general tendency to offer computational tools that would facilitate the task of practitioners. John of Lignères worked in strict adherence to Ptolemaic astronomy, as almost all medieval astronomers working in Latin. In particular, John of Lignères depended heavily on the Toledan Tables, and his set can be understood as a bridge between them and the Parisian Alfonsine Tables, which had been shaped in Paris based on the Castilian Alfonsine Tables compiled in Toledo a few decades beforehand.

67 José. Chabás, in collaboration with Antoni Roca and Xavier Rodríguez, L’astronomia de Jacob ben David Bonjorn (Barcelona: Institut d’Estudis Catalans, 1992), pp. 217–222.

39

Edition of the Tables with Comments

42

edi t i on of the tab les with comments

1. Sine Tabula sinus (MS B, 62r–v) Titles in other manuscripts MS C, 74v–75r: Tabula sinus MS E, 41v: Tabula sinus MS P, 25r–v: Tabula sinus MS V, 33v–34r: Tabula sinus tantum Description The table has two columns for the argument, downwards from 0;30º to 90;0º and upwards from 90;0º to 179;30º, at intervals of half a degree. There is another column for the values of the sine of the argument, here called corde mediate, given in degrees, minutes, and seconds. The entries correspond to the sinus of the argument, with a unit radius of 60, and they reach a maximum value of 60;0,0º at 90º. This table is usually presented as six blocks of 15º each. The same table is found in the Toledan Tables (see Pedersen 2002, pp. 957–959), ultimately derived from that in the zij of al-Battānī (see Nallino, 1899–1907, 2: 55–56). It is also mentioned in Chapters 1–6 and 10 in the canons beginning Cuiuslibet arcus (see Saby 1987, pp. 75–77.). Recomputation The entries can be recomputed by means of the following modern formula: y = 60 · sin α, where α is the argument. The table displays 180 entries, each of which consists of three sexagesimal numbers. In 39 cases out of 540, the entries in MS B, taken as the base manuscript, differ from the recomputed values according to the following distribution of differences (T – C: text – computation). T–C Freq.

–60” 2

…  

–2’’ 1

–1” 24

0 500

+1” 8

…  

+3” 2

…  

+9’’ 1

+10” 2

There are two cases in which the difference between the text and computation amounts to –1’, corresponding to arguments 26;0º and 72;30º. The others concern the seconds. In 32 cases, the differences are –1” or +1”, most probably due to the rounding procedure.

sine

John of Lignères. Sine (1) Base: Basel F II 7, 62r-62v Tabula sinus Arcus augmentati per dimidium graduum G

a

M 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15

b

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

c

Corde mediate

M 179 179 178 178 177 177 176 176 175 175 174 174 173 173 172 172 171 171 170 170 169 169 168 168 167 167 166 166 165 165

d

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

e

M 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 14 14 15 15

f

S 31 2 34 5 37 8 39 11 42 13 45 16 47 18 49 21 52 23 54 25 56 26 57 28 59 29 0 30 1 31

g

25 50 14 38 2 25 45 7 28 46 2 18 32 44 53 1 7 10 10 8 3 54 43 29 11 49 24 55 22 45

43

44

edi t i on of the tab les with comments

Variant readings Comparisons between recomputed entries and those in the various manuscripts show that the best choice for a base manuscript is Basel, Universitätsbibliothek, F II 7 (MS B), for it presents the lowest number of discrepancies with respect to computation. Compared to B, there are seven variants in C, seventeen in E, twenty-five in P, and nine in V. None are common variants. A closer look at the variants provided interesting hints. All manuscripts consulted have 17’ for the entry at argument 26;0º, except MS V, which has 18’. The recomputed value is 26;18,8º. Now, when we turn to the Toledan Tables we note that Pedersen’s edition refers to three manuscripts displaying this faulty entry in the minutes, 17’ instead of 18’: two late-thirteenth-century manuscripts (Erfurt, Universitätsbibliothek, CA 8º 82 and Paris, Bibliothèque nationale de France, lat. 16655), and an early-fourteenth-century manuscript (El Escorial, Biblioteca del Monasterio, O.II.10) containing glosses written by John of Murs. One may think that any of these manuscripts could be at the origin of the faulty entry in John of Lignères’ table. A second variant reading in all manuscripts containing the Tables of 1322 concerns the entry for the seconds at argument 38;0º, where MSS Basel and Cracow have 32”, and MSS Erfurt, Paris, and Vatican have 33’’. The recomputed value is 36;56,23º. In almost all manuscripts of the Toledan Tables edited by Pedersen, the entry is indeed 23”, but Erfurt, Universitätsbibliothek, CA 8º 82 and Paris, Bibliothèque nationale de France, lat. 16655 have 32’’, and El Escorial, Biblioteca del Monasterio, O.II.10, has 31’. Thus, for the two variant readings considered here, the known manuscripts of the Toledan Tables sharing both variants with the Tables of 1322 are the manuscripts in Erfurt and in Paris. Moreover, BnF lat. 16655 was in Paris at this time. It was owned by Peter of Limoges and was one of the 120 manuscripts he left at the College de Sorbonne upon his death in 1306. It is plausible that John of Lignères might have used this particular copy, or a close copy, for his table of sines, or maybe the manuscript El Escorial, O.II.10, which also was in Paris in the early fourteenth century. As mentioned above, this is only a hint, revealing itself through the detailed edition of the Toledan Tables by Fritz S. Pedersen in 2002. Any further conclusion would require additional systematic analysis.

sine

Tabula sinus Arcus augmentati per dimidium graduum G

a

M 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30

b

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

c

Corde mediate

M 164 164 163 163 162 162 161 161 160 160 159 159 158 158 157 157 156 156 155 155 154 154 153 153 152 152 151 151 150 150

d

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

e

M 16 16 17 17 18 18 19 19 20 20 21 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30

f

S 2 32 2 32 2 32 2 32 1 31 0 30 59 28 57 26 55 24 52 21 49 17 46 14 42 10 37 5 32 0

g

3 18 27 33 32 28 18 2 42 16 45 7 24 35 39 38 30 15 54 25 50 8 19 22 18 6 46 19 43 0

45

46

edi t i on of the tab les with comments

Tabula sinus Arcus augmentati per dimidium graduum G

a

M 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45

b

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

c

Corde mediate

M 149 149 148 148 147 147 146 146 145 145 144 144 143 143 142 142 141 141 140 140 139 139 138 138 137 137 136 136 135 135

d

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

e

M 30 30 31 31 32 32 33 33 33 34 34 35 35 36 36 36 37 37 38 38 38 39 39 40 40 40 41 41 42 42

f

S 27 54 21 47 14 40 6 33 59 24 50 16 41 6 31 56 21 45 9 34 58 21 45 8 32 55 18 40 3 25

g

8 8 0 43 17 42 58 5 4 52 32 2 23 32 32 32 3 33 54 2 1 49 26 51 7 12 4 46 16 35

sine

Tabula sinus Arcus augmentati per dimidium graduum G

a

M 45 46 46 47 47 48 48 49 49 50 50 51 51 52 52 53 53 54 54 55 55 56 56 57 57 58 58 59 59 60

b

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

c

Corde mediate

M 134 134 133 133 132 132 131 131 130 130 129 129 128 128 127 127 126 126 125 125 124 124 123 123 122 122 121 121 120 120

d

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

e

M 42 43 43 43 44 44 44 45 45 45 46 46 46 47 47 47 48 48 48 49 49 49 50 50 50 50 51 51 51 51

f

S 47 9 31 52 14 35 56 16 37 57 17 37 57 16 36 55 16 32 50 8 26 44 1 19 36 52 9 25 41 57

g

42 37 24 52 12 19 18 57 27 46 50 43 23 50 4 5 53 28 49 59 51 31 59 13 12 58 30 48 52 42

47

48

edi t i on of the tab les with comments

Tabula sinus Arcus augmentati per dimidium graduum G

a

M 60 61 61 62 62 63 63 64 64 65 65 66 66 67 67 68 68 69 69 70 70 71 71 72 72 73 73 74 74 75

b

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

c

Corde mediate

M 119 119 118 118 117 117 116 116 115 115 114 114 113 113 112 112 111 111 110 110 109 109 108 108 107 107 106 106 105 105

d

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

e

M 52 52 52 52 53 53 53 53 54 54 54 54 55 55 55 55 55 56 56 56 56 56 56 57 57 57 57 57 57 57

f

S 13 28 43 58 13 27 41 55 9 22 35 48 1 13 25 34 49 0 12 22 33 43 53 3 12 22 31 40 49 57

g

17 38 45 37 14 37 45 40 18 42 51 46 25 49 58 52 30 53 1 53 30 52 58 48 23 41 45 31 4 20

sine

Tabula sinus Arcus augmentati per dimidium graduum G

a

M 75 76 76 77 77 78 78 79 79 80 80 81 81 82 82 83 83 84 84 85 85 86 86 87 87 88 88 89 89 90

b

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

c

Corde mediate

M 104 104 103 103 102 102 101 101 100 100 99 99 98 98 97 97 96 96 95 95 94 94 93 93 92 92 91 91 90 90

d

G 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0 30 0

e

M 58 58 58 58 58 58 58 58 58 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 59 60

f

S 5 13 20 27 34 41 47 53 59 5 10 15 20 24 29 33 36 40 43 46 48 51 53 55 56 57 58 59 59 0

g

20 4 32 44 40 21 44 51 53 18 38 40 27 58 12 13 51 16 25 18 54 14 17 4 35 49 46 27 52 0

Variants from B in C, E, P, and V: f26;0: V18; f30;30: E7 P17; f36;30: C50 EPV32; f51;0: P16; f51;30: P37; f52;0: P55; f52;30: C37 P13; f53;0: P32; f53;30: B marg. CEV13 P50; f54;0: P lac.; f54;30: P16; f55;0: P44; f55;30: EV 16 P1; f56;0: P19; f56;30: P26; f57;30: E26; f59;0: EP35; f61;0: EP24; f62;30: E12 P11; f68;0: B marg. CV37 EP41; f68;30: P59; f72;30: B could be read 13; g19;30: EPV45; g20;0: EP42; g36;30: C32; g37;0: EP2; g38;0: EPV33; g42;0: P41; g48;30: B marg. V14 CEP18; g51;30: E33; g55;0: B marg. 57; g61;0: CEV28; g62;30: EP18.

49

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edi t i on of the tab les with comments

2. Shadow Tabula umbre (MS B, 63r) Titles in other manuscripts MS C, 76r: Tabula umbre MS E, 42r: Tabula umbre MS P, 26r: Tabula umbre MS V, 35r: Tabula umbre Description The argument is presented in a single column, from 1º to 90º, at intervals of 1º. There is another column for the values of the cotangent of the argument, here called umbra, given in parts and minutes with a norm of 12. The entry for argument 45º is 12;0º, for cotan 45º = 1. The same table is already found in the Toledan Tables (Pedersen 2002, pp. 993) and in the zijes of al-Battānī (Nallino 1899–1907, 2: 60) and al-Khwārizmī (Suter 1914, p. 174). This table is mentioned in Chapters 16–19 in the canons beginning Cuiuslibet arcus (see Saby 1987, pp. 109–113). Recomputation The entries can be recomputed by means of the following modern formula: s = 12 · cotan h, where s is the length of the shadow on the horizontal plane cast by a gnomon of 12 units, as a function of the solar altitude h. For further details, we invite the reader to consult Chabás and Goldstein 2012, pp. 24–25. Except for the entry for argument 3º, the differences (T – C: text – computation) between the entries in MS B and the recomputed values only affect the minutes, according to the following distribution. T – C –3’ Freq. 1

–2’ 1

–1’ 5

0 163

+1’ 4

…  

+3’ 1

…  

+5’ 1

+6’ 1

…  

+27’ 1

The third entry, 203;28, differs substantially from recomputation, 228;58. We note that the value given by Suter is 228;58, with variant readings of 243 and 273 for the puncta and 47 for the minutes. Both Nallino and Pedersen give 228;58, and Pedersen adds variant readings of 291, 293, 243, and 273 for the puncta, and 38, 28, and 47 for the minutes. Variant readings The base manuscript used for this table is Basel, Universitätsbibliothek, F II 7 (MS B). There are 270 sexagesimal numbers, including those for the argument. Compared to B, there are ten variants in C, six in E, five in P, and five in V. Of these, two are common to MSS C, E, P, and V, thus identifying scribal errors in B.

sha dow

John of Lignères. Shadow (2) Base : Basel F II 7, 63r Tabula umbre Umbra

Gradus altitudinis

a

P

Gradus altitudinis

M

Umbra P

M

1

687

26

31

19

58

2

343

39

32

19

12

3

203

28

33

18

29

4

171

42

34

17

47

5

137

10

35

17

8

6

114

10

36

16

30

7

97

44

37

15

55

8

85

28

38

15

21

9

75

46

39

14

49

10

68

30

40

14

18

11

61

44

41

13

48

12

46

27

42

13

20

13

51

59

43

12

52

14

48

8

44

12

26

15

44

46

45

12

0

16

41

51

46

11

35

17

39

15

47

11

11

18

36

54

48

10

48

19

34

51

49

10

26

20

32

58

50

10

4

21

31

19

51

9

43

22

29

42

52

9

22

23

28

16

53

9

3

24

26

57

54

8

43

25

25

44

55

8

24

26

24

36

56

8

6

27

23

33

57

7

48

28

22

34

58

7

30

29

21

40

59

7

13

30

20

47

60

6

56

b

c

a

b

c

51

52

edi t i on of the tab les with comments

For argument 3°, the manuscripts reviewed here read 293 (MSS E, P, and V) and 303 (MS C) for the puncta and 28 for the minutes. The value of 203 found in MS B could be a misreading of 303 or 293, which is most ubiquitous in the manuscripts containing the Toledan Tables. For argument 10°, MS B reads 68;30, whereas all others, as well as recomputation, give 68;3: this corresponds to a difference of +27’. In MS B, the marginal note, al. 3, was added to restore the correct value.

sha dow

Tabula umbre Gradus altitudinis

a

Umbra P

M

62 63 64 65 66 67

6 6 5 5 5 5

23 7 51 36 21 6

68

4

51

69

4

36

70

4

22

71

4

8

72

3

55

73

3

40

74

3

26

75

3

13

76

3

0

77

2

46

78

2

33

79

2

20

80

2

7

81

1

54

82

1

41

83

1

28

84

1

16

85

1

3

86

0

50

87

0

38

88

0

25

89

0

13

90

0

0

b

c

Variants from B in C, E, P, and V: b44: P13; c9: C26; c10: B marg. CEPV3; c21: EPV 16; c26: CE38; c49: C46; c51: B marg. CEV23; c57: C46; c61: C30; c65: CEPV39; c66: CV12 V corr.21; c72: C53 P58; c89: V13 corr.16.

53

54

edi t i on of the tab les with comments

3. Solar declination Tabula declinationis solis ab equatore (MS B, 63r) Titles in other manuscripts MS C, 75v: Tabula declinationis solis ab equatore MS E, 41v: Tabula declinationis solis ab equatore MS P, 26r: Tabula declinationis solis ab equatore MS V, 34v: Tabula declinationis solis ab equatore Description The first column is for the argument, from 1º to 90º, at intervals of 1º. The entries correspond to the solar declination and are displayed in three columns, each of 30º, in degrees, minutes, and seconds. The maximum entry, 23;33,30º, is the obliquity of the ecliptic, and it occurs at argument 90º. The same table, with the same value for the obliquity, is already found in the Toledan Tables (Pedersen 2002, p. 964). This table is mentioned in Chapter 11 in the canons beginning Cuiuslibet arcus (see Saby 1987, pp. 101–103), where the maximum value, 23;33,30º, is explicitly given and in Chapter 19 in the canons beginning Priores astrologi (see Saby 1987, p. 205). Recomputation The entries for the solar declination can be recomputed by means of the following modern formula: δ = arcsin (sin λ sin ε), where λ is the solar longitude and ε = 23;33,30º is the obliquity of the ecliptic. For further details, see Chabás and Goldstein 2012, pp. 22–24. For MS B, the differences between text and computation (T – C) are irregularly distributed, as can be seen in the following table. T–C Freq.

[–3’’, –50’’] –5’’ 3 1

–4’’ 4

–3’’ 8

–2’’ 7

–1’’ 16

0 23

+1’’ 14

+2’’ 7

+3’’ 1

[9’’, 70’’] 6

This does not necessarily mean that the table in MS B was carelessly copied. When compared with the table for solar declination in the Toledan Tables (Pedersen 2002, p. 964), only seven differences are found, indicating that John of Lignères’ table, as extant in MS B, was borrowed from the Toledan Tables, and that the table he was copying had already accumulated many scribal errors. This is easily seen from the distribution of the differences between text and computation for the table presented by Pedersen. T–C Freq.

–16’’ 1

–5’’ 1

–4’’ 5

–3’’ 10

–2’’ 7

–1’’ 17

0 26

+1’’ 14

+2’’ 7

+3’’ 1

34’’ 1

sola r declination

55

John of Lignères. Solar declination (3) Base : Basel F II 7, 63r Tabula declinationis solis ab equatore Numerus Linee graduum numeri G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

a

Arietis Virginis Libre Piscium G

89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 b

M 0 0 1 1 1 2 2 3 3 3 4 4 5 5 5 6 6 7 7 7 8 8 8 9 9 10 10 10 11 11

c

Numerus Linee graduum numeri S

24 48 11 35 59 23 47 11 35 58 22 46 9 32 56 19 42 5 28 51 14 36 59 21 43 5 27 48 10 31 d

G 0 0 56 51 47 40 30 19 5 46 28 0 30 55 15 29 38 40 34 25 5 49 3 30 28 26 14 52 19 36

e

G 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

a

Tauri Leonis Scorpii Aquarii G

59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 b

M 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16 16 17 17 17 18 18 18 18 19 19 19 19 20 20

c

S 52 13 24 54 15 35 55 14 33 53 12 30 49 7 25 42 59 16 33 49 6 21 36 51 6 21 35 48 2 15

d

45 40 23 53 11 16 7 41 6 11 3 41 2 8 17 29 45 44 25 46 48 29 53 40 41 6 7 48 1 0 e

56

edi t i on of the tab les with comments

The distributions above indicate that MS B added a few more errors to those already present in the corresponding table for the solar declination in the Toledan Tables. Variant readings The base manuscript used here is also Basel, Universitätsbibliothek, F II 7. Most variant readings concern the column for the minutes, and MS P shows an upward shift of entries between arguments 37º and 43º. We note that all five manuscripts examined have 23;1,59º for argument 78º, a value that differs by 70’’ from the recomputed value, 23;0,49º. No such difference appears in the copies of the Toledan Tables examined by Pedersen, indicating that this faulty value most probably originated from John of Lignères’ tables, or soon after them.

sola r declination

57

Tabula declinationis solis ab equatore Numerus Linee graduum numeri G

a

G 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

b

Geminorum Cancri Sagitarii Capricorni G

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

c

M 20 20 20 21 21 21 21 21 21 22 22 22 22 22 22 22 22 23 23 23 23 23 23 23 23 23 23 23 23 23

d

S 27 39 51 3 14 24 35 45 54 3 12 20 28 35 42 49 55 1 5 10 15 18 22 25 27 29 31 33 33 33

e

36 48 39 9 12 56 11 2 25 35 16 29 17 38 36 6 44 59 10 46 5 56 21 19 49 50 44 2 13 30

Variants from B in C, E, P, and V: d2: C28; d3: CV12; d5: C58; d11: EP2; d12: C4; d14: E33; d22: C38; d24: E31; d33: CEV34; d37: P14; d38: P33; d39: P53; d40: P12; d41: P30; d42: P49; d43: P7; d48: C18; d88: CEP32; e24: C20; e46: C25; e59: CEPV7.

58

edi t i on of the tab les with comments

4. Ascensional difference Tabula diversitatis differentie ascensionum in universa terra (MS B, 63v) Titles in other manuscripts MS C, 76r: Tabula diversitatis differentie ascensionum umbra (sic) MS E, 41v–42r: Tabula diversitatis differentie ascensionum in universa terra et certificata MS P, 26r: Tabula diversitatis differentie ascensionum in universa terra et certificata MS V, 35r: Tabula diversitatis differentie ascensionum in universa terra et certificata Description The purpose of this table is to find the ascensional difference, γ; that is, the difference between the right ascension and the oblique ascension of a point of longitude λ. One may think that this table should come after those for computing right and oblique ascensions (Tables 5 and 6, below). However, this is not the case, for the table under examination here does not directly provide the difference γ. Instead, the entries represent 5 · tan δ(λ), where δ is the declination of the point considered, which is related to γ through the modern expression tan δ = sin γ tan (90 – φ). The coefficient (5 = 60/12) derives from the fact that a sine with unit radius of 60 is involved, and a table for shadows, normed 12, is to be used. We note that the entries for the ascensional difference do not depend on the observer’s geographical latitude, thus justifying the “universa terra” found in the title. Further details can be found in Chabás and Goldstein 2012, pp. 30–31, and Figure 4, p. 26. Thus, to obtain sin γ, it is necessary to divide the entry already obtained by tan (90 – φ) or multiply it by cotan (90 – φ), which is directly found in the table for shadows (see Table 2, above). Therefore, to compute the ascensional difference, only tables for the sine, shadow, and declination are needed (Tables 1, 2, and 3). There is one column for the argument, λ, displayed for all integer numbers from 1 to 90, and another column for the entry, which is dimensionless, but given in degrees, minutes, and seconds in the manuscripts. The table is usually presented as three blocks of 30º each. A similar table with the same purpose is attributed to al-Khwārizmī (Neugebauer 1962, pp. 50–53) and has the same layout and a maximum value of 5;31,34 at argument 90º (see Pedersen 2002, p. 988). The table was computed with Ptolemy’s value of the obliquity of the ecliptic, ε = 23;51º. We note that this table is normed 150. When converted into a table with norm 60, the resulting maximum is 2;12,38. This is exactly the maximum value in another similar table among the Toledan Tables (Pedersen 2002, p. 990), indicating the strong relationship between the two. There is a third table of this type based on a different value of ε. It is found in the Almanac of Azarquiel (Millás 1943–50, p. 225) and uses ε = 23;33º. Its maximum, 2;10,46, is reached at argument 90º. This table is mentioned in Chapter 27 in the canons Cuiuslibet arcus (see Saby 1987, pp. 129–135).

a scensiona l difference

John of Lignères. Ascensional difference (4) Base : Basel F II 7, 63v Tabula diversitatis differentie ascensionum in universa terra Linee numeri G

Porciones Arietis Virginis Libre Piscium G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a

M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

b

S 2 4 6 8 10 12 14 16 18 20 22 24 26 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 1

c

Linee numeri G

5 10 15 20 25 30 35 40 44 48 52 55 58 2 5 8 11 14 17 20 20 19 18 17 16 15 15 14 13 11 d

Porciones Tauri Leonis Scorpii Aquarii G

31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 a

M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

b

S 3 5 6 8 10 12 14 16 17 19 21 23 24 26 28 29 31 33 34 36 37 39 40 42 44 45 46 48 49 50

c

7 2 55 47 48 30 22 10 57 42 26 8 50 32 12 51 29 7 45 33 55 26 58 29 0 31 56 15 28 35 d

59

60

edi t i on of the tab les with comments

Recomputation The entries in John of Lignères’ table, with a maximum of 2;10,50, are very similar to those in the Almanac of Azarquiel, and one may think that they follow somehow from Azarquiel’s table. A close comparison indicates that John of Lignères’ table does not derive directly from it. Rather, the table seems to have been computed for a value of the obliquity of the ecliptic of 23;33,30º, which was already used by John of Lignères in his table for solar declination (see Table 3, above) and taken from the Toledan Tables. Therefore, this is the first table in this set that can be attributed to John of Lignères. Recomputation of the 90 entries using the following modern formula for the declination: δ = arcsin (sin λ · sin 23;33,30), yields the following differences (T – C: text – computation): T–C Freq.

11” 1

The negative differences clearly outnumber the positive, a feature that is spread all over the table and shows no clear pattern. We have not found a plausible explanation for it. It is worth noting that the entries in the 30º–40º range display a smooth pattern, as they concentrate on six zeros and three values of T – C = ±1. Computation with other values of ε yields worse results. Variant readings For this table, the base manuscript is MS B. Except in one case, all variant readings detected in the other manuscripts concern the column for the seconds. As compared to B, there are five variants in C, two in E, seventeen in P, and four in V. Two of the variants are common in all four collated manuscripts. Perhaps the only salient feature is that the copyist of MS P shifted all entries upwards by one cell for the seconds for arguments 72º–85º.

a scensiona l difference

Tabula diversitatis differentie ascensionum in universa terra Linee numeri G

a

Porciones Geminorum Cancri Sagittarii Capricorni G

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

c

M 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

d

S 51 52 54 55 56 57 58 59 0 1 2 3 4 4 5 6 6 7 7 8 8 9 9 9 10 10 10 10 10 10

e

45 58 10 21 31 33 34 33 33 30 14 14 1 47 33 10 44 18 52 26 49 12 34 52 7 22 34 42 46 50

Variants from B in C, E, P, and V: d9: B marg.45 CEPV44; d35: B marg.CEPV 48; d47: CV26; d50: C38; d62: C48; d68: V31; d70: P14; d72-d85: P one-cell shift upwards.

61

62

edi t i on of the tab les with comments

5. Right ascension Tabula ascensionum signorum in circulo recto (MS B, 63v–64r) Titles in other manuscripts MS C, 76v–77r: Tabula ascensionum signorum in circulo recto MS E, 42r: Tabula elevationum signorum in circulo recto MS P, 26v: Tabula elevationum signorum in circulo recto MS V, 35v–36r: Tabula ascensionum signorum in circulo recto in omni regione Description There is one column for the argument, from 1º to 30º at intervals of 1º. The entries represent the right ascension, displayed in 12 columns, one for each zodiacal sign, beginning in Capricorn. Actually, the entries are for the right ascension increased by 90º, α´ = α + 90º, which is also called normed right ascension. The entries are given in zodiacal signs, degrees, and minutes. This table is the same as that found in the Toledan Tables (Pedersen 2002, pp. 957–959), ultimately derived from that in the zij of al-Battānī (Nallino 1899–1907, 2: 55–56), except for the fact that it does not display a specific column for the equation of time. It may seem odd to a modern reader that John of Lignères reproduced a table in al-Battānī’s zij computed for an obliquity of 23;35º, whereas he used a solar declination table with a maximum value of 23;33,30º. In this case, however, inconsistency does not affect the results significantly, and hardly at all in the minutes, as is shown in the example below. This table is mentioned in Chapters 20–22, 37, and 40, and in others of the canons beginning Cuiuslibet arcus (see Saby 1987, pp. 113 ff.). Recomputation The entries, α´ = α + 90º, can be recomputed by means of the following modern expressions: sin α = tan δ / tan ε and δ = arcsin (sin λ sin ε), where δ is the declination of the point considered, λ its longitude, and ε the obliquity of the ecliptic. Further details are available in Chabás and Goldstein 2012, pp. 24–28. The formulas above allow us to gauge the accuracy obtained with different values of the obliquity of the ecliptic. For example, for λ = 60º we obtain α´ = 117;47,25º, which rounds off at 117;47º, when ε = 23;35º, al-Battānī’s value for the obliquity of the ecliptic; whereas we obtain α´ = 117;47,43º, which rounds off to 117;48º, when ε = 23;33,30º. The value actually given in the table is 147;47º.

63

rig ht a scension

John of Lignères. Right ascension (5) Base : Basel F II 7, 63v-64r Tabula ascensionum signorum in circulo recto Linee Ascensiones numeri Capricorni G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 17 18 19 20 21 22 23 24 25 26 28 29 30 31 32

M 6 11 16 22 28 33 38 43 48 53 58 3 8 13 18 23 27 31 35 39 43 47 51 55 58 1 4 7 10 13

Ascensiones Aquarii G 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

M 15 17 19 21 23 24 26 27 28 29 29 30 30 30 30 30 29 28 27 26 25 24 22 20 18 16 14 12 10 7

Ascensiones Piscium G 63 64 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 78 79 80 81 82 83 84 85 86 87 88 89 90

M 4 1 58 55 52 48 45 41 37 33 29 25 21 17 12 8 3 59 54 49 44 40 35 30 25 20 15 10 5 0

Ascensiones Arietis G 90 91 92 93 94 95 96 97 98 99 100 101 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 115 116 117

M 55 50 45 40 35 30 25 20 16 11 6 1 57 52 47 43 39 35 31 27 23 19 15 12 8 5 2 59 56 53

Ascensiones Tauri G 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147

M 50 48 46 44 42 40 38 36 35 34 33 32 31 30 30 30 30 30 31 31 32 33 34 36 37 39 41 43 45 47

Ascensiones Geminorum G 148 149 150 151 153 154 155 156 157 158 159 160 161 162 163 164 165 166 166 169 170 171 172 173 174 175 176 177 178 180

M 50 53 56 59 2 5 9 13 17 21 25 29 33 37 42 45 52 57 2 7 12 17 22 27 33 38 43 49 54 0

a b c d e f g h i j k l m Variants from B in C, E, P, and V: e4: E22; e25: CEP19; e26: P17; e28: P13; g17: P59; g18: P54; g19: P49; g20: E45 P44; g21: P40; g22: P35; g23: P30; g24: P25; g25: P20; h10: CP32; h13: P22; i20: C29; m16: CV47.

64

edi t i on of the tab les with comments

Variant readings This table contains 1560 sexagesimal numbers, including those for the argument. For this table, the base manuscript used is Basel, Universitätsbibliothek, F II 7 (MS B). By comparison, there are eleven variants in C, five in E, sixteen in P, and one in V. The only salient feature is an upwards shift of the entries for the minutes in Pisces in MS P.

rig ht a scension

65

Tabula ascensionum signorum in circulo recto Linee Ascensiones numeri Cancri G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

M

Ascensiones Leonis G

M

Ascensiones Virginis G

M

Ascensiones Libre G

M

181 6 213 15 243 4 270 55 182 11 214 17 244 1 271 50 183 17 215 19 244 58 272 45 184 22 216 21 245 55 273 40 185 28 217 23 246 52 274 35 186 33 218 24 247 48 275 30 187 38 218 26 248 45 276 25 188 43 220 27 249 41 277 20 48 221 28 250 37 278 16 189 190 53 222 29 251 33 279 11 191 58 223 29 252 29 280 6 193 3 224 30 253 25 281 0 194 8 225 30 254 21 281 57 195 13 226 30 255 17 282 52 196 18 227 30 256 12 283 48 197 23 228 30 257 8 284 43 198 27 229 29 258 3 285 39 199 31 230 28 258 59 286 35 54 287 31 200 35 231 27 259 201 39 232 26 260 49 288 27 202 43 233 25 261 44 289 23 203 47 234 24 262 40 290 19 204 51 235 22 263 35 291 15 205 55 236 20 264 30 292 12 206 58 237 18 265 25 293 8 208 1 238 16 266 20 294 5 209 4 239 14 267 15 295 2 210 7 240 12 268 10 295 59 211 10 241 10 269 5 296 56 212 13 242 7 270 0 297 53 n o p q r s t u v Variants from B in C, E, P, and V: p30: C14; r30: C29; v17: CE37 P35; x21: C33; x25: CEP38; z27: C41; z29: C59.

Ascensiones Scorpionis G 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 w

M

x

50 48 46 44 42 40 38 36 35 34 33 32 31 30 30 30 30 30 31 31 32 33 34 36 37 39 41 43 45 47

Ascensiones Sagitarii G 328 329 330 331 333 334 335 336 337 338 329 340 341 342 343 344 345 346 348 349 350 351 352 353 354 355 356 357 358 360 y

M

z

50 53 56 59 2 5 9 13 17 21 24 29 33 37 42 47 52 57 2 7 12 17 22 27 32 38 43 49 54 0

66

edi t i on of the tab les with comments

6. Oblique ascension Ascensiones septimi climatis cuius latitudo est 48 graduum / Ascensiones septimi climatis cuius dies longior 16 horarum equalium (MS B, 65v–66r) Titles in other manuscripts MS C, 77v–78r: Ascensiones signorum in septimo climate cuius latitudo est 48 graduum MS E, 43v: Elevationes signorum in septimo climate cuius latitudo est 48 graduum et hore equales 15 graduum et 56 minutorum MS P, 28r–v: Elevationes signorum in 7º clymate cuius latitudo est 48 graduum et hore equales 15 et 56 minutorum MS V, 37v–38r: Tabula ascensionum signorum septimi climatis cuius latitudo 48 graduum et dies longior 15 horarum et 53 minutorum In the manuscripts containing the Tables of 1322 by John of Lignères, other tables for specific latitudes are found in addition to that for latitude 48º. Manuscript B also has a table for the sixth climate, latitude 45;22º, and manuscripts E and P have tables for the latitude of Cremona (45º). Manuscript V has four additional tables, for the sixth climate (latitude 47;33º and longest daylight 15;30h), seventh climate (latitude 50º and longest daylight 16;10h), and latitudes 51º and 52º. Among the various possibilities, we have chosen to transcribe and comment on the table for latitude 48º, which is the latitude traditionally assigned to Paris in medieval manuscripts. Titles in other tables for oblique ascensions (not edited) in these manuscripts: MS B, 64v–65r: Ascensiones sexti climati cuius latitudo est 45 graduum 22 minutorum MS E, 42v–43r: Elevationes signorum secundum latitudinem civitatis Cremone 45 graduum existentem MS P, 27r–v: Elevationes signorum secundum latitudinem civitatis Cremone 45 graduum existentem MS V, 36v–37r: Tabula ascensionum signorum sexti climatis cuius latitudo 47 graduum et 33 minutorum et dies eius longior 15 horarum et 30 minutorum MS V, 38v–39r: Tabula ascensionum signorum ultra medium septimi climatis cuius latitudo est 50 graduum et eius dies longior 16 horarum equalium et 10 minutorum MS V, 39v–40r: Tabula elevationis signorum ad latitudinem 51 graduum MS V, 40v–41r: Tabula elevationis signorum ad latitudinem 52 graduum Description The format of this table is the same as for the right ascension: one column for the argument, from 1º to 30º at intervals of 1º, and entries given in zodiacal signs, degrees, and minutes, displayed in 12 columns, one for each zodiacal sign. There are two notable differences: the entries begin in the sign of Aries, not in Capricorn, and there is an additional column for the length of the diurnal seasonal hour. This quantity, called partes horarum by medieval astronomers, is one twelfth of the time from sunrise to sunset, and it is given in

67

oblique a scension

John of Lignères. Oblique ascensions (6) Base : Basel F II 7, 65v-66r Ascensiones septimi climatis cuius latitudo est 48 graduum Arietis Linee Ascensiones numeri G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

M 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 8 9 9 10 10 11 11 12 13 13 14 14

28 57 26 54 23 50 18 45 12 40 8 37 5 33 2 31 0 29 59 28 58 28 58 29 59 30 1 32 3 33

Tauri

Partes horarum G

Ascensiones

M 15 15 15 15 15 15 15 15 15 15 15 15 15 16 16 16 16 16 16 16 16 16 16 16 16 16 17 17 17 17

4 8 13 17 22 26 31 36 40 45 50 54 58 2 7 11 16 20 24 29 33 38 42 47 51 55 0 4 8 13

G 15 15 16 16 17 17 18 19 19 20 20 21 21 22 23 23 24 25 25 26 26 27 28 28 29 30 31 31 32 33

M 6 38 12 45 18 52 27 2 36 11 46 21 56 32 6 45 24 2 40 18 58 38 18 58 39 21 4 47 31 14

Geminorum Partes horarum G

Ascensiones

M 17 17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 19 19

17 21 25 29 33 37 41 45 49 53 57 1 5 9 13 16 20 24 28 32 35 39 42 45 49 52 56 59 2 5

G 34 34 35 36 37 37 38 39 40 41 42 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

M 0 46 32 18 4 53 42 31 20 9 2 55 48 41 33 31 28 25 21 18 17 17 16 15 15 18 22 25 28 32

Partes horarum G

M 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19

8 11 13 16 19 21 23 26 28 31 33 35 37 39 41 42 44 45 46 48 49 50 51 52 52 53 53 54 54 54

a b c d e f g h i j k l m Variants from B in C, E, P, and V: b27: P12; c1 : E58 corr.28; c3: E56; c9: V13; c13: CEP6; c25: CEP50; d27: E16; d28: E16; d29: E16; d30: E16 corr.17; e2: E18; e13: CEPV59; e14: CEPV3; e23: CEPV43; f28: V32; g2: CEPV39; g5: P12; g10: E7; e13: V51; e16: C55; e19: C44 EP14; g23: EP28; g25: CEPV38; k2: V26; k8: EPV32; k9: V50 corr.20; k15: CEV34; k24: CEPV16; k25: CV17; k30: E22; m1: CEP11; m2: CEP13; m3: CEP17; m4: CEP19V17; m5-m22: EP one-cell shift upwards; m16: V43.

68

edi t i on of the tab les with comments

degrees and minutes. The maximum value in this column is 19;54º at longitudes 87º–92º. This value corresponds to the maximum length of daylight announced in the title for the length of the diurnal seasonal hour. Indeed, 19;54º converted into hours is 15;55h ≈ 16h; see the variety of values in the titles. The same table for latitude 48º (Climate 7) is found among the Toledan Tables (Pedersen 2002, pp. 1067–1070). This table is not mentioned in John of Lignères’ canons. Recomputation The entries for oblique ascension result from adding the corresponding entries for right ascension displayed in Table 5 to those for the ascensional difference given in Table 4, with a coefficient cotan (90º – φ) for the specific value of 42º = 90º – 48º found in Table 2. The entries for the diurnal seasonal hours are obtained by means of the following expression: (α(λ + 180) – α(λ))/12, where α is oblique ascension as a function of longitude, λ. Variant readings For this table, the base manuscript used is also Basel, Universitätsbibliothek, F II 7 (MS B). This table contains 1560 sexagesimal numbers, including those for the argument. Compared with B, there are 89 variants in C, 106 in E, 144 in P, and 69 in V. There are 11 common variants in the four manuscripts we collated with B -errors likely due to this particular copyist- that do not reflect the high volume of mistakes in them. The poor score obtained by the scribe of MS P (over 9% errors) results from shifts of entries in various columns. The expression (α(λ + 180) – α(λ))/12 for computing the length of the diurnal seasonal hour makes it possible to detect scribal errors. Consider, for example, the case when the longitude is Aries 30º. All manuscripts consulted agree on the entry 17;13º. This value should result from dividing the difference between the oblique ascensions for Libra 30º and Aries 30º by 12. While for Aries 30º all manuscripts consulted give an oblique ascension of 14;33º, for Libra 30º we are offered different entries: 220;7º (MS B), 221;7º (MSS E and P), and 222;7º (MS C). With these three values, we obtain 17;8º, 17;13º, and 17;18º, respectively. Therefore, the scribes of manuscripts B and C miscopied this entry or copied it from erroneous original copies.

69

oblique a scension

Ascensiones septimi climatis cuius latitudo est 48 graduum Cancri Linee Ascensiones numeri G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

61 62 63 65 66 67 68 69 70 72 73 74 75 77 78 79 80 82 83 84 85 87 88 89 91 92 93 95 96 97

M 41 49 58 7 15 26 36 47 57 8 21 35 48 1 14 30 45 1 17 33 52 10 29 48 6 26 46 6 26 47

Leonis

Partes horarum G

Ascensiones

M 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19

54 54 53 53 52 52 51 50 49 48 46 45 44 42 40 39 37 35 33 31 29 26 23 21 19 16 13 11 8 5

G 99 100 101 103 104 105 107 108 109 111 112 114 115 116 118 119 120 122 123 125 126 127 129 130 131 133 134 136 137 138 f

M 7 28 48 9 30 52 14 36 58 20 42 5 27 49 11 35 59 23 47 11 33 54 15 37 58 21 44 7 30 53

Virginis

Partes horarum G

Ascensiones

M 19 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 17 17 17 17 17 17 17 17 17 17 17 17

2 58 55 52 49 45 42 39 35 32 28 24 20 16 13 9 5 1 57 53 49 45 41 37 33 29 25 21 17 13

G 140 141 143 144 145 147 148 149 151 152 154 155 156 158 159 160 162 163 164 166 167 169 170 171 173 174 175 177 178 180 j

M 16 40 3 27 49 11 33 54 16 36 1 24 46 9 32 54 15 37 59 20 43 5 28 51 13 35 56 17 39 0

Partes horarum G

M 17 17 17 16 16 16 16 16 16 16 16 16 16 16 16 16 15 15 15 15 15 15 15 15 15 15 15 15 15 15

8 4 0 55 51 47 42 38 33 29 24 20 16 11 7 2 58 54 50 45 40 36 31 26 22 17 13 8 4 0 m

a b c d e g h i k l Variants from B in C, E, P, and V: c5: C16; c7: C26; c11: CEPV22; c12: E25; c15: CP18; e4: E52; e5: E53; e6: E53; e16: C30; e25: E20; g23: E11 PV13; h1: E18; i1: P3; i13: V20 corr.19; j8: C150; j10: C153; k2: EPV14 V corr.40; k3: EPV13 V corr.3; k4: E17; k6 C33 V11 corr.23; k7: C54 V33 corr.54; k8: C16 V54 corr.16; k9: CP38 V16 corr.36; k10: CP1 V38 corr.1; k11: CP24 V1 corr.34; k12: C46 V24; corr.45; k13: C9 V46 corr.9; k14: C32 V9 corr.32; k15: C54 V32 corr.54; k16: C15 V54 corr.15; k17: C37 V15 corr.37; k18: C59 V37 corr.59; k19: C20 V59 corr.20; k20: C43 V20 corr.43; k21: C5 V43 corr.5; k22: C28 V5 corr.28; k23: C51 V28 corr.51; k24: C13 V51 corr.13; k25: C37 EP23 corr.35; k26: C56 V35 corr.56; k27 C17 V56 corr.17; m10: CEPV28; m15: CEPV6; m27: CPV14; m28: CPV9.

70

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Ascensiones septimi climatis cuius dies longior 16 horarum equalium Libre Linee Ascensiones numeri G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

M

Scorpionis Partes horarum G

M

Ascensiones G

M

Sagitarii

Partes horarum G

M

Ascensiones G

M

Partes horarum G

M

181 21 14 56 222 30 12 43 263 34 10 53 182 43 14 51 223 53 12 39 264 54 10 49 184 4 14 47 225 16 12 35 266 14 10 47 185 25 14 43 226 39 12 31 267 34 10 44 186 45 14 39 228 2 12 27 268 54 10 41 188 9 14 34 229 23 12 23 270 12 10 39 189 32 14 29 230 45 12 19 271 31 10 37 190 55 14 24 232 6 12 15 272 50 10 34 17 14 20 233 27 12 11 274 9 10 32 192 193 40 14 15 234 49 12 7 275 27 10 29 195 1 14 10 236 12 12 3 276 43 10 27 196 23 14 6 237 37 11 59 277 59 10 25 197 45 14 2 239 1 11 55 279 15 10 23 199 6 13 58 240 25 11 51 280 30 10 21 200 28 13 53 241 49 11 47 281 45 10 19 201 51 13 49 243 11 11 44 282 59 10 18 203 14 13 44 244 35 11 40 284 12 10 16 204 36 13 40 245 55 11 36 285 25 10 15 18 11 32 286 39 10 14 205 59 13 36 247 207 22 13 32 248 40 11 28 287 52 10 12 208 44 13 27 250 11 11 25 289 3 10 11 210 5 13 22 251 24 11 21 290 13 10 10 211 27 13 18 252 46 11 18 291 24 10 9 212 49 13 13 254 8 11 15 292 34 10 8 214 11 13 9 255 30 11 11 293 45 10 8 215 34 13 5 256 51 11 8 294 53 10 7 216 57 13 0 258 12 11 5 296 2 10 7 218 20 12 56 259 32 11 2 297 11 10 6 19 10 6 219 44 12 52 260 53 10 58 298 220 7 12 47 262 13 10 55 299 28 10 6 a b c d e f g h i j k l m Variants from B in C, E, P, and V: b26: C214; b30: C222 EP221; c5: CEV47; c7: EV33 corr.32; c19: EP57 V50 corr.59; c20: P23; c26: C14; d28 to d30: E13; e1: CV58 corr.56; e4: C49 V44 corr.43; e23 corr.17; f24: C253; f27: CV257; f28: V258; g6: EP13; g11: CEP13; g18: C45; g21: CEP2; g22: C2; g23: C24; g24: C8; g25: C8; g26: C30; g27: C51; g28: C12; g29: CE53; g30: CE14; h10: E11; h11: E11; h29: E11; h30: E11; i5: E23 P25; i6: EP21; i15: EP49; i16: E47 P45; i19: CEP33; i20: CEP29; i24: C12; j27: P295; k4: P31; k9: CEPV8; k15: CEPV46; k24: EP32; k25: C55; m12: E24 corr.25.

71

oblique a scension

Ascensiones septimi climatis cuius dies longior 16 horarum equalium Capricorni Linee Ascensiones numeri G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

M

Aquarii

Partes horarum G

M

Ascensiones G

M

Piscium

Partes horarum G

M

Ascensiones G

M

Partes horarum G

M

300 32 10 6 327 29 10 58 345 57 12 52 301 35 10 6 328 13 11 2 346 28 12 56 302 38 10 7 328 56 11 5 346 59 13 0 303 43 10 7 329 29 11 8 347 30 13 5 304 45 10 8 330 22 11 12 348 1 13 9 305 44 10 8 331 2 11 15 348 31 13 13 306 44 10 9 331 42 11 18 349 2 13 18 307 43 10 10 332 22 11 21 349 32 13 22 43 10 11 333 2 11 25 350 2 13 27 308 309 42 10 12 333 42 11 28 350 32 13 31 310 39 10 13 334 20 11 32 351 1 13 36 311 35 10 15 334 58 11 36 351 31 13 40 312 31 10 16 335 37 11 40 352 0 13 44 313 29 10 18 336 15 11 44 352 29 13 49 314 26 10 19 336 53 11 47 352 58 13 53 315 19 10 21 337 24 11 51 353 27 13 58 316 12 10 23 338 3 11 55 353 55 14 2 317 5 10 25 338 39 11 59 354 23 14 6 14 12 3 354 52 14 10 317 58 10 27 339 318 51 10 29 339 49 12 7 355 22 14 15 319 40 10 32 340 34 12 11 355 48 14 20 320 29 10 34 340 58 12 15 356 15 14 24 321 18 10 37 341 33 12 19 356 42 14 29 322 7 10 39 342 8 12 23 357 10 14 34 322 56 10 41 342 42 12 27 357 37 14 39 323 42 10 44 343 15 12 31 358 6 14 43 324 28 10 47 343 48 12 35 358 34 14 47 325 14 10 49 344 21 12 39 359 3 14 51 32 14 56 326 0 10 52 344 53 12 43 359 326 46 10 55 345 27 12 47 360 0 15 0 a b c d e f g h i j k l m Variants from B in C, E, P, and V: b18: P316; b24: P321; b29: P325; c4: CV42 E34 P40; c7-c29: EP one-cell shift upwards; c15: C20 V24; c30: P36; c25: V58 corr.56; e25: P42; f9: C332; f12: V335; f13: V336; f15: V337; f16: V338; f18: V339; f20: V340; f21: C330; f22: V341; f23: V342; f24: P341; f25: V343; g4: CEP39 V30 corr.39; g15: C59; g21: CPV24; g25: EP22; g26: P35; g30: V17 corr.27; h19: C11; h20: P11; i1: E28; i14: E4 P41; i15: P42; i16: E41 P43; i17: P45; i18: P50; i19-i30: P two-cell shift downwards; i22: C12; i29: E47; i30: E43; j3: C340; j9: C349; k19: EP53; k23: P43; l2: C13; l3: P12; l4: P12; l17: P13; l18: P13; m1: P43: m2: P47; m3-m7: P two-cell shift downwards; m9-m21: P twocell shift downwards; m20: C12; m22: E34; m23: E39; m24: E44.

72

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7.

Equation of time

Tabula equationis dierum cum noctibus suis (MS B, 68v) Titles in other manuscripts MS C, 78v: Tabula equationis dierum cum suis noctibus MS E, 46v: Tabula equationis dierum cum noctibus suis MS P, 54r: Tabula equationis dierum cum noctibus suis MS V, 33r: Tabula equationis dierum cum noctibus suis Description The equation of time is the difference between apparent and mean time, where apparent time is counted from the moment the Sun crosses the meridian and mean time is counted from mean noon. In this set, the equation of time is not displayed as a column in a table for right ascensions, but as a table by itself, with entries in degrees and minutes for each degree of the ecliptic, beginning in Capricorn 1º. The relative extremal values are: Min: 0; 0º (at Aqu 18º –25º), max: 5;21º (at Tau 25º–27º), min: 2;49º (at Leo 5º), Max: 7;57º (at Sco 8º–9º). The equation of time is a complex function that depends on several variables: i) the longitude of the solar apogee, which varies slowly with time; ii) the eccentricity of the solar model used, closely related to the maximum value of the solar equation, a parameter that astronomers often modified since Ptolemy (a list of historical values can be found in Chabás and Goldstein 2012, p. 66); iii) the obliquity of the ecliptic, also frequently updated through time (for a list of historical values of it, see Chabás and Goldstein 2012, p. 23); and iv) an epoch constant introduced to make the entries of the equation of time always positive. In contrast to most tables in a set, this time-dependency made some astronomers to adjust this table to their own parameters. An illustration of this table can be found in Chabás 2019, p. 182. The entries do not agree with those in the Toledan Tables, but they do agree with the description given by Peter of Saint Omer, who was active in Paris around 1294, for his own table for the equation of time. The figure below represents the curves corresponding to the equation of time in the tables by John of Lignères and the editio princeps of the Parisian Alfonsine Tables (Ratdolt 1483), which is the same as in the Toledan Tables and the zij of al-Battānī. The units on the Y-axis are time-degrees. This table is mentioned in Chapters 31 and 46 of the canons beginning Priores astrologi (see Saby 1987, pp. 220–222 and 275–277).

73

equation of time

John of Lignères. Equation of time (7) Base : Basel F II 7, 68v Tabula equationis dierum cum noctibus suis Linee numeri Gradus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Equatio dierum

Equatio dierum

Equatio dierum

Equatio dierum

Equatio dierum

Equatio dierum

Capricorni

Aquarii

Piscium

Arietis

Tauri

Geminorum

G

M 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0

46 37 29 22 14 7 0 52 45 38 31 24 17 10 3 57 51 45 39 33 27 22 16 10 5 1 57 52 47 42

G

M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

38 34 31 27 23 21 19 16 14 12 10 8 6 4 3 2 1 0 0 0 0 0 0 0 0 1 2 3 4 6

G

M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 2

9 11 13 15 18 21 23 26 30 33 37 40 44 48 52 56 1 5 9 14 19 23 28 33 37 42 47 52 57 2

G

M 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4

7 12 17 22 27 33 38 43 48 53 59 4 9 15 21 27 31 36 40 45 50 55 0 4 9 13 17 21 25 29

a b c d e f g h i Variants from B in C, E, P, and V: c2: C47; c5: E4; c8: C55; c18: E59; g9: E29 V29 corr.30; g25: EP38; i19: E49.

G

j

M 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

k

33 36 39 43 46 49 53 56 58 1 3 6 9 10 12 14 15 17 18 19 19 20 20 20 21 21 21 20 20 20

G

l

M 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

19 17 15 14 13 12 10 8 6 4 1 59 57 55 52 49 46 43 40 37 34 31 28 25 22 19 16 13 10 7 m

74

edi t i on of the tab les with comments

Variant readings For this table, the base manuscript used is again Basel, Universitätsbibliothek, F II 7. It contains 780 sexagesimal numbers, including those for the argument. Compared to B, there are eighteen variants in C, nine in E, two in P, and sixteen in V. Of these, only one is common to MSS C, E, P, and V, indicating a scribal error in B.

75

equation of time

Tabula equationis dierum cum noctibus suis Linee numeri Gradus 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Equatio dierum

Equatio dierum

Equatio dierum

Equatio dierum

Equatio dierum

Equatio dierum

Cancri

Leonis

Virginis

Libre

Scorpionis

Sagittarii

G

G

G

M 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2

4 1 58 54 50 47 44 41 38 35 32 28 25 22 19 16 13 11 9 7 5 3 1 59 57 56 55 54 53 52

G

M 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

G 51 50 50 50 49 50 51 51 52 52 53 54 56 58 59 0 2 5 7 9 11 14 17 20 23 26 29 32 35 39

M 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 6

43 47 51 55 59 4 8 12 17 22 26 31 36 40 46 50 56 1 6 11 16 20 25 30 35 40 45 50 55 0

G

M 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

5 10 15 20 25 30 35 40 44 48 53 57 1 5 8 12 16 20 23 26 29 32 35 38 41 43 45 47 49 51

M 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

53 54 54 55 55 56 56 57 57 56 56 55 55 54 53 51 50 48 46 44 41 38 36 32 27 25 21 17 14 10

M 7 7 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 3

5 0 55 50 45 40 34 28 22 16 10 2 58 52 45 38 30 23 16 9 2 55 48 40 32 24 17 9 2 54

a b c d e f g h i j k l m Variants from B in C, E, P, and V: c3: CE57 V57 corr.59; c18: C22 E12 ; e11: C54 V54; e12: CV56; e15: CV0 E58; e16: C12 V2; e17: CV5 ; e18: CV7; e19: CV9; e20: CV11; e21: CV14; e22: CV17; e23: CV20; e24: CV23; e25: CV26; e26: CV29; i28: E vac.; k25: CEPV28; m12: C4.

76

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8. Planetary latitudes Tabula latitudinis quinque planetarum (MS B, 69v) Titles in other manuscripts Since this table is not found in MS Erfurt, we have used London, British Library, Egerton 889 (henceforth MS L) as a fourth comparison manuscript. MS C, 79r: Tabula latitudinis quinque planetarum MS L, 39r: Tabula latitudinum 5 planetarum MS P, 24v: Tabula latitudinum 5 planetarum que est longitudo eorum a via solis in circulo signorum MS V, 44v: Tabula latitudinis quinque planetarum que est longitudo a via solis in circulo signorum Description The argument is presented in two columns (from 6º to 180º, and their complement in 360º) at intervals of 6º. Most manuscripts divide the table in two halves, leaving a blank line between arguments 90º and 96º. For each of the five planets, two columns are displayed. In the case of the three superior planets, the headings are effregion septentrionale and effregion meridionale, corresponding to northern and southern latitudes, respectively. For the two inferior planets, the columns are headed declinatio and reflexus, referring to the first two components of latitude: inclination and slant. We note that the third component of latitude, deviation, is not addressed. There is yet another column for purposes of interpolation, headed minuta proportionalia (minutes of proportion). All entries in the table are given in degrees and minutes, except in the last column, where they are displayed in minutes and seconds. This table is already found in the Toledan Tables (Pedersen 2002, pp. 1325–1326) and in the zij of al-Battānī (Nallino 1899–1907, 2: 140–141). It ultimately derives from Almagest XIII.5 (Toomer 1984, pp. 632–634). It is mentioned in Chapters 21–24 of the canons beginning Priores astrologi (see Saby 1987, pp. 206–211). Variant readings For this table, the base manuscript used is Basel, Universitätsbibliothek, F II 7. There are 720 sexagesimal numbers, including those for the argument. Compared to B, there are 40 variants in C, 31 in L, 24 in P, and 61 in V. Of these, two are common to MSS C, L, P, and V, both in the minutes, indicating scribal errors in B.

pla neta ry latitudes

John of Lignères. Planetary Latitudes (8) Base : Basel F II 7, 69v Tabula latitudinis quinque planetarum Linee Latitudo Saturni per numeri augmentum 50 graduum super augmentate centrum per 6 gradus Effregion septentrionale G 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

G 354 348 342 336 330 324 318 312 306 300 294 288 282 276 270

96 102 108 114 120 126 132 138 144 150 156 162 168 174 180 a

264 258 252 246 240 234 228 222 216 210 204 198 192 186 180 b

G

c

M

Effregion meridionale G

M

Latitudo Jovis per diminutionem 20 graduum a cuspide Effregion septentrionale G

M

Effregion meridionale G

M

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

4 5 6 7 8 10 11 12 14 16 18 21 24 27 30

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 3 4 5 6 7 8 10 13 15 18 21 24 27 30

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

7 8 8 9 10 11 12 13 14 16 18 21 24 27 30

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

5 6 6 7 8 9 10 11 Superior 13 16 18 21 24 27 30

2 2 2 2 2 2 2 2 2 2 2 3 3 3 3

34 36 39 42 45 47 50 53 55 57 59 0 1 2 2

2 2 2 2 2 2 2 2 2 2 3 3 3 3 3

33 36 39 42 45 48 51 54 56 58 0 2 3 4 5

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

33 36 39 42 45 48 51 54 57 0 3 5 6 7 8

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2

33 36 39 42 45 48 51 54 57 0 3 5 6 7 8

d

e

f

g

h

i

j

Inferior

y

77

78

edi t i on of the tab les with comments

Tabula latitudinis quinque planetarum Linee numeri augmentate per 6 gradus

Latitudo Martis et est absque augmento et diminutione

Effregion septentrionale G 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

G 354 348 342 336 330 324 318 312 306 300 294 288 282 276 270

96 102 108 114 120 126 132 138 144 150 156 162 168 174 180 a

264 258 252 246 240 234 228 222 216 210 204 198 192 186 180 b

G

k

M

Effregion meridionale G

M

Latitudo Veneris

Declinatio

Reflexus

G

G

M

M

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 9 11 13 14 16 18 21 24 28 32 36 41 46 52

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 4 5 6 7 9 12 15 18 22 26 32 36 42 49

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

2 1 0 59 57 55 51 46 41 36 29 23 16 8 0

0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

8 16 24 33 41 49 57 5 13 20 28 35 43 50 57

Superior

0 1 1 1 1 1 2 2 2 2 3 3 4 4 4

59 6 14 23 34 47 1 16 34 55 16 38 0 14 21

0 1 1 1 1 1 2 2 2 3 4 4 5 6 7

56 4 13 24 37 51 10 33 56 29 9 55 43 36 30

0 0 0 0 0 1 1 1 2 3 3 4 5 6 7

10 20 32 45 59 13 38 57 23 3 43 26 24 14 22

2 2 2 2 2 2 2 2 2 2 2 1 1 0 0

3 9 15 20 25 28 30 30 28 22 12 55 27 48 0

Inferior

l

m

n

o

p

q

r

y

pla neta ry latitudes

79

Tabula latitudinis quinque planetarum Linee numeri augmentate per 6 gradus

G 6 12 18 24 30 36 42 48

G 354 348 342 336 330 324 318 312

54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180 a

Latitudo Mercurii

Partes latitudinis quinque planetarum

Declinatio

Reflexus

G

G

M

Minuta proportionalia

M

M

S

1 1 1 1 1 1 1 1

45 44 43 0 39 30 23 15

0 0 0 0 0 1 1 1

11 22 33 44 55 6 17 27

59 58 57 54 52 48 44 40

306 300 294 288 282 276 270

1 0 0 0 0 0 0

8 59 44 38 26 16 0

1 1 1 2 2 2 2

35 43 52 0 7 14 20

35 30 24 18 12 6 0

264 258 252 246 240 234 228 222 216 210 204 198 192 186 180 b

0 0 0 1 1 1 2 2 2 3 3 3 3 4 4

15 31 48 6 25 45 6 26 47 7 26 42 54 2 5

2 2 2 2 2 2 2 2 2 1 1 1 0 0 0

25 28 29 30 29 26 20 11 0 45 29 10 48 24 0

6 12 18 24 30 35 40 44 48 52 54 56 58 59 60

s

t

u

v

w

36 36 0 36 0 24 24 0 Superior 12 0 24 24 24 24 0 24 24 24 24 0 12 0 24 Inferior 24 0 36 0 36 36 0 x y

Variants from B in C, L, P, and V: b162: C298; b163: C292; d24 to d84: LV one cell shift upwards; d72: C20; d90: LV32; d132: C54 PV8; d180: CV2 P3; f24: C4 V5; f30: C5 V6; f144: CLPV55; f180: V4; f150: CV2 P1; h66: V17; h72: CV21 P22; j84: V28; k18, 30, 48, 60, 78: P vac.; l6: CV7; l60: CP29 V28; 78: C41 PV42; l120: C24 PV34; l162: CLV33; m18, 36, 48, 60, 72: P vac. ; m150: CV3 P2; m156: CV4 P3; m168: CV5 P4; m174: CV6 P5;m180: CV7 P6; n24: L7; n30: L8; n72: CP30 V32; n174: CPV26; n180: CV30 P20; o6 to 12 : CV1; o18, 36, 48, 60, 78, 108: P vac.; o150 to 156: V2; o162 to 168: V3; o174 to 180: V4; p12: C2 V1; p30: C58 V57; p66: LV31; p72: C33 P22 V23; p138: CV 57 P59; p150: V33; p180: CPV12; q18, to q30: P vac.; r78: CV43 P42; p138: V38; r36: L46; s102: P vac.; t12: V34; t24: CLPV40; t30: CP36 V39; t42: CP24 V23; t48: CP16 V15; t66: CP49 V44; t72: V30; u18: P vac.; u66: C2 PV1; v12: C12; v54: CPV44; v60: L44; v96: V26; v102: V18; v132: V21; v174: CLV28; w150: C;50 in C, one more column z with tertia: y6 to y90 are: 4,10,16, 24, 26, 36, 40, 44, 48, 52, 56, 60, 60, 60, 0.

80

edi t i on of the tab les with comments

9. Lunar latitude Tabula latitudinis lune et est distancia eius a via solis (MS B, 70r) Titles in other manuscripts Since this table is not found in MSS Erfurt and Paris, we have used London, British Library, Egerton 889 (MS L) and Oxford, Bodleian Library, Can. Misc. 27 (henceforth MS Ox) as comparison manuscripts. MS C, 83v: Tabula latitudinis lune MS L, 40r: Tabula latitudinis lune MS Ox, 91r: Tabula latitudinis lune MS V, 45v: Tabula latitudinis lune Description The entries in this table represent the latitude of the Moon at intervals of 1º, counted from the lunar ascending node from 1º to 180º. The entries are given to seconds, and they are symmetrical about 90º, where they reach a maximum value of 5;0,0º. In the table for the lunar equation in Almagest V.8 (Toomer 1984, p. 238), Ptolemy displayed a column for lunar latitude with a maximum of 5;0º. This is also the case in the zij of al-Battānī, where the entries are displayed to seconds, with a maximum of 5;0,0º, for all integers from 1 to 180 (Nallino 1899–1907, 2: 78–83). The Toledan Tables reproduce al-Battānī’s values for lunar latitude with a column in the table for lunar equations (Pedersen 2002, pp. 1253–1258). John of Lignères adhered to this long tradition, but he displayed the latitude of the Moon in a separate table. The entries in his table agree with those in the Toledan Tables. This table is mentioned in Chapter 20 of the canons beginning Priores astrologi (see Saby 1987, p. 205). Recomputation The entries in this table can be recomputed by means of the following modern formula: β = arcsin (sin i · sin ω), where β is the lunar latitude, i is the inclination of the lunar orb -here taken as 5;0º- and ω is the argument of latitude, that is, the angular distance between the lunar ascending node and the true position of the Moon on its orb, although many medieval astronomers took the argument of latitude to be the angular distance on the ecliptic between the ascending node and the foot of the perpendicular drawn from the Moon to the ecliptic. Variant readings For this table, the base manuscript is Basel Universitätsbibliothek, F II 7 (MS B). The table contains 660 sexagesimal numbers, including those for the argument. Compared to

luna r latitude

John of Lignères. Lunar latitude (9) Base : Basel F II 7, 70r Tabula latitudinis lune et est distancia eius a via solis Signa

0

1

2

Linee numeri

Latitudo lune et est ascendens

Latitudo lune et est ascendens

Latitudo lune et est ascendens

G

G

G

G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Signa

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 16 12 11 10 9 8 7 6 5 4 3 2 1 0

M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2

5 10 15 20 26 31 36 41 46 52 57 2 7 12 17 22 27 32 37 42 47 52 57 1 6 11 16 20 25 29 11

S 13 27 40 53 7 19 31 42 52 1 9 16 23 30 36 35 33 31 29 27 23 17 8 56 40 22 2 40 17 52

M 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4

34 38 43 47 51 56 0 4 8 12 16 20 24 28 32 35 39 42 46 49 53 56 59 2 5 8 11 14 17 19 10

S 24 21 17 39 57 10 21 29 35 39 39 35 26 15 0 41 17 49 17 40 0 16 28 37 38 37 34 22 7 47

M 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5

S

22 24 27 29 31 33 36 38 40 41 43 45 46 48 49 51 52 53 54 55 56 57 57 58 58 59 59 59 59 0 9

a b c d e f g h i j k Variants from B in C, L, Ox, and V: c12: C0 V1; d17: Ox25; e26: C42 V22; g1: CLV33; g3: CLV44; g6: COxV57; h2: CLV52; h3: CLV57; i30: C4 V5.

22 51 14 34 49 59 4 4 0 52 38 18 52 20 44 3 17 35 28 25 17 4 45 25 55 15 35 50 58 0

81

82

edi t i on of the tab les with comments

B, there are eleven variants in C, nine in L, four in O, and eleven in V. Of these, seven are common to MSS C, L, and V, which can thus be considered as possible scribal errors in B. As indicated above, the entries are symmetrical about 90º, and thus β(ω) = β(180º – ω). However, this is not always the case in the table of MS B, for this equality does not hold for 13 pairs, all of them in the seconds, out of 90 entries. We note that this lack of symmetry in a few entries also occurs in the Toledan Tables, as transcribed by F. S. Pedersen 2002, pp. 1253–1258, where six non-symmetrical pairs can be identified. In the table by John of Lignères, the greatest difference within a pair amounts to four seconds, and corresponds to the entries 4;11,34º (at argument 1s 27º) and 4;11,30º (at 4s 3º). With the formula above, recomputation yields 4;11,30º. The six non-symmetrical pairs are reproduced in the five manuscripts we have collated. We thus conclude that John of Lignères did not compute the table for lunar latitude in his own set. Instead, he borrowed it directly from the Toledan Tables and reproduced the erroneous entries found there. It is worth highlighting that in the editio princeps of the Parisian Alfonsine Tables (Ratdolt 1483), no different values are found in any of these six pairs, indicating that the entries in this table were carefully reviewed before going to press or were taken from a non-erroneous manuscript of a different tradition.

luna r latitude

Tabula latitudinis lune et est distancia eius a via solis Signa

3

4

5

Linee numeri

Latitudo lune et est descendens

Latitudo lune et est descendens

Latitudo lune et est descendens

G

G

G

G

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Signa

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 16 12 11 10 9 8 7 6 5 4 3 2 1 0

M 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

59 59 59 59 58 58 57 57 56 55 54 53 52 51 49 48 46 45 43 41 40 38 36 33 31 29 27 24 22 19 8

S 58 50 35 15 15 21 45 4 17 25 28 25 17 3 44 20 51 18 38 52 0 4 4 59 49 34 14 41 22 46

M 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2

17 14 11 8 5 2 59 56 53 49 46 42 39 35 32 28 24 20 16 12 8 4 0 56 51 47 43 38 34 29 7

S 7 22 30 37 38 35 28 16 0 40 17 49 17 41 0 15 26 36 39 30 30 39 21 10 57 39 17 21 24 52

M 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

S

25 20 16 11 6 1 57 52 47 42 37 32 27 22 17 12 7 2 57 52 46 41 36 31 26 20 15 10 5 0

17 40 2 22 40 56 8 17 23 27 29 31 33 35 34 30 23 16 9 1 52 42 31 19 6 53 40 27 13 0

6

l m n o p q r s Variants from B in C, L, Ox, and V: n5: CLV51 Ox55; n6: Ox25; n28: CLV51; p9: V53; q20: LOx39; q22: CLV29; q29: L29.

t

83

84

edi t i on of the tab les with comments

10. Daily unequal motion of the planets Tabula diversorum motuum planetarum in una die (MS B, 69r) Titles in other manuscripts MS C, 79v: Tabula diversorum motuum planetarum in una die MS E, 46v: Tabula diversorum motuum planetarum in una die MS P, 55r: Tabula diversorum motuum planetarum in una die MS V, 45r: Tabula diversorum motuum planetarum in una die Description This table is used to determine the daily velocity of the planets at any time. The argument is given at steps of 6º in two columns from 6º to 180º, and their complement in 360º. For each of the five planets, there are two columns, one for each of the two variables on which planetary velocities depend: its centre and anomaly. The entries represent the arcs travelled in a day in both components and they are given to seconds. In many manuscripts there is an additional column for the hourly (not daily) mean motion in lunar longitude and, in some cases, there are two other columns for the daily progress of the Moon, in degrees (see Goldstein, Chabás, and Mancha 1994). In the table, R stands for “retrogradus”. Chapter 27 of the canons to the Castilian Alfonsine Tables gives a detailed description of this table. It is undoubtedly of Andalusian origin and many copies exist in Arabic, Hebrew, and Latin. Therefore, it was not authored by John of Lignères. A complete edition based on about 50 manuscripts and printed editions can be found in Chabás and Goldstein 2003, pp. 170–182; see also Pedersen 2002, pp. 1433–1436. The table is mentioned in Chapter 44 in the canons beginning Priores Astrologi (see Saby 1987, pp. 268–273). For a transcription of the Castilian Chapter 27 and John of Lignères’ Chapter 44 and their translations into English, see Goldstein, Chabás, and Mancha 1994, pp. 73–84, with a comparison of both texts. Recomputation The mathematical analysis of the table for the true velocity of the planets is found in Goldstein, Chabás, and Mancha 1994, pp. 68–72. There was no full agreement between entries and recomputation, although a reasonably close agreement was reached for the velocities of the planets when using the parameters in Ptolemy’s models, and for the lunar velocity when using al-Battānī’s correction table.

85

daily unequa l motion of the pla nets

John of Lignères. Daily unequal motion of the planets (10) Base : Basel F II 7, 69 r Tabula diversorum motuum planetarum in una die Linee numeri Motus Saturni diversus in Motus Jovis diversus in communes una die una die augmentate Motus Motus Motus Motus per 6 puncti centri portionis puncti centri portionis G

G

M

S

M

S

M

S

M

S

Motus Martis diversus in una die Motus puncti centri M

S

Motus portionis M

S

6 354 1 44 5 43 4 32 8 50 25 43 11 5 12 348 1 45 5 36 4 34 8 42 25 50 11 0 18 342 1 46 5 24 4 35 8 30 26 0 10 58 24 336 1 46 5 12 4 36 8 18 26 15 10 45 30 330 1 47 5 0 4 38 8 7 26 30 10 42 36 324 1 48 4 46 4 39 7 50 26 45 10 36 30 42 318 1 48 4 32 4 41 7 34 27 0 10 48 312 1 49 4 16 4 43 7 24 27 15 10 24 54 306 1 50 3 50 4 44 6 50 27 30 10 11 60 300 1 51 3 20 4 46 6 20 27 50 10 0 66 294 1 52 2 52 4 48 5 45 28 25 9 48 72 288 1 53 2 22 4 50 5 5 29 0 9 30 78 282 1 55 1 50 4 53 4 45 29 40 9 10 84 276 1 56 1 15 4 55 3 34 30 20 8 50 90 270 1 58 0 36 4 58 2 22 31 0 8 25 96 264 2 0 R0 R0 5 0 1 10 31 35 7 55 3 R0 R0 32 10 7 10 102 258 2 1 0 36 5 108 252 2 3 1 20 5 6 1 15 32 55 6 20 114 246 2 4 2 0 5 10 2 30 33 30 5 20 120 240 2 6 2 40 5 13 3 45 34 0 4 0 126 234 2 7 3 10 5 16 5 0 34 50 2 10 132 228 2 8 3 50 5 19 6 15 35 35 0 0 138 222 2 9 4 30 5 21 7 25 36 10 2 18 144 216 2 10 5 0 5 23 8 32 36 35 7 45 150 210 2 11 5 20 5 25 9 40 37 0 13 0 156 204 2 12 5 50 5 27 10 40 37 20 25 20 198 2 12 6 15 5 28 11 35 37 40 29 30 162 168 192 2 13 6 40 5 29 12 35 38 0 39 29 174 186 2 14 7 0 5 30 12 50 38 20 49 20 180 180 2 14 7 15 5 30 13 0 38 40 53 50 a b c d e f g h i j k l m n Variants from B in C, E, P, and V f6:V48; f60: V10; h12: CEP33; j12: P22; j168: CEPV25; k162: EP38; l90: V10; l96: P45; n18: EP55; n24: EP48; n180: P59.

86

edi t i on of the tab les with comments

Variant readings For this table, the base manuscript used for collation is MS B. The variants in the other manuscripts examined only contain scribal errors. Since the columns for the hourly mean motion of the Moon are not found in MS B, for this specific column we have used MS C as base manuscript to collate MSS E, P, and V. The table for the planets contains 660 sexagesimal numbers, including those for the argument, and there are 60 additional sexagesimal numbers in the sub-table for the Moon. For the planets, compared to B, there are five variants in C, nine in E, eleven in P, and eight in V. Of these, only two are common to MSS C, E, P, and V, indicating possible errors in B. For the Moon, compared with C, there are eight variants in E, fifteen in P, and nine in V.

daily unequa l motion of the pla nets

Tabula diversorum motuum planetarum in una die Linee numeri Motus Veneris diversus in Motus Mercurii diversus in communes una die una die augmentate Motus Motus Motus Motus per 6 puncti centri portionis puncti centri portionis G

G

M

S

M

S

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180 a

354 348 342 336 330 324 318 312 306 300 294 288 282 276 270 264 258 252 246 240 234 228 222 216 210 204 198 192 186 180 b

57 57 57 57 57 57 57 57 57 58 58 58 58 58 59 59 59 59 59 60 60 60 60 60 60 61 61 61 61 61 o

8 10 13 14 21 27 34 42 50 0 12 24 36 50 10 20 34 47 59 11 23 36 41 50 54 0 5 9 13 15 p

15 15 15 15 15 15 15 14 14 14 14 14 13 13 12 12 10 10 8 7 2 R0 5 7 14 25 42 56 87 96 k

30 28 26 23 18 12 4 56 46 30 20 2 40 10 48 18 30 20 50 5 0 R0 0 20 10 50 30 0 0 0 r

M

s

56 56 56 56 56 56 56 57 57 57 57 58 58 58 58 59 59 59 59 60 60 60 61 61 61 61 61 61 62 62

Motus lune in una hora

S

M

S

M

S

10 15 21 26 36 46 57 8 22 34 46 0 14 28 44 0 16 34 42 10 28 44 0 14 26 38 45 53 0 5 t

51 51 50 50 49 49 47 45 43 40 37 34 31 28 25 19 12 5 R0 11 19 31 44 51 71 83 94 103 108 112 u

10 5 58 30 50 0 30 30 0 0 0 0 0 0 0 0 0 0 R0 0 0 0 0 0 0 0 0 0 0 0 v

29 29 29 29 29 29 29 30 30 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 37 37 37 37 37 w

0 4 8 13 22 33 46 2 19 37 48 18 39 3 26 49 15 44 17 49 17 44 11 38 1 19 33 44 52 56 x

Variants from B in C, E, P, and V p66: V3; p144: CEPV54; p150: V59; q114: V7; q168: EP65; t24: CEP28; t48: C28; t156: EP36; u48:C48; u114: E30; w72: P30; w84: P31; w96: EPV32; w102: P32; w108: EPV33; w114: P33; w120: EPV34; w126: P34; w132: EPV35; w138: P35; w144: EPV36; w150: P36; x66: EPV58; x102: V19; x126: EPV17; x180: EPV46.

87

88

edi t i on of the tab les with comments

11. Retrogradation of the planets Tabula retrogradationis planetarum stationum et directionum et intratur cum argumento equato planetarum (MS B, 74r) Titles in other manuscripts MS C, 80r: Tabula retrogradationis planetarum stationum et directionum et intratur in ipsum cum argumento equato MS E, 46r: Tabula retrogradationum planetarum stationum et directionum MS P, 53v: Tabula retrogradationum planetarum et stationum et directionum MS V, 46r: Tabula retrogradationis planetarum stationum et directionum et intratur cum argumento vero eorum Description In this short table for each of the five planets we are given four values corresponding to the first and second stations, both at apogee and perigee. The entries are given in signs, degrees, and minutes. We note that the values for the stationis initium prime (first station at apogee) and for the stationis secunde finis (second station at apogee) should add up to 12 signs. Similarly, the values for the retrogrationis initium (first station at perigee) and for the retrogrationis finis (second station at perigee) add up to 12 signs. As we will see, this table contains information for planetary stations, which is developed in the next table (Table 12). This short table is not extant in the Toledan Tables, but it is often found among medieval sets of tables. The oldest known version of it, with the same values and format, was compiled by the Andalusian astronomer Ibn al-Kammād, active in Córdoba in the early twelfth century, whose zij had a limited diffusion in Latin (Chabás and Goldstein 2015, p. 218). Manuscript B has 3s 24;44º for the first station at apogee of Saturn, whereas it should be 3s 22;44º, in order to add up to 12 signs along with the second station at apogee (8s 7;16º). This difference occurs in all other manuscripts we consulted, and it indicates that it was introduced at an early stage of the transmission of the table, or at the very beginning. There is another peculiar characteristic of this small table in the set by John of Lignères: for Mercury, the values of the first two entries are switched, and the same happens with the last two entries. Again, all other manuscripts consulted share this characteristic, which does not occur, for example, in the analogous table by Ibn al-Kammād. This fact also points to the beginning of this tradition. The first (or at least an early) copyist realized that for Saturn, Jupiter, Mars, and Venus, the second entry (first station at perigee) exceeds the first one (first station at apogee), as their respective functions increase monotonically over the whole range. However, this is not the case for Mercury, and the copyist probably decided thus to switch the values, possibly to maintain ‘consistency’. All these repeated differences lead one to believe that this is a literary remnant, repeated over and over, on which the table-user had no real grasp. This table is not mentioned in the canons by John of Lignères.

retrog ra dation of the pla nets

Variant readings The base manuscript for this table is Basel, Universitätsbibliothek, F II 7 (MS B). The variants found in the other manuscripts examined (none in MS P) are trivial scribal errors in a table consisting of only 20 entries. For the edition of the table, we have introduced an extra column to facilitate the identification of the variants. John of Lignères. Retrogradation of the planets (11) Base : Basel F II 7,74 r Tabula retrogrationum planetarum stationum et directionum et intratur cum argumento equato planetarum Portio equata 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Saturni Initium stationis prime Retrogrationis initium Retrogrationis finis Stationis secunde finis Jovis Stationis initium prime Retrogrationis initium Retrogrationis finis Stationis secunde finis Martis Initium stationis prime Retrogrationis initium Retrogrationis finis Stationis secunde finis Veneris Initium stationis prime Retrogrationis initium Retrogrationis finis Stationis secunde finis Mercurii Initium stationis prime Retrogrationis initium Retrogrationis finis Stationis secunde finis a

Signa 3 3 8 8 Signa 4 4 7 7 Signa 5 5 6 6 Signa 5 5 6 6 Signa 4 4 7 7 b

Variants from B in C, E, P, and V: c6: E11; d10: V25; d12: V33; c16: CE24.

G 24 25 4 7 G 4 7 22 25 G 7 19 10 22 G 15 18 11 34 G 24 27 2 5 c

M 44 30 30 16 M 5 11 49 55 M 28 15 45 32 M 51 21 39 9 M 42 14 46 18 d

89

90

edi t i on of the tab les with comments

12. Planetary stations Tabula de stationibus quinque planetarum (MS B, 70v) Titles in other manuscripts MS C, 89v–90r: Tabula stationum et directionum planetarum MS Ox, 89v: Tabula de stationibus MS P, 56r: Tabula de stationibus planetarum MS V, 50v: Tabula stationum et directionum planetarum Description All planets are addressed in a single table. For each of the planets, there are two columns, for the first and second stations. The argument is given at intervals of 6º (or at intervals of 1º in one of the manuscripts), from 6º to 180º and their complement in 360º. The entries are displayed in degrees and minutes. We note that, in contrast, the entries for the stations in the preceding table for retrogradation were given in signs, degrees, and minutes, which is a probable indication of their different origin. For each planet and for each argument, the two entries add up to 360º. No such table appears in the Toledan Tables, where only the column for the first station of each planet is tabulated in the respective tables for the planetary equations. The separation of an old table into two tables mimics the case of the lunar latitude (see Table 9, above). The entries for the first station displayed here by and large agree with those in the Toledan Tables (Pedersen 2002, pp. 1267–1305), as well as with those in the zij of al-Battānī (Nallino 1899–1907, 2:138–139), which are also given at intervals of 6º. This table is not mentioned in the canons by John of Lignères. We further note that chapter 18 of the Priores refers to one table for equations and stations, which differs from Tables 11 and 12. Variant readings For this table, the base manuscript used is Basel, Universitätsbibliothek, F II 7 (MS B), which has been collated with four other manuscripts (MSS C, Ox, P, and V). Altogether, there are 660 sexagesimal numbers, including those for the argument. In comparison with MS B, there are 40 variants in MS C, 11 in MS Ox, 27 in MS P, and 36 in MS V. Since the values for the first and the second station of each planet should add up to 360º, it is easy to identify scribal errors. Overall, there are 150 pairs of entries for which the sum should be 360º. In MS B, this does not occur in twenty-one cases, and only for two of them do the other four manuscripts display entries adding up correctly to 360º. These are certainly errors in MS B. For the rest of the entries, no clear correlation emerges.

pla neta ry stations

91

John of Lignères. Planetary stations (12) Base : Basel F II 7, 70 v Tabula de stationibus quinque planetarum Centrum linee numeri

Saturni

Jovis

Martis

Veneris

Mercurii

statio statio statio statio statio statio statio statio statio statio prima secunda prima secunda prima secunda prima secunda prima secunda

G

G

G

M

G

M

G

M

G

M

G

M

G

M

G

M

G

M

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180

348 348 342 336 330 324 318 312 306 300 294 288 282 276 270 264 258 252 246 240 234 228 222 216 210 204 198 192 186 180

112 112 112 112 112 113 113 113 113 113 113 113 113 114 114 114 114 114 114 114 114 115 115 115 115 115 115 115 115 115 a

45 47 49 52 56 2 8 14 20 27 36 42 55 4 11 19 27 35 43 50 57 0 9 15 19 22 25 27 29 30 b

247 247 247 247 247 246 246 246 246 246 246 246 246 245 245 245 245 245 245 245 245 244 244 244 244 244 244 244 244 244 c

15 13 11 8 4 58 52 46 30 33 24 16 5 59 49 41 33 25 17 10 3 57 51 47 41 38 35 32 31 30 d

124 124 124 124 124 124 124 124 124 124 125 125 125 125 125 125 125 126 126 126 126 126 126 126 127 127 127 127 127 127 e

5 7 8 11 17 23 29 35 44 53 1 12 22 32 42 50 58 8 14 24 32 42 48 54 0 4 7 10 11 12 f

235 235 235 235 235 235 235 235 235 235 234 234 234 234 234 234 234 233 233 233 233 233 233 233 232 232 232 232 232 232 g

55 53 52 49 43 39 35 25 16 7 59 48 38 28 18 10 2 54 46 36 28 18 12 6 0 56 52 0 49 47 h

157 33 202 27 165 52 194 8 157 37 202 23 165 54 194 6 157 48 202 13 165 57 194 3 157 59 202 1 165 59 194 1 158 15 201 45 166 1 193 59 158 33 201 27 166 6 193 54 158 57 201 3 166 12 193 48 159 20 200 40 166 18 193 42 159 51 200 9 166 24 193 36 160 21 199 39 166 30 193 30 160 56 199 4 166 36 193 24 161 31 198 29 166 43 193 17 162 9 197 51 166 53 193 7 162 47 197 13 167 2 192 58 163 25 196 35 167 11 192 49 163 57 196 3 167 17 192 42 164 31 195 49 167 23 192 37 164 51 195 55 167 30 192 31 165 39 194 21 167 37 192 23 166 11 193 40 167 45 192 15 166 41 193 19 167 52 192 8 167 11 192 49 167 56 192 4 167 37 192 23 168 2 191 58 168 1 191 59 168 6 191 54 168 21 191 39 168 10 191 50 168 41 191 19 168 14 191 46 168 53 191 7 168 19 191 43 169 5 190 55 168 19 191 41 169 11 190 40 168 20 191 40 169 15 190 45 168 21 191 39 i j k l m n o p

G

M

G

M

147 147 147 146 146 146 146 145 145 145 145 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 q

12 8 0 49 39 21 6 50 34 19 5 55 49 43 37 34 32 30 30 29 29 30 32 34 36 38 39 40 42 42 r

212 212 213 213 213 213 213 214 214 214 214 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 s

48 52 0 11 24 39 54 10 26 42 55 5 11 17 23 26 28 30 30 31 31 30 28 26 24 22 21 20 18 18 t

Variants from B in C, L, Ox, and V: a78: V114; a132: P114; b132: V3; b180: C3; d54: CPV40; d66: C34; d72: V18; d84: CV56; d144: V45; f108: CPV6; f126: P33; g150: CPV233; h42:CPV31; h132: C38; h162: V53; h168: CV50; h180: CV 48 P49; i78: CPV161; i108: CV165; i114: C167 V166; i120: CV167; i126: CV168; i132: CV168; i138: PV168; i150: CV169; i156: CV169; i162: CPV169; j18: V38; j126: CV51; k72: CV199; k108: CP194; l6: P23; l12: P13; l18: P1 V22; l24: P45; l30:P27 V19; l36: C47 V17; l42: P7; l102: V29; l108: V19; l126: V9; l138: CV29 P33; l174: C49; l180: C35; n78: P54; n162: P17; p96: PV43; p114: C33; q24: C147; q42: C145; q66: C144; r18: P4; r30: CPV37; r36: CPV31; s12: C213; s42: C214; s66: C215; t60: CV41; t156: CP20; t162: C22; t168: C23 P21; t174: CP19.

92

edi t i on of the tab les with comments

13. Planetary phases Tabula de exitu planetarum de (sic) sub radiis solis et eorum ingressu (MS B, 71r) Titles in other manuscripts MS C, 80v: Tabula de exitu planetarum sub radiis solis et eorum ingressu MS E, 46v: Tabula de exitu planetarum sub radiis solis MS P, 54v: Tabula de exitu planetarum de (sic) sub radiis solis et eorum ingressu MS V, 46v: Tabula de exitu planetarum sub radiis solis et eorum ingressu Description This table for the visibility of the planets refers to two phases (morning rising and evening setting) of the three superior planets and four phases (evening rising, morning setting, morning rising, and evening setting) of the inferior planets. In each case, there are entries for the beginnings of the 12 zodiacal signs. The same table can be found in previous sets, including the Toledan Tables (Pedersen 2002, pp. 1530–1537) and the zij of al-Battānī (Nallino 1899–1907, 2:142–143). It probably derives from a table in Ptolemy’s Handy Tables computed for climate IV, for a geographical latitude of about 36º (Chabás and Goldstein 2012, pp. 124–126) and, as was the case with the table for retrogradation, it seems to be an unuseful remnant. We note that the evening rising of Mercury refers to 152º, rather than to 112º. A description of this table is found in Chapter 42 in the canons beginning Priores astrologi (see Saby 1987, pp. 266–267). Variant readings For this table, the base manuscript used is Basel, Universitätsbibliothek, F II 7 (MS B). For the edition of the table, we have introduced an extra column, with integer numbers from 1 to 12 for the 12 signs, to facilitate the identification of the variants. The variant readings in the four other manuscripts consulted allow us to identify several scribal errors in MS B, none of them considered relevant.

pla neta ry pha ses

93

John of Lignères. Planetary phases (13) Base : Basel F II 7, 71 r Tabula de exitu planetarum sub radiis solis et eorum ingressu

Signa

1 2 3 4 5 6 7 8 9 10 11 12

Visio Saturni

Occultatio eius

Visio Jovis

Occultatio eius

Visio Martis

Occultatio eius

Ortus matutinus

Occasus vesperinus

Ortus matutinus

Occasus vesperinus

Ortus matutinus

Occasus vesperinus

Aries 29 28 13 46 19 33 9 28 29 0 14 12 Taurus 26 26 14 7 18 21 9 38 27 11 15 8 Gemini 22 10 15 5 14 15 10 36 22 14 16 7 Cancer 17 18 17 9 11 41 11 44 18 15 18 15 Leo 14 8 19 48 9 44 13 32 16 7 22 14 Virgo 13 8 22 0 9 7 15 23 15 8 27 11 Libra 12 5 22 32 9 0 16 7 14 12 29 0 11 Scorpius 12 1 21 30 9 7 15 23 15 8 27 Sagitarius 13 47 18 35 9 44 13 32 16 7 22 14 Capricornus 16 36 16 36 11 44 11 44 18 15 18 15 Aquarius 21 16 14 40 14 44 10 16 22 14 16 7 Pisces 26 46 14 0 18 11 9 38 27 11 15 8 a b c d e f g h i j k l m Variants from B in C, E, P, and V: a8: V Scorpio; b3: C12; b8: EP13 V12 corr.13; c7: CEP15; c9: E41; e8: CE20 V30 corr.20; f12: V16; g4: V41 corr.44; i3: V16;i9: P33; j5: CE6; k8: V8 corr.7; k9: V7 corr. 18; k11: C4; l3: V16 corr.15; m11: V17 corr.7. Tabula de exitu planetarum sub radiis solis et eorum ingressu Visio Veneris et occultatio eius

Visio Mercurii et occultatio eius

Ortus Occasus Ortus Occasus Ortus Occasus Ortus Ortus vesperinus matutinus a matutinus a vesperinus vesperinus matutinus a matutinus a vesperinus ab uno 224 in 360 180 in 223 a 138 in 180 ab uno 249 in 360 180 in 248 a 112 in 180 gradu in 137 gradu in 152

1 2 3 4 5 6 7 8 9 10 11 12

15 31 4 25 3 36 2 27 24 10 12 24 22 43 12 9 13 48 4 29 4 9 3 30 21 15 12 18 24 23 12 12 10 39 7 38 5 14 8 47 17 10 12 37 22 28 14 44 8 38 8 58 10 22 10 40 14 9 14 9 18 48 19 48 7 5 8 59 17 45 11 30 12 53 16 29 15 8 23 25 6 53 10 46 23 40 7 43 12 8 20 23 13 15 26 37 11 9 22 27 6 40 12 10 23 50 12 29 25 38 6 57 7 11 11 26 15 14 6 17 12 43 23 49 12 10 20 35 7 56 12 27 7 1 5 12 14 3 20 44 11 16 17 41 9 18 9 18 2 18 2 18 16 19 16 19 12 15 12 40 12 47 8 29 1 36 1 14 20 25 14 7 14 25 11 32 15 20 7 43 2 43 1 31 24 38 12 14 18 22 11 47 a b c d e f g h i j k l m n o p Variants from B in C, E, P, and V: b11: C49; d4: V58 corr. 59; d6: E40; d11: CV36; f4: CV12; f8: V19 corr.14; h4 : CEV44; h5: V30 corr.43; i12: C34; j8: CEPV41; j9: V30 corr.3; j11: CEPV15; l5: CEPV39; l6: V33 corr.23; l8: V40 corr.49; p8: V35 corr.45; p10: EP30 V31 corr.30.

94

edi t i on of the tab les with comments

14. Mean syzygies for collected years The comments for the following three tables are presented jointly, below (see Table 16). Tabula medie coniunctionis solis et lune in annis Christi collectis super meridianum parisiensem and Tabula medie oppositionis solis et lune in annis Christi collectis ad parisius (MS B, 71v) Titles in other manuscripts MS C, 81r: Tabula medie coniunctionis solis et lune in annis Christi collectis super meridianum parisiensem and Tabula medie oppositionis solis et lune in annis Christi collectis MS E, 44r: Tabula medie coniunctionis solis et lune in annis Christi collectis supra meridianum parisiensem and Tabula medie oppositionis solis et lune in annis Christi collectis ad meridianum parisiensem MS P, 48r: Tabula medie coniunctionis solis et lune in annis Christi collectis super meri­ dianum parisiensem and Tabula medie oppositionis solis et lune in annis Christi collectis super meridianum parisiensem MS V, 26v: Tabula medie coniunctionis solis et lune in annis collectis and Tabula medie oppositionis solis et lune in annis collectis

me a n syzyg ies for collected yea rs

95

John of Lignères. Mean syzygies for collected years (14) Base : Basel F II 7, 71 v Tabula medie coniunctionis solis et lune in annis Christi collectis super meridianum parisiensem Anni Christi collecti 1 2 3 4 5 6 7 8 9 10 11 12 13

Tempus medie coniunctionis solis et lune

Anni

D

H

M

S

1321 1345 1369 1393 1417 1441 1465 1489 1513 1537 1561 1585 1609 a

28 3 8 12 17 22 26 1 6 10 15 20 24 b

18 19 9 23 13 3 17 18 8 23 13 4 17 c

2 21 24 27 30 33 36 50 58 1 4 8 11 d

6 9 15 21 26 32 38 41 46 52 58 4 9 e

Medius motus solis et lune Signa 10 9 9 10 10 10 10 9 9 10 10 10 10 f

Medium argumentum lune

G

M

S

T

16 21 26 1 6 10 15 21 25 0 5 9 14 g

21 56 38 20 1 43 25 0 42 24 6 47 29 h

14 39 21 7 52 38 23 45 30 16 1 27 32 i

29 51 25 0 34 8 42 5 39 13 47 21 58 j

Signa 10 1 4 8 0 3 7 9 1 5 8 0 3 k

G

M

S

17 8 26 14 1 19 6 28 16 3 21 9 26 l

5 52 28 3 39 15 51 3 14 50 26 2 27 m

20 13 6 59 52 45 38 30 23 16 9 2 55 n

Argumentum latitudinis lune Signa 0 3 6 10 2 5 9 0 3 7 11 2 6 o

G

M

S

20 9 28 17 6 25 14 3 22 11 0 19 8 p

38 6 14 22 31 39 47 15 23 31 39 47 53 q

41 37 46 55 4 13 23 18 27 36 46 55 4 r

Variants from B in C, E, P and V : a7: E1456; c10 : C13; c12: EPV3; d8: CV55; P45; e1: C9; e2: C6; g1: C6; h10: V34 corr.24; i2: CPV35; i5: V54 corr.52; i12: EP 427; i13: C33; j1: V39 corr.29; j5: V34 corr.35; j11: EP42; j13: EPV56; l8: E18; m5: V39 corr.35; m7: C52; m8: CEP38; m13: EPV37; q7: EP40; q13: EPV56; on the right of r, one more column in C for tertia with 0. Tabula medie oppositionis solis et lune in annis Christi collectis ad Parisius Anni Christi collecti 1 2 3 4 5 6 7 8 9 10 11 12 13

Tempus medie oppositionis

Anni

D

H

M

S

1321 1345 1369 1393 1417 1441 1465 1489 1513 1537 1561 1585 1609 a

13 18 23 27 2 7 11 16 21 25 0 5 9 b

23 13 3 17 19 9 23 13 3 17 18 8 22 c

40 43 46 49 8 11 14 17 20 23 42 46 49 d

5 11 16 22 25 31 36 42 48 52 56 2 8 e

Medius motus solis et nadiir lune

Medium argumentum lune

Signa

Signa

10 10 10 10 9 9 10 10 10 10 9 9 9 f

G

M

S

1 6 11 15 21 26 0 5 10 14 20 25 29 g

48 29 11 53 28 11 52 33 15 57 32 14 56 h

2 47 33 19 40 26 11 57 42 28 49 35 20 i

23 53 31 5 28 2 36 11 45 19 42 16 50 j

4 7 11 2 5 9 0 4 7 11 2 5 9 k

G

M

S

4 21 9 26 18 6 23 11 29 16 8 26 13 l

10 46 22 58 45 21 57 33 8 44 31 7 43 m

50 43 36 29 21 14 7 0 53 46 39 32 25 n

Argumentum latitudinis lune Signa 6 9 1 5 7 11 2 6 10 1 4 8 11 o

G

M

S

5 24 13 2 21 10 29 18 7 26 15 4 23 p

18 26 34 43 10 19 27 35 43 51 19 27 35 q

35 44 53 2 57 7 16 25 34 43 39 48 57 r

Variants from B in C, E, P and V : b3: EP22; b13: V6; c4: V17 corr.18; d4: V49 corr.55; e6: P32; e8: C41; e9: C42; e10: CEPV54; f5-f6 : C10; g1: V11; h6: CEPV10; j2: CEPV57; j4: C2 V15 corr.5; l1: V14 corr.4; m1: V50 corr.10; n1: V58 corr.50.

96

edi t i on of the tab les with comments

15. Mean syzygies for expanded years Tabula medie coniunctionis et oppositionis solis et lune in annis Christi expansis (MS B, 72r) Titles in other manuscripts MS C, 81r: Tabula medie coniunctionis et oppositionis solis et lune ad annos Christi expansos MS E, 44r: Tabula medie coniunctionis et oppositionis solis et lune ad annos Christi expansos MS P, 48r: Tabula medie coniunctionis et oppositionis solis et lune ad annos Christi expansos MS V, 27r: Tabula medie coniunctionis et oppositionis solis et lune in annis expansis et mensibus

mea n syzyg ies for expa nded yea rs

97

John of Lignères. Mean syzygies for expanded years (15) Base : Basel F II 7, 72 r Tabula medie coniunctionis et oppositionis solis et lune in annis Christi expansis Anni Christi expansi 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

b

b

b

b

b

b a

Tempus medie coniunctionis et oppositionis

Medius motus solis et lune

Argumentum medium lune

D

H

M

S

Signa G

M

S

T Signa G

10 21 2 14 24 5 16 29 9 19 0 12 23 4 14 26 7 18 28 10 21 2 13 b

15 6 8 0 15 17 8 0 2 17 20 11 2 4 20 11 13 5 20 22 13 16 7 c

11 22 50 1 12 40 51 3 30 41 9 20 31 29 10 21 49 0 12 39 50 18 29 d

23 47 7 31 54 14 38 1 21 45 5 29 52 12 36 59 20 43 6 27 50 10 34 e

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11

16 33 56 13 30 53 10 27 50 7 30 47 4 27 44 1 24 41 58 21 38 11 18 h

50 40 55 45 35 40 50 31 45 35 50 40 31 45 35 26 40 31 21 36 26 40 31 i

20 39 10 30 49 21 40 10 31 51 22 41 1 32 52 11 42 2 21 53 12 43 3 j

f

19 8 26 16 5 23 13 2 20 10 28 17 7 25 14 4 22 11 0 19 8 27 16 g

k

10 8 7 6 4 3 2 0 11 9 9 7 5 4 3 1 0 11 9 8 6 6 4

M

9 48 19 38 25 23 5 1 14 49 20 26 0 14 10 2 15 40 25 25 1 5 10 53 20 41 26 18 6 6 5 54 21 32 0 20 11 8 16 45 26 33 2 10 11 58 l m

Argumentum latitudinis lune S

Signa G

M

S

7 14 22 29 36 43 50 57 5 12 19 26 33 41 48 55 2 9 16 24 31 38 45 n

0 0 1 2 2 3 3 4 5 5 7 7 7 8 9 9 10 11 11 0 0 2 2

2 5 48 51 54 36 39 42 25 28 11 13 16 59 2 5 48 50 53 36 39 22 25 q

45 30 29 15 0 59 44 29 28 13 12 58 43 42 27 12 11 57 42 41 26 25 10 r

o

8 16 24 2 10 19 27 5 14 22 1 9 17 25 4 12 20 28 6 15 23 2 10 p

Variants from B in C, E, P, and V: a4: P b; a8: P b; a12 ; P b; a16: P b; a20: P b; b8: CP28 V29 corr.28; b12: E11; b16: P25; b22: C22 EP3; d2: EP23; d14: CEPV59; d19: E13; e2: P46; e20: P29; g6: C26; h22: EP1; i6: EV50; i7: CEPV40; i20: CPV35; i21: V36 corr.26; j2: E 19 corr.39; j8: EP0 V10 corr.0; j10: V33 corr.51; j20: V52 corr.53; k11: V7 corr.9; l16: CEPV15; l18: CEP1; m2: CEPV36; m3: CEPV13; m10: CEPV28; n15: V58; n18: C0; o11: P6; o18: E17; o22: P1; q14: V29 corr.59; r5: EV12; r6: P50.

98

edi t i on of the tab les with comments

16. Mean syzygies for months in a year Tabula medie coniunctionis et oppositionis solis et lune in mensibus (MS B, 72r) Titles in other manuscripts MS C, 81v: Tabula medie coniunctionis et oppositionis solis et lune in mensibus MS E, 44r: Tabula medie coniunctionis et oppositionis solis et lune in mensibus MS P, 48v: Tabula medie coniunctionis et oppositionis solis et lune in mensibus MS V, 27r: Tabula mensium medie conunctionis et oppositionis solis et lune et coniunctionibus in mensibus Description Tables 14, 15, and 16 are for determining mean syzygies. The first one consists of two sub-tables, one for conjunctions and another for oppositions. The other two tables are valid for conjunctions and oppositions. In all cases, there are columns for four quantities: time (in days, hours, minutes, and seconds), mean solar and lunar longitudes (in signs and degrees, to thirds), mean lunar anomaly (in signs and degrees, to seconds), and mean argument of lunar latitude (in signs and degrees, to seconds). The same format and quantities were used in the Toledan Tables (Pedersen 2002, pp. 1327–1340), although with a different calendar, as well as in other previous sets of tables. The argument differs in each table. In Table 14, it is the number of collected years since Incarnation from 1321 to 1609, at intervals of 24 years. In this case, the given time refers to the day and hour of the first occurrence of a syzygy in that year, beginning in January. The first entries for 1321 are 28d 18;2,6h (conjunction) and 13d 23;40,5h (opposition), indicating that the first opposition took place on January 13 at 23;40,5h, after noon, and the first conjunction on January 28 at 18;2,6h, after noon, that is, half a mean synodic month later. These times were computed for the meridian of Paris, and could have been derived from the corresponding tables, with the same structure compiled by John of Murs for the meridian of Toledo by applying a shift of about 0;48h between Toledo and Paris. We also note that the Tables of Toulouse, derived from the Tables of Toledo but adapted to the Julian calendar, also use intervals of 24 years as well (Chabás 2019, pp. 108­–110). In Table 15, the argument is the number of expanded years from 1 to 23, and in Table 16 it is the accumulated time in each month of the year. The entry for time corresponding to January in Table 16 is 29d 12;44,3h, which is the length of the mean synodic month used here. This value is given, although not explicitly, in canon 27 of the Priores astrologi as the difference between 31 days and 1d 11;15,57h. The entries for the rest of the months are multiples of the mean synodic month, up to October, where an extra second has been added. This means that in 10 months, the number of seconds is increased by one, indicating that the table was computed for a slightly more precise value: 29d 12;44,3,3h. This is the value used from then onwards in Alfonsine astronomy. Table 16 has an extra row for the values corresponding to half a mean synodic month and an extra column for the accumulated number of days in a year.

1 2 3 4 5 6 7 8 9 10 11 12 13

12 1 14 2 15 4 17 5 18 7 20 8 18

c

b

H

29 59 88 118 147 177 206 236 265 295 324 354 14

D

d

44 28 12 56 40 24 8 52 36 20 4 48 22

M

Tempus medie coniunctionis et oppositionis

e

3 6 9 12 15 18 21 24 27 31 34 37 2

S

f

0 1 2 3 4 5 6 7 8 9 10 11 6

Signa

g

29 28 27 26 25 24 23 22 21 21 20 19 14

G

h

6 12 19 25 32 38 44 51 57 4 10 16 33

M

i

24 48 12 36 0 25 49 13 37 1 26 50 12

S

j

T

Medius motus solis et lune

0 0 0 0 0 0 0 0 0 0 0 0 0 k

0 1 2 3 4 5 6 6 7 8 9 10 6

Signa

l

25 21 17 13 9 4 0 26 22 18 13 9 12

G

m

49 38 27 16 5 54 4 3 31 10 59 48 54

M

n

1 1 2 2 3 3 4 5 5 6 6 7 30

S

Medium argumentum lune

o

1 2 3 4 5 6 7 8 9 10 11 0 6

Signa

p

0 1 2 2 3 4 4 5 6 6 7 8 15

G

q

40 20 0 40 21 1 41 21 2 42 22 2 20

M

r

14 28 41 55 9 23 36 50 4 18 31 45 7

S

Argumentum latitudinis lune

Variants from B in C, E, P and V: d8: C51; f13: P0; h4: C35; i11: EPV36; m7: EPV43; m8: EPV32; m9: EPV21; n3: V1; n8: V2; n8: V4; p4: P41; r10: V18 corr.17.

januarius februarius martius aprilis maius junius julius augustus september october november december media lunacio a

Menses

Nomina mensium

Tabula medie coniunctionis et oppositionis solis et lune in mensibus

John of Lignères. Mean syzygies for a month in a year (16) Base : Basel F II 7, 72 r

s

31 59 90 120 151 181 212 243 273 304 334 365

Numerus dierum mensium agregatorum

mean syzyg ies for months in a yea r 99

10 0

edi t i on of the tab les with comments

These tables were adapted by John of Lignères to the meridian of Paris and later integrated in his Tabule magne (dated 1325). They are addressed in canons 25–29 of the Priores astrologi (see Saby 1987, pp. 213–219). Variant readings For these tables, the base manuscript used is Basel, Universitätsbibliothek, F II 7 (MS B). A second hand in MS V adds several columns, in particular a column for the thirds in the mean motion of the Moon. For the edition of Table 14, we have introduced an extra column, with integer numbers from 1 to 13 for the years 1321–1609, to facilitate the identification of the variants. In Table 14, MS C adds, at the far right, a column for thirds, filled in with zeros. In Table 16, cell a13, below the names of the months, MSS E and P read tempus medie lunationis and MS V reads oppositio supra media lunatio (sic). In MSS E and P, no entries are given from j1 to 12, whereas in MS V another hand gives the following values for the thirds: 11, 13 corr. 23, 34, 46, 59, 9, 21, 33, 44, 56, 7, and 19.

mean syzyg ies for months in a yea r

Figure 6. Vatican, Pal. lat. 1374, 27r.

1 01

102

edi t i on of the tab les with comments

17. Mean motion in elongation Tabula medie elongationis lune a sole in annis, Tabula medie elongationis lune a sole in diebus, and Tabula medie elongationis lune a sole in horis (MS C, 90r) Titles in other manuscripts MS E, 44r: Tabula medii motus lune in elongatione sui a sole in annis collectis et expansis, Tabula medii motus lune in elongatione sui a sole in diebus, and Tabula medii motus lune in elongatione sui a sole in horis et minutis horarum MS L, 42r: Tabula medii motus lune in elongatione sui a sole in annis collectis et expansis, Tabula medii motus lune in elongatione sui a sole in diebus, and Tabula medii motus lune in elongatione sui a sole in horis, minutis et secundis horarum MS P, 49r: Tabula medii motus lune in elongatione sui a sole in annis collectis et expansis, Tabula medii motus lune in elongatione sui a sole in diebus, and Tabula medii motus lune in elongatione sui a sole in horis et minutis horarum MS V, 26r: Tabula medii motus lune in elongatione sui a sole in annis collectis et expansis et cetera, Tabula medii motus lune in elongatione sui a sole in diebus et cetera, and Tabula medii motus lune in elongatione sui a sole in horis et cetera Description This table is for the mean elongation between the Sun and the Moon, that is, the mean angular distance between the two luminaries, and it is to be used for the computation of mean syzygies. This table, with its three sub-tables, is not found in all manuscripts containing this set. The first sub-table is for expanded and collected years, for each year from 1 to 20, from 20 to 100 at intervals of 20 years, from 100 to 1000 at intervals of 100 years, and finally for 2000 years. The second sub-table is for the days in a month, from 1 to 30, and the entry for one day is 12;11,26,42º/d, which is the key value characterizing the table. Indeed, this value results from subtracting two standard parameters in Alfonsine astronomy: the lunar mean motion in longitude (13;10,35,1,15º/d) and the solar mean motion (0;59,8,19,37º/d). The third sub-table is for the hours in a day, from 1 to 30 (or 24 in some manuscripts). Madrid, Biblioteca Nacional, 10002, 41r, adds two other sub-tables, for the months in a year, whether a leap year or not. It seems worth noting that the interval of 20 years for the mean motions used by John of Lignères for the first time here was not a novelty in medieval astronomy: the canons of the Castilian Alfonsine Tables mention that this was the interval at which the tables for mean motions compiled in Toledo were built, and it was also used in some pre-Alfonsine tables (see Chabás 2019, pp. 121 and 128). In MS V, this is the first table of the set, opening Incipiunt tabule magistri Johannis de Lineriis. This table is not mentioned in the canons by John of Lignères. It was later integrated in his Tabulae magne (dated 1325).

mea n motion in elong ation

John of Lignères. Mean motion in elongation (17) Base : Cracow BJ 551, 90r Tabula medii motus lune in elongatione sui a sole in annis collectis et expansis 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Numerus dierum

a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 17 20 40 60 80 100 200 300 400 500 600 700 800 900 1000 2000

Signa

b

G 4 8 0 5 10 2 6 11 3 8 0 5 9 1 6 10 3 7 11 4 8 1 5 10 8 6 4 2 1 11 9 8 6 0

c

M 9 19 28 20 0 9 19 11 20 0 10 2 11 21 0 22 2 11 21 13 26 10 23 7 14 21 28 5 12 19 26 3 10 21

d

S 37 14 52 40 18 55 33 21 59 37 14 2 40 17 55 43 21 58 36 24 49 14 35 4 9 14 19 24 29 34 39 43 48 37

e

T 23 46 9 59 22 46 9 59 22 42 9 59 22 45 8 58 22 41 8 58 57 55 54 52 45 38 31 24 17 10 3 56 49 38

f

15 31 47 43 58 14 29 26 41 57 12 9 24 40 55 52 7 13 38 35 10 45 20 55 50 44 39 24 29 33 28 23 18 35

Variants from C in E, L, P, and V: a0: EP anni collecti et expansi VL anni; a19: ELPV 19; b13: V6; b28: LV3; d10: ELPV36; d23: EL39 V34; e10: ELPV45; e18: ELP45; e21: ELP57; f3: ELPV46; f9: V21; f18: ELPV23; f23: V10; f28: L34 V4; f30: L23; f32: L13; f34: L15.

1 03

104

edi t i on of the tab les with comments

Variant readings This table is not extant in MS B, and the base manuscript taken here is MS C. Maybe the most striking variant reading is found both in MSS E and P, where the entry for one day is given as 14;11,26,42º/d rather than 12;11,26,42º/d. We note that in MS C, the argument of the three sub-tables is erroneously headed numerus dierum.

1 05

mea n motion in elong ation

Tabula medii motus lune in elongatione sui a sole in diebus 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Numerus Signa dierum

a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

b

0 0 1 1 2 2 2 3 3 4 4 4 5 5 6 6 6 7 7 8 8 8 9 9 10 10 10 11 11 0

G

c

12 24 6 18 0 13 25 7 19 1 14 26 6 20 2 15 27 9 21 3 16 28 10 22 4 16 29 11 23 5

M

S

11 22 34 45 57 8 20 31 43 54 5 17 28 40 51 3 14 26 37 48 0 11 23 34 46 57 9 20 31 43 d

26 53 20 40 13 40 6 33 0 26 53 20 47 13 40 7 23 0 27 53 20 47 13 40 7 34 0 27 54 20 e

T

f

42 23 5 47 28 10 51 33 15 56 38 20 1 43 2 6 48 29 11 53 34 16 58 39 21 2 44 26 7 49

Variants from C in E, L, P, and V: b7: L3; b29: EL11; c1: EP14; c7: L15; c13: EP8; d7: L30; d10: L14; d13: LV24; e3: L27; e12: L30; e17: EPV33; f11: L37; f12: L29; f15: LEP24; f17: L47; f19: L10; f20: L54 f22: L15; f23 : L57; f25: L20; f30: L48; in L one column for quarta, from 1 to 33 : 38, 16, 54, 31, 9, 47, 25, 3, 41, 19, 56, 34, 12, 50, 28, 6, 44, 22, 59, 37, 15, 53, 31, 9, 57, 24, 2, 40, 18, 56.

Tabula medii motus lune in elongatione sui a sole in horis et minutis horarum Numerus dierum

a

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

G

M

S

T

Q

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 b

30 0 31 1 32 2 33 3 34 4 35 5 36 6 37 7 38 8 39 9 40 10 40 11 41 12 42 13 43 14 c

28 57 25 54 22 51 20 48 17 46 14 43 11 40 9 37 6 35 3 32 0 29 58 26 55 23 52 21 49 18 d

36 13 50 26 3 40 17 53 30 7 44 20 57 34 11 47 24 1 37 14 51 28 4 41 18 55 31 8 45 22 e

44 28 12 56 40 24 8 53 37 21 5 49 33 17 1 45 39 13 57 42 26 10 54 38 22 6 50 34 22 6 f

Variants from C in E, L, P, and V: a0: EP numerus horarum cum minutis P add. et ceteris LV hore; d2: E55; d5: L23 V24; d17: P5; e1: E13 corr.36; e2 to 29 : E one cell shift upwards; e30: E lac.; f7: L9; f9: L36; f17: LP29; f22: E20; e30: L2.

10 6

edi t i on of the tab les with comments

18. Corrections of the hourly lunar motion Tabula diversitatis lune in una hora (MS C, 84r) Titles in other manuscripts MS E, 44v: Equatio diversi motus lune in una hora MS L, 41v: Tabula diversitatis lune in una hora MS Ox, 94v: Equatio lune MS P, 49r: Equatio diversi motus lune in una hora Description This table only applies when the Sun and the Moon are close to true syzygy. The elongation between the two luminaries is displayed at intervals of 1º from 1 to 7. The entries range from 0;0,0º to 0;0,6º and are to be added to or subtracted from the superatio, that is, the difference between the lunar and solar hourly velocities. For an explanation of its use, see Chabás and Goldstein 1992, pp. 269–270, and Chapter 22 of the canons to the Parisian Alfonsine Tables by John of Saxony (Poulle 1984, pp. 80–87). This is the smallest table in John of Lignères’ set, with only 28 sexagesimal numbers, included those for the argument. In parallel, the table is found among those in the Almanac of Azarquiel (Millás 1943–1950, p. 233) and in the Toledan Tables (Pedersen 2002, p. 1414). It differs slightly from that of the zij of al-Battānī (Nallino 1899–1907, 2: 88), where the entries range instead from 0;0,1º to 0;0,7º. It is not mentioned in the canons by John of Lignères. Variant readings As this table is not extant in MS B, the base manuscript taken here is MS C. It has just seven entries, and no variants in the other manuscripts have been recorded. A lack of scribal mistakes in a table consisting of two columns with successive integer numbers and two columns filled in with zeros is not surprising.

cor r ectio ns of the hourly luna r motion

John of Lignères. Corrections of the hourly lunar motion (18) Base : Cracow BJ 551, 84r Tabula diversitatis lune in una hora Longitudo inter solem et lunam

Equatio diversi motus lune in una hora

Numerus

G 1 2 3 4 5 6 7

M 0 0 0 0 0 0 0

No variants from C in E, L, Ox, and P

S 0 0 0 0 0 0 0

0 1 2 3 4 5 6

1 07

10 8

edi t i on of the tab les with comments

19. Equations and hourly velocities of the Sun and the Moon Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora (MS B, 72v–73v) Titles in other manuscripts MS C, 82r–83r: Tabula equationis et lune et ad sciendum motus solis et lune in una hora MS L, 43v–44v: Tabula equationis solis et lune ad sciendum motum utriusque in una hora MS Ox, 80r–81r: Tabula equationis solis et lune. Et ad sciendum motum solis et lune in una hora MS V, 27v–28v: Tabula equationis solis et lune ad sciendum motus (sic) solis et lune in una hora Description In this combined table, the argument consists of all integer degrees from 1º to 30º for signs from 0 to 5. There are two columns for the equations, one for the Sun, and another for the Moon, and two others for the hourly velocities of each of the two luminaries. This is the largest table in the set, with 2,160 sexagesimal numbers, including those for the argument. The solar equation has a maximum value of 2;10,0º at 92º–94º, and the equation in lunar anomaly a maximum value of 4;56,0º at 95º. The entries for the equations of the Sun and the Moon, and thus also their characteristic values, differ from those in the zij of al-Battānī (Nallino 1899–1907, 2: 78–83) and the Toledan Tables (Pedersen 2002, pp. 1245–1258). However, they were present in Paris at the time John of Lignères compiled his set of tables for 1322. In fact, they are embedded (in the case of the Sun: see p. 118, below) or explicitly given (in that of the Moon) in the corresponding tables computed by John Vimond shortly beforehand (see Chabás and Goldstein 2004, pp. 222–226). John of Lignères mentions tables for the equations of the Sun and the Moon in Chapters 13, 14, and 20 of his canons beginning Priores astrologi (Saby 1987, pp. 199–200, 219–220). We also note that no tables for the equation of the planets were included in his Tables of 1322. The two other columns in this table display entries for the hourly velocities of the Sun and the Moon, ranging from 0;2,23º/h to 0;2,33º/h and from 0;30,18º/h to 0;36,4º/h, respectively. This is probably the table mentioned by John of Saxony in Chapter 22 of his Tempus est mensura, in which he refers to a table compiled by John of Lignères, which is ‘more exact than I have seen’ (verior quam vidi; see Poulle 1984, p. 82). Possibly, by ‘verior’ John of Saxony meant that the table by John of Lignères had six times as many entries as the corresponding one in the Toledan Tables. The columns for the solar and lunar velocities differ in the number of entries from those for the same purpose in the zij of Battānī (Nallino 1899–1907, 2: 88) and the Toledan Tables (Pedersen 2002, pp. 1410–1412), only given at intervals of 6º, but the common entries agree. The lunar velocity in this table can be recomputed by means of the procedure explained by Goldstein 1996, pp. 181–183, based on indications given by Ptolemy in Almagest V.4. It is worth mentioning that the table is based on Ptolemy’s simple model and predates the one compiled by John of Genoa computed according to Ptolemy’s complete lunar model.

1 09

eq uation s an d hour ly velo cities of the sun a nd the moon

John of Lignères. Equations and hourly velocities of the Sun and the Moon (19) Base : Basel F II 7, 72v-73v Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora Signa Linee numeri Numerus graduum

0 minue Equatio solis

G

G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

a

b

c

Equatio lune

Motus solis in una hora

Motus lune in una hora

G M S G M S M S M S 2 4 6 8 10 12 15 17 19 21 23 25 27 30 23 34 36 38 40 42 44 46 48 51 53 55 57 59 0 2

10 0 4 19 0 9 27 0 14 36 0 19 44 0 23 53 0 28 2 0 33 10 0 37 19 0 42 28 0 47 36 0 52 45 0 56 53 1 1 1 1 5 8 1 10 16 1 15 23 1 19 30 1 25 37 1 29 43 1 33 49 1 38 55 1 42 59 1 46 4 1 51 4 1 55 2 2 0 1 2 4 59 2 8 57 2 13 54 2 17 11 adde d e f g

Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora Signa Linee numeri Numerus graduum G

G

46 31 15 0 44 28 11 54 27 19 0 41 20 59 38 15 51 24 0 32 3 33 1 27 52 15 37 57 14 29

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 24 24 24 24

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30

18 18 18 19 19 19 20 20 20 21 21 21 22 22 23 23 24 24 25 25 26 27 27 28 29 31 32 33 35 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

h

i

j

k

l

a

b

Variants from B in C, L, Ox, and V: d15: CLV32; d30: V12; h12: LV14.

1 minue Equatio solis

Equatio lune

Motus solis in una hora

Motus lune in una hora

G M S G M S M S M S 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 c

4 6 8 10 12 13 15 17 19 20 22 24 25 27 29 30 32 33 35 37 38 39 41 42 44 45 46 48 49 50

46 2 21 37 2 25 28 2 30 19 2 34 9 2 38 58 2 42 41 2 46 24 2 50 6 2 54 48 2 58 29 3 1 10 3 5 50 3 9 29 3 13 46 3 16 46 3 19 23 3 23 59 3 27 30 3 30 0 3 34 30 3 37 58 3 40 57 3 44 54 3 47 14 3 50 34 3 53 53 3 56 10 3 59 28 4 2 45 4 5 10 adde d e f g

46 55 5 12 17 21 22 19 14 7 46 46 31 13 26 26 59 30 57 20 40 57 17 20 26 29 30 26 17 4

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

24 24 24 24 24 24 24 24 24 24 24 24 24 24 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25

30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31

37 38 40 42 43 44 45 46 47 48 50 51 53 54 58 0 0 1 3 5 7 8 10 12 14 16 18 20 22 24

h

i

j

k

l

Variants from B in C, L, Ox, and V: d7: C17; d11: C24; d15: CLV30; d24: LV43; e1: V36; e11: CLV10; e22: LV59; e28: V18; g8: V30; g11: CLV5; g15: CL19; g16: C10; g19: C34; h3: L58; h8: L24; h9: L15; h10: L14; h14: V33; h19: C20; h20: V10; h23: CLV19; h25: LV27; h28: L36; j15: V24.

edi t i on of the tab les with comments

The following graph displays the hourly lunar velocity in MS B, once the faulty values for arguments 5s 23º – 5s 30º have been corrected. 37 36

(minutes per hour)

35 34 33 32 31 30 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

110

(degrees)

Variant readings As compared with MS B, the base manuscript, we detected many variants in the manuscripts used for collating the entries. There are 55 in MS C, 54 in MS L, 43 in MS Ox, and 76 in MS V. This high number is due to the fact that there are 24 common variants in these four manuscripts, indicating that MS B has at least 24 faulty entries. One of them is obvious: for arguments 5s 23º – 5s 30º, MS B has, erroneously, 0;35,4º/h for the hourly velocity of the Moon, rather than the standard value, 0;36,4º/h, found in all other manuscripts examined. Three tables for the same purpose We have identified three tables for the velocities of the Sun and the Moon in this set, which are presented and described under numbers 19, 20, and 21. The first one is a combined table displaying the equations and the velocities of the two luminaries, the second only has entries for velocities in two separate sub-tables, with different intervals, and the third is a single table for the hourly velocities of the two celestial bodies at 6º-intervals. As will be seen below, the entries in all three tables differ. Moreover, the use of signs also differs: Table 19 displays signs of 30º, Table 20 uses signs of 60º, and Table 21 has no signs. Likewise, different steps for the argument are used: 1º in Table 19, 3º in Table 20, and 6º in Table 21.

111

eq uation s an d hour ly velo cities of the sun a nd the moon

Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora Signa Linee numeri

2 minue Equatio solis

G

G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

a

b

c

Equatio lune

Signa Numerus Motus Motus graduum solis lune Linee in una in una numeri hora hora

G M S G M S M S M S 51 52 54 55 56 57 58 58 59 0 1 2 2 3 3 4 5 5 6 6 7 7 8 8 8 9 9 9 9 9

51 4 7 56 4 10 9 4 13 6 4 15 9 4 18 11 4 20 2 4 22 52 4 25 41 4 27 26 4 29 13 4 31 2 4 33 41 4 35 21 4 36 59 4 38 36 4 40 16 4 41 48 4 43 17 4 44 45 4 46 12 4 47 36 4 48 2 4 49 27 4 50 45 4 51 1 4 52 17 4 53 32 4 53 45 4 54 57 4 54 9 adde d e f g

Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora

47 27 3 35 30 27 47 2 12 18 20 18 11 59 43 23 58 28 53 13 26 35 38 41 38 28 11 50 25 58

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

25 25 26 26 26 26 26 26 26 26 26 26 26 26 27 27 27 27 27 27 27 27 27 27 27 27 28 28 28 28

31 31 31 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32

26 29 31 33 37 38 41 43 46 48 51 53 56 58 1 3 6 8 11 14 17 19 22 25 28 31 34 36 38 42

h

i

j

k

l

Variants from B in C, L, Ox, and V: d8: CLV58; d9: CLV59; d13: L3; e15: L50; e21: CL12; e22: C37; g23: V45; h5: CLV3; h14: V50; h15: C49; h19: V51; h26: V38; j14: V27; k14: V32; l5: CLV36.

3 minue Equatio solis

G

G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1

a

b

c

Equatio lune

Motus solis in una hora

Motus lune in una hora

G M S G M S M S M S 9 10 10 10 9 9 9 9 9 8 8 8 7 7 6 6 5 5 4 4 3 2 1 0 59 58 57 56 55 54

59 4 55 0 4 55 0 4 55 0 4 55 57 4 56 51 4 55 36 4 55 20 4 55 2 4 55 45 4 54 25 4 54 6 4 53 41 4 52 14 4 52 46 4 51 18 4 50 48 4 49 14 4 48 42 4 46 5 4 45 27 4 44 37 4 42 45 4 40 51 4 39 53 4 37 55 4 35 57 4 33 57 4 31 57 4 29 57 4 27 8 adde d e f g

18 37 49 55 0 56 43 25 4 41 12 38 59 14 22 22 17 10 54 33 7 34 56 15 29 37 41 34 20 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

28 28 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29

32 32 32 32 32 32 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 34 34 34 34 34

45 48 51 53 56 59 4 8 13 18 22 27 29 30 32 32 35 36 39 42 46 49 52 55 58 1 5 8 11 14

h

i

j

k

l

Variants from B in C, L, Ox, and V: e7: V56; e14: C16; e18: L18 V16; g6: L56; h18: L20; k1 to k6: C33; l10: V16; l16: CV 54 L33; l23: CV54.

112

edi t i on of the tab les with comments

New values for the solar equation The column for the solar equation displays values which are characteristic of Alfonsine astronomy. As mentioned above, their first appearance in Paris is found embedded in a table for the true solar anomaly compiled by John Vimond, ca. 1320, and extant in Paris, Bibliothèque nationale de France, lat. 7286C, 2r (see below). The solar equation results from the difference between the true solar anomaly, here called motus completus, and the mean solar anomaly, which is the argument of the table, as shown in below (see Chabás and Goldstein 2004, p. 224). I

II

argumentum

motus completus

s

(º) 0 0 … 2 3 3 3 3 … 5 6 6 … 8 8 8 9 9 … 11 12

s

II – I (º)

(°)

3 6

0 0

2;54 5;47

–0; 6 –0;13

27 0 3 6 9

2 2 3 3 3

24;51 27;50 0;50 3;50 6;51

–2; 9 –2;10 –2;10 –2;10 –2; 9

27 0 3

5 6 6

26;53 0; 0 3; 7

–0; 7 0; 0 0; 7

21 24 27 0 3

8 8 8 9 9

23; 9 26;10 29;10 2;10 5; 9

2; 9 2;10 2;10 2;10 2; 9

27 0

11 12

27; 6 0; 0

0; 6 0; 0

113

eq uation s an d hour ly velo cities of the sun a nd the moon

Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora Signa Linee numeri Numerus graduum G

G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

a

b

4 minue Equatio solis

Equatio lune

Signa Linee Motus Motus numeri solis lune Numerus in una in una graduum hora hora

G M S G M S M S M S 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 c

53 52 52 50 48 47 46 44 43 41 40 38 37 35 34 32 31 29 27 26 24 22 20 18 17 15 13 11 9 7

46 4 24 35 4 22 24 4 19 12 4 16 59 4 14 46 4 11 20 4 8 53 4 5 26 4 2 57 3 59 27 3 56 57 3 53 25 3 49 53 3 45 20 3 42 46 3 38 12 3 34 37 3 31 50 3 27 3 3 23 16 3 19 58 3 15 40 3 10 51 3 6 0 3 2 8 2 57 16 2 53 13 2 48 10 2 44 7 2 39 7 adde d e f g

Tabula equationis solis et lune et ad sciendum motum solis et lune in una hora

38 7 38 58 13 23 28 23 30 20 5 47 23 52 17 37 53 3 10 12 9 2 50 35 15 51 23 51 15 35

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 31 31 31 31 32 32 32 32 32 32 32

34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 35 35 35 35 35 35 35 35 35 35 35 35 35 35

17 20 23 26 29 32 35 38 41 43 46 49 52 54 57 59 2 4 7 9 11 13 16 18 20 22 25 27 29 31

h

i

j

k

l

Variants from B in C, L, Ox, and V: d12: V39; e4: CLV15; e12: V55; e14: V43; e16: LV26; e17: LV13; e22: CLV28; e24: V11; e30: CL0; g10: C52; h8: CLV31; h26: C21; j12 to j17: C32 LV30; j18 to j23: CLV32; l5: V20; l6: V22; l27: LV22.

5 minue Equatio solis

G

G

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

a

b

c

Equatio lune

Motus solis in una hora

Motus lune in una hora

G M S G M S M S M S 5 2 0 58 56 54 52 50 48 45 43 41 39 37 35 32 30 28 26 23 21 19 16 14 11 9 7 4 2 0

1 2 34 54 2 30 57 2 25 40 2 20 33 2 15 25 2 10 17 2 5 9 2 0 11 1 55 54 1 49 44 1 44 35 1 39 26 1 34 16 1 28 6 1 23 51 1 18 35 1 12 19 1 7 1 1 1 42 0 56 22 0 50 1 0 44 40 0 39 19 0 33 58 0 28 36 0 22 12 0 16 48 0 11 24 0 5 0 0 0 6 adde d e f g

52 6 16 23 26 26 22 17 9 54 44 27 9 49 26 1 34 6 36 5 32 58 23 47 10 33 56 18 40 0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

32 32 32 32 32 32 32 32 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33

35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35

33 35 37 39 41 43 45 47 48 49 51 52 52 53 53 53 54 54 55 57 58 59 1 2 2 3 3 3 4 4

h

i

j

k

l

Variants from B in C, L, Ox, and V: h10: C58; h19: L37; h27: C57 V53; j9: V32; j21 to j30: CV32; k23 to 30: CLV36; l29 to l30: V2 corr.4.

114

edi t i on of the tab les with comments

20. Velocities of the Sun and the Moon in a minute of a day Tabula ad inveniendum motum solis in uno minuto diei and Tabula ad sciendum motum lune in uno minuto diei (MS L, 52r–v) Titles in other manuscripts MS K, 13v–14r: Tabula ad inveniendum motum solis in uno minuto diei and Tabula ad inveniendum motum lune in uno minuto diei MS Ox, 81v–82r: Tabula ad inveniendum motum solis in uno minuto diei and Tabula ad sciendum motus solis in uno minuto diei MS W, 109r–v: Tabula ad inveniendum motum lune in uno minuto diei and Tabula ad inveniendum motum lune in uno minuto diei MS X, 127r–128r: Tabula motus solis in uno minute diei and Tabula ad inveniendum motum lune in uno minuto diei Description This second table for velocities is presented as two sub-tables, one for the velocity of the Sun and the other for the Moon. The argument for the Sun is given at intervals of 3º, and for the Moon it is displayed at intervals of 1º. The entries are in minutes of arc per minute of a day, that is, sixtieths of a day, to two significant places. The use of a sexagesimal sub-multiple of a day is consistent with the fact that in all the manuscripts we have examined, signs of 60º are used for the argument. We note, however, that minutes of a day is a unit not found anywhere else in the Tables of 1322 by John of Lignères. For the Sun, the extreme values are 0;57’/mn at argument 3º and 1;2’/mn at argument 3s (= 180º), where ‘mn’ stands for minute of a day, whereas for the Moon the extreme values are 12;9’/mn at 1º and 14:25’/mn at 180º. These entries can be compared with those for the velocities of the two luminaries in Table 19 by multiplying those given in degrees per hour by 24 to convert them into minutes of arc per minute of a day. For the extreme values, we find: 0;2,23 · 24 = 0;57’/mn and 0;2,23 · 24 = 1;1’/mn, for the Sun, and 0;30,18 · 24 = 12;7’/mn and 0;36,4 · 24 = 14;26’/mn, for the Moon, indicating that the entries in this table are compatible with those in Table 19. However, they are not exactly the same. For the Moon, we spot-checked the 180 entries by selecting the entries for arguments that are multiples of 10º. Out of the 18 pairs of entries in both tables, only five agree, 12 differ in 0;1’/mn, and one differs in 0;2’/mn. This seems an indication that Table 20 was not directly computed from Table 19 in this set, or that it was carelessly computed. In some manuscripts, we find instead a single table for velocities with the argument at integer numbers for both luminaries. A major difference is that the entries are given to four significant places, the extreme values for the Sun being 0;57,1,12’/mn (at 1º) and 1;1,30,15’/ mn (at 180º) and those for the Moon, 11;51,9,1’/mn (at 1º) and 14;44,47,8’/mn (at 180º). For both luminaries, the entries differ from those in Tables 19, 20 (two-sub-table version), and

veloc ities of the sun an d the moon in a minute of a day

John of Lignères. Velocities of the Sun and the Moon in a minute a day (20) Base: London, Egerton 889, 52r-52v Tabula ad inveniendum motum solis in uno minuto diei Argumentum solis

Signa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

a

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1

G 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 0 3 6 9 12 15 18 21 24 27 30 b

Signa

c

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4

Motus solis in uno minuto diei G 57 54 51 48 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 0 57 54 51 48 45 42 39 36 33 30 d

M

e

S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Variants from L in K, Ox, W and X: g10 to 30: Ox3.

f

Argumentum solis

Signa 57 57 57 57 57 57 57 57 57 57 57 57 57 58 58 58 58 58 58 58 58 58 58 59 59 59 59 59 59 59

g

1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

G 33 36 39 42 45 48 51 54 57 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 0 h

Signa

i

4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Motus solis in uno minuto diei G

j

27 24 21 18 15 12 9 6 3 0 57 54 51 48 45 42 39 36 33 30 27 24 21 18 15 12 9 6 3 0

M

k

S 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

l

59 59 59 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

115

116

edi t i on of the tab les with comments

21. In this case, the entries were taken from John of Montfort’s velocity table dated 1332, where the entries are also given in minutes of arc per minute of a day and were computed according to Ptolemy’s complete lunar model (Goldstein 1992, pp. 11–13). It is likely that this one-table version superseded the two-sub-table version, for it was considered more precise and accurate. Table 20 has a few features that set it aside from the rest of the tables in this set: low frequency in the corpus of manuscripts examined, use of sixtieths of a day, and two different versions of the same table. At first sight, these features may prompt its removal from the set of tables compiled by John of Lignères. Nevertheless, the fact remains that this particular table is found among the tables attributed to him by those who copied or assembled the codices containing them. Variant readings For this table, the base manuscript is London, British Library, Egerton 889 (MS L), and its entries have been collated with those in manuscripts Cologne (MS K), Oxford (MS Ox), and two manuscripts at the Vatican (MSS W and X). The numerous variants add no relevant information, but for the column for the seconds in the lunar velocity between 2s 1º and 2s 30º. Indeed, this table was frequently miscopied, for parts of it show shifts by one or two cells.

veloc ities of the sun an d the moon in a minute of a day

Tabula ad sciendum motum lune in uno minuto diei Argumentum lune

Signa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

a

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 b

Signa

c

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Motus lune in uno minuto diei G 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 d

M

e

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

S

f

Argumentum lune

Signa 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 11 11 12 12 12 13 13 13 14 14 15 15 15

Variants from L in K, Ox, W and X: a20 to a30: Ox 0 W0; l13: Ox23; g30: K,Ox,X1.

g

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

G 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 0 h

Signa

i

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

Motus lune in uno minuto diei G

j

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

M

k

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12

S

l

16 16 17 17 18 18 19 19 20 20 21 22 22 23 24 24 25 25 26 27 28 29 29 30 31 31 32 33 34 35

117

118

edi t i on of the tab les with comments

Tabula ad sciendum motum lune in uno minuto diei Argumentum lune

Signa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

a

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 b

Signa

c

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Motus lune in uno minuto diei G 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 d

M

e

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13

S 36 37 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 58 58 58 0 1 2 3 4 5 f

Argumentum lune

Signa

g

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2

G 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 0 h

Signa

i

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

Variants from L in K, Ox, W, and X: f20: X55; f28: X4; l6: W13; l8 to l13: one-cell shift upwards in W, X.

Motus lune in uno minuto diei G

j

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

M

k

13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13

S

l

6 8 9 9 11 12 14 16 17 18 19 20 22 22 24 25 25 27 28 29 31 32 33 34 35 36 38 40 41 41

veloc ities of the sun an d the moon in a minute of a day

Tabula ad sciendum motum lune in uno minuto diei Argumentum lune

Signa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

a

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 b

Signa

c

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Motus lune in uno minuto diei G 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 d

M

e

13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 14 14 14 14 14 14 14 14 14 14 14 14 14 14

S 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 1 2 3 4 5 6 7 8 9 10 11 12 12 f

Argumentum lune

Signa

g

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

G 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 0 h

Signa

i

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Motus lune in uno minuto diei G

j

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

M

k

14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14

S

l

13 14 14 15 16 17 17 18 18 19 20 20 20 21 21 22 22 22 23 23 23 24 24 24 24 24 24 24 25 25

Variants from L in K, Ox, W, and X: e16: Ox14; e17: W13; f9: Ox 52; f4-f18: one-cell shift upwards in X; f10 to f28 : one-cell shift upwards in Ox; g30: K,Ox,W,X3.

119

12 0

edi t i on of the tab les with comments

21. Velocities of the Sun and the Moon at intervals of 6º Tabula diversi motus solis et lune in horis (MS B, 74r) Titles in other manuscripts MS C, 84r: Tabula diversi motus solis et lune in horis MS E, 44v: Tabula diversi motus solis et lune in horis MS P, 49r: Tabula diversi motus solis et lune in horis MS V, 29r: Tabula diversitatis motus solis et lune in horis Description The hourly velocities of the Sun and the Moon are presented here in a single table. In this case, the argument is displayed at steps of 6º, in contrast to Table 19, above, where the hourly velocities were given for each integer degree. However, the ranges of the entries in both tables agree: 0;2,23º/h – 0;2,33º/h for the Sun and 0;30,18º/h – 0;36,4º/h for the Moon. The common entries also agree. We note that the Toledan Tables (Pedersen 2002, pp. 1410–1412), and other prior sets, such as the zij of al-Battānī (Nallino 1899–1907, 2:88) and the Almanac of Azarquiel (Millás 1943–1950, p. 174), feature the same table, also at steps of 6º. A recomputation of the lunar velocity in this table is available in Goldstein 1996, pp. 181–183. This table is mentioned in Chapter 31 in the canons beginning Priores astrologi (see Saby 1987, pp. 220–222), where it is explicitly said that the argument increases at intervals of 6º. Variant readings For this table, the base manuscript is MS B. There are no significant variants in the manuscripts examined here. We also compared the entries in this table with those in the zij of al-Battānī and the Toledan Tables, and the only feature worth mentioning is that the entries for arguments 108º and 114º are 0;33,35º/h and 0;33,56º/h, respectively, in all collated manuscripts, whereas the two Arabic zijes give 0;33,36º/h and 0;33,55º/h. These are not striking variants, but they may be considered as a characteristic feature of Table 21. This is all the more evident when the corresponding entries in Table 19 do not agree with those given in Table 21, but with the entries in these two Arabic zijes. All three tables for solar and lunar velocities identified in the Tables of 1322 by John of Lignères (Tables 19, 20, and 21) are based on Ptolemy’s first model. Nevertheless, they are quite different. One reproduces the standard table in the Toledan Tables and the zij of al-Battānī (Table 21), while another builds on it to give six times as many entries (Table 19), and the third one changes the unit (Table 20) to accommodate it to the pure sexagesimal counting of days described, among others, by John of Lignères in his canons Quia ad inveniendum loca planetarum. This is also a nice example of the versatility of the tables for the same purpose compiled at the time. The simultaneous presence of the three tables in the same set is not easy to justify, and maybe it tells us that all of the 32 different tables in this set were not assembled at the same time.

veloc ities of the sun a nd the moon at interva ls of 6º

John of Lignères. Velocities of the Sun and the Moon at intervals of 6° (21) Base : Basel F II 7, 74r Tabula diversi motus solis et lune in horis Linee numeri communes argumenti solis et lune G

a

G 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180

b

Motus solis diversus in una hora M

360 354 348 342 336 330 324 318 312 306 300 294 288 282 276 270 264 258 252 246 240 234 228 222 216 210 204 198 192 186 180

c

S 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

d

Motus lune diversus in una hora M

23 23 23 23 23 24 24 24 25 25 25 26 26 27 27 28 28 29 29 29 30 30 31 31 32 32 33 33 33 33 33

e

S 30 30 30 30 30 30 30 30 31 31 31 31 31 32 32 32 32 33 33 33 34 34 34 35 35 35 35 35 35 36 36

Variants from B in C, E, P, and V: e42: V31; e72: V32; e96: V33; e114: V34; e132: V35; e168: EV36 P36 corr.35; f120: CEPV14.

f

18 19 21 24 28 35 43 51 1 12 24 38 53 8 25 42 59 17 35 56 19 32 49 4 18 31 43 52 58 2 4

1 21

12 2

edi t i on of the tab les with comments

22. Parallax Tabula diversitatis aspectus lune in septimo climate cuius latitudo est 48 graduum dies eius longior 16 horarum equalium qui supponit lunam esse in longitudine longiori (MS B, 75r) Titles in other manuscripts MS C, 85v: Tabula diversitatis aspectus lune in 7° climate qui supponit lunam [esse] in longitudine longiori cuius latitudo 48 graduum et eius dies longior 16 horarum equalium. MS E, 45r: Tabula diversitatis aspectus lune in septimo climate qui supponit lunam esse in longitudine longiori cuius latitudo est 48 graduum et eius hore 16 MS P, 50v: Tabula diversitatis aspectus lune in septimo clymate qui supponit lunam esse in longitudine longiori cuius latitudo est 48 graduum et eius hore 16 MS V, 30r: Tabula diversitatis aspectus lune in septimo climate cuius latitudo est 48 graduum et hore 16 et supponit lunam esse in longitudine longiori Titles in other tables for parallax, not edited, in these manuscripts MS B, 74v: Tabula diversitatis aspectus lune in climate sexto cuius latitudo est 45 graduum 24 minuta et hore eius 15 et 28 minuta MS C, 85r: Tabula diversus (sic) aspectus lune in climate 6° cuius latitude est 45 graduum minutorum et hore eius 15 et 28 minuta hore MS E, 44v (supra): Tabula diversitatis aspectus lune in longitudine longiori in climate quinto cuius latitudo est 41 graduum 44 minutorum et eius hore 15 MS E, 44v (infra): Tabula diversitatis aspectus lune in climate sexto cuius longitudo est 45 graduum et 4 minutorum et hore eius 15 et 28 minuta hore MS P, 49v: Tabula diversitatis aspectus lune in longitudine longiori in clymate quinto cuius latitudo est 41 graduum 44 minutorum et eius hore 15 MS P, 50r: Tabula diversitatis aspectus lune in clymate sexto cuius latitudo 45 graduum minutorum et hore eius 15 et 28 minuta hore MS V, 29v: Tabula diversitatis aspectus lune in sexto climate cuius latitudo est 45 graduum et 24 minuta hore 15 et 28 minuta Description Parallax, called diversitas aspectus by medieval astronomers, is the angle between the true position of the celestial object and its apparent position for an observer. It depends both on the altitude of the object at a given time and the geographical latitude of the observer. Table-makers generated tables for various latitudes, normally one for each of the seven climates. In manuscripts containing the Tables of 1322 by John of Lignères, one or more such tables are found, but all of them have a table for a latitude of 48º, which corresponds to Paris according to medieval astronomers. This is the table edited here and already used in the Toledan Tables, among other sets of tables. It consists of 12 sub-tables, one for each zodiacal sign, where the Sun and the Moon are at the beginning of each of the signs. It follows the pattern established in Ptolemy’s

1 23

pa ra lla x

John of Lignères. Parallax (22) Base : Basel F II 7, 75 r Tabula diversitatis aspectus lune in septimo climate cuius latitudo est 48 graduum dies eius longior 16 horarum equalium qui supponit lunam esse in longitudine longiori Cancri

H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

longi­ tudo

M

8 7 6 5 4 3 2 1 Recessus 1 2 3 4 5 6 7 8 a b

M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

39 38 34 32 29 24 17 9 0 9 17 24 29 32 34 38 39

c

lati­ tudo

M 42 38 35 31 28 25 24 22 21 22 24 25 28 31 35 38 42 d

Leonis

H

M 7 40 7 0 6 0 5 0 4 0 3 0 2 0 1 0 Recessus 1 0 2 0 3 0 4 0 5 0 6 0 7 0 7 40 e f

longi­ tudo

lati­ tudo

M

M

g

39 40 40 39 35 30 22 14 5 3 11 18 22 25 25 23 21

h

33 32 28 25 25 23 22 22 23 26 29 32 36 39 42 45 46

longi­ tudo

Virginis

H

M 6 54 6 0 5 0 4 0 3 0 2 0 1 0 0 0 Recessus 0 0 1 0 2 0 3 0 4 0 5 0 6 0 6 54 i j

M

k

45 45 43 40 35 27 19 0 11 0 3 5 11 15 18 18 17

lati­ tudo

M 24 23 22 21 20 24 26 0 29 0 32 36 44 43 45 46 48 l

Variants from B in C, E, P, and V: a10 to a17: P inversed 8 to 1; b2 to b16: CEP vac. V0; c1: CP30; f2 to f16: CEP vac. V40; h4: E23; h9: EP33; h10: E36; j2 to j16: CEP vac. V54; k7: V19 corr.14; l5: V20 corr.22. Capricorni H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

M 4 3 2 1 0 0 0 0 Recessus 0 0 0 0 1 2 3 4 a b

longi­ lati­ tudo tudo M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c

30 24 17 9 0 0 0 0 0 0 0 0 0 9 17 24 30

M 42 45 47 48 0 0 0 0 49 0 0 0 0 48 47 45 42 d

Aquarii H

M 4 20 4 0 3 0 2 0 1 0 0 0 0 0 0 0 Recessus 0 0 0 0 0 0 1 0 2 0 3 0 4 0 4 20 e f

longi­ lati­ tudo tudo M

g

21 20 14 6 2 0 0 0 10 0 0 0 19 26 23 37 39

M 46 47 48 49 48 0 0 0 47 0 0 0 44 41 38 34 33 h

Piscium M 5 5 4 3 2 1 0 0 Recessus 0 0 1 2 3 4 5 5 i j

longi­ lati­ tudo tudo

H

M 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6

k

17 17 13 7 0 8 0 0 17 0 0 24 31 37 41 45 45

M 48 48 48 47 46 44 6 0 41 0 0 37 34 31 27 25 24 l

Variants from B in C, E, P, and V: b2 to b16 : CEP vac. V0; f2 to f16 : CEP vac. V20; g1: V21 corr.22; j2 to j16: CEP vac.; k12: V24 corr.34; l12: V37 corr.38.

12 4

edi t i on of the tab les with comments

Handy Tables. This table is strictly valid only for conjunctions, and it only applies to solar eclipses. The argument is the time before or after noon, in integer number of hours, except for the extremes, which correspond to the times of sunrise and sunset in each case throughout the year, given to minutes. The entries display the adjusted parallax, that is, the difference between the lunar parallax at syzygy and the solar parallax. It is presented in two columns, one for each of its two components, longitudinal and latitudinal, both given in minutes of arc. The Latin word recessus refers here to the nonagesimal, which is the point on the ecliptic closest to the zenith and at 90º along the ecliptic from the eastern and western horizons. This point indicates when the longitudinal component has to be added to or subtracted from the lunar longitude. The table ascribed to John of Lignères for latitude 48º and a longest daylight of 16h is already found, with scribal errors, in the Toledan Tables (Pedersen 2002, p. 1403), for latitude 48;13º and a longest daylight of 16h. It was borrowed from the zij of al-Battānī (Nallino 1899–1907, 2: 101), which was in turn derived from the Handy Tables by Ptolemy (Stahlman 1959, p. 280), who compiled it for climate VII (Borysthenes), latitude 48;32º and the longest daylight of 16h. In short, this table for the mouths of Borysthenes, where the river Dnieper flows into the Black Sea, was reused about 12 centuries later for Paris. This table is mentioned in Chapter 46 of the canons beginning Priores astrologi (see Saby 1987, p. 275). Variant readings For this table, the base manuscript is MS B: Basel, Universitätsbibliothek, F II 7. We note that in the first sub-table (Cancer) the entry for the longitudinal component for 8h is 30’ in the Handy Tables, the zij of al-Battānī, and the Toledan Tables. However, of the five manuscripts we examined containing the Tables of John of Lignères, three have 39’ and two have 30’. Surprisingly, the entry for the longitudinal component at time 4h in Capricorn, which should be the same as that for time 8h in Cancer, is 30’ in all manuscripts we have collated. Likewise, in the same sub-table for Cancer, the value of the longitudinal component for 7h is 33’ in the Handy Tables, the zij of al-Battānī, and the Toledan Tables, whereas it is 38’ in all five manuscripts we examined containing the tables of John of Lignères. Most probably, these characteristics originated at the very beginning in the compilation of the Tables of 1322.

1 25

pa ra lla x

Tabula diversitatis aspectus lune in septimo climate cuius latitudo est 48 graduum dies eius longior 16 horarum equalium qui supponit lunam esse in longitudine longiori Libre

longi­ tudo

H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

M 6 5 4 3 2 1 0 0 Recessus 0 0 1 2 3 4 5 6 a b

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

M 46 45 42 37 31 23 0 0 5 0 0 7 0 6 11 14 16 c

lati­ tudo

M

d

21 22 23 25 26 28 0 0 30 0 0 35 38 42 45 48 48

longi­ tudo

Scorpii

H

M 5 5 4 3 2 1 0 0 Recessus 0 0 1 2 3 4 5 5 e f

M 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6

g

45 45 42 37 31 24 0 0 17 0 0 8 0 7 13 17 17

lati­ tudo

M 24 25 27 31 34 38 0 0 41 0 0 44 46 47 48 48 48 h

Sagittarii

H

M 4 20 4 0 3 0 2 0 1 0 0 0 0 0 0 0 Recessus 0 0 0 0 0 0 1 0 2 0 3 0 4 0 4 20 i j

longi­ tudo

lati­ tudo

M

M

27 29 33 26 19 0 0 0 10 0 0 0 2 6 18 20 21

k

l

31 34 38 41 44 0 0 0 47 0 0 0 48 49 48 47 46

Variants from B in C, E, P, and V: b2 to b16 : CEP vac. V0; c9: V5 corr.15; c15: V7 corr.11; d9: V30 corr.51; f2 to f16 : CEP vac. V6; j2 to j16: CEP vac. V20; k1: C29; k2: C27; k15: CV14; l1: C33 V33 corr.31; l16: E44 corr.47. Arietis

H 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

longi­ tudo

M

6 5 4 3 2 1 0 0 Recessus 0 0 1 2 3 4 5 6 a b

M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c

16 14 11 6 0 7 0 0 15 0 0 23 31 37 42 45 46

lati­ tudo

M 48 48 45 42 38 35 0 0 31 0 0 28 26 25 23 22 21 d

Tauri

H

M 6 54 6 0 5 0 4 0 3 0 2 0 1 0 0 0 Recessus 0 0 1 0 2 0 3 0 4 0 5 0 6 0 6 54 e f

longi­ tudo

M

g

17 18 18 15 11 5 3 0 11 0 19 27 35 40 43 45 45

lati­ tudo

M 44 47 45 43 40 36 32 0 29 0 46 24 20 21 22 23 24 h

Geminorum

H

M 7 40 7 0 6 0 5 0 4 0 3 0 2 0 1 0 Recessus 1 0 2 0 3 0 4 0 5 0 6 0 7 0 7 40 i j

Variants from B in C, E, P, and V: b2 to b16 : CEP vac.; f2 to f16 : CEP vac. ; j2to j16: CP vac.; k13: V35 corr.30.

longi­ tudo

M

k

21 23 25 25 22 18 11 3 5 14 22 30 35 39 40 40 39

lati­ tudo

M 46 45 42 39 36 32 29 26 23 22 22 23 23 25 28 32 33 l

126

edi t i on of the tab les with comments

23. Proportions for correcting lunar parallax Tabula diversitatis aspectus lune ad solem (MS B, 75v) Titles in other manuscripts MS C, 86r: Tabula diversitatis equationis aspectus lune ad solem MS E, 45r: Tabula equationis diversitatis aspectus lune ad solem MS P, 51r: Tabula equationis diversitatis aspectus lune ad solem MS V, 30v: Tabula equationis diversitatis aspectus lune ad solem Description The purpose of this table of proportions is to correct parallaxes for intermediate positions of the Moon on its epicycle. The argument is lunar anomaly, and it is given at steps of 6º from 0º to 180º and their complement in 360º. The entries range from 0 to 12 minutes of arc and are the same as those in the column headed circulus brevis (epicycle) displayed at steps of 6º in the table for corrections (also called tabula actacium; see Table 30 below). The same table is found among the Toledan Tables (Pedersen 2002, p. 1442) and derives from Ptolemy’s Handy Tables (Stahlman 1959, p. 257). One wonders why this table of proportions for intermediate positions of the Moon on its epicycle was separated from the other for correcting parallax when the centre of the lunar epicycle is not at the apogee of the deferent, as in Table 30, where it is displayed as a specific column. We note that the 31 entries in this table agree with those in Table 30. This table is not mentioned in John of Lignères’ canons. Variant readings For this short table, the base manuscript used is Basel, Universitätsbibliothek, F II 7. No significant differences are reported.

p rop ortion s for correcting luna r pa ra lla x

John of Lignères. Proportion for correcting lunar parallax (23) Base : Basel F II 7, 75 v Tabula diversitatis equationis aspectus lune ad solem Proportionalia G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

G 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180

M 360 354 348 342 336 330 324 318 312 306 300 294 288 282 276 270 264 258 252 246 240 234 228 222 216 210 204 198 192 186 180

a b Variants from B in C, E, P, and V: b12: C394; c28: V14 corr. 12.

c

0 0 0 0 1 1 1 2 2 2 3 3 4 4 5 6 6 7 8 8 9 9 10 10 11 11 11 12 12 12 12

1 27

12 8

edi t i on of the tab les with comments

24. Solar eclipses with argument of lunar latitude as argument Tabula eclipsis solis ad longitudinem longiorem and Tabula eclipsis solis ad longitudinem propinquiorem (MS B, 76r) Titles in other manuscripts MS C, 86r–v: Tabula eclypsis solis ad longitudinem longiorem and Tabula eclypsis solis ad longitudinem propinquiorem MS E, 45r: Tabula eclipsis solis ad longitudinem longiorem and Tabula eclipsis solis ad longitudinem propinquiorem MS P, 51r: Tabula eclipsis solis ad longitudinem longiorem and Tabula eclipsis solis ad longitudinem propinquiorem MS V, 31r: Tabula eclipsis solis ad longitudinem longiorem and Tabula eclipsis solis ad longitudinem propinquiorem Description John of Lignères included four tables for the digits of eclipses, two for solar eclipses (Tables 24 and 25) and two for lunar eclipses (Tables 26 and 27). In both cases, he used different variables to enter each of the tables: the argument of lunar latitude and lunar latitude. In the Almagest, Ptolemy used the argument of lunar latitude to compile his tables for the digits of eclipse and this practice was followed by al-Khwārizmī and many other medieval table-makers. It is also found in the Toledan Tables, as well as in the present table and Table 26. In contrast, in the Handy Tables, Ptolemy switched to lunar latitude as the variable to enter these specific tables. This tradition was continued by al-Battānī and many others, such as the compilers of the Toledan Tables and John of Lignères in Tables 25 and 27. We note that the Toledan Tables included both types of tables, in a clear example of the coexistence of the two traditions, as did John of Lignères in his set. The conversion between the argument of lunar latitude and lunar latitude is made possible by means of Table 29. Table 24 is presented in four sub-tables, two for longest distance (ad longitudinem longiorem) and two for shortest distance (ad longitudinem propinquiorem), both for positive and negative values of the argument of lunar latitude, that is, north and south of the ecliptic, respectively. We note that longitudinem longiorem refers to the situation where the Moon is at the apogee of its epicycle, whereas longitudinem propinquiorem corresponds to the Moon at perigee. In the four sub-tables, besides the two complementary columns for the argument (in degrees and minutes), there are two columns. One is for the digits of the eclipse (in puncta and minutes of puncta), where the solar disc is divided into 12 linear digits, and the other is for the immersion, here called minuta casus (in minutes and seconds), that is, the difference in minutes of arc between the centre of the Moon at the beginning of the eclipse (first contact) and at mid-eclipse. The presentation is the same as that in the Toledan Tables (Pedersen 2002, pp. 1461–1462), and the entries agree, but for a few cases.

1 29

s ol ar eclip ses with argumen t of luna r latitude a s a rg ument

John of Lignères. Solar eclipses with argument of lunar latitude as argument (24) Base : Basel F II 7, 76r Tabula eclipsis solis ad longitudinem longiorem

Tabula eclipsis solis ad longitudinem longiorem

Portio latitudinis septentrionalis

Puncta eclipsis

Minuta casus

G

P

M

M

G

M

M

Argumentum latitudinis meridiane

S

G

M

1

6

37

173

32

0

0

0

0

1 359

2

6

30

173

30

0

11

5

30

2 359

0

3

6

0

174

0

1

5

13

7

3 358

4

5

30

174

30

1

55

17

10

4 358

5

5

0

175

0

2

45

20

10

5

6

4

30

175

30

3

37

22

41

6

7

4

0 176

0

4

29

24

8

3

30 176

30

5

21

9

3

0

177

0

6

10

2

30

177

30

11

2

0

178

12

1

30

13

1

14

0

15

G

Minuta casus

P

M

M

S

30

11

32

30

51

181

0

9

29

30

27

30

181

30

8

28

30

7

0

182

0

7

27

29

28

357

30

182

30

6

26

28

39

357

0

183

0

6

23

27

21

41

7 356

30

183

30

5

21

26

15

26

15

8 356

0 184

0

4

29

24

41

13

27

21

9

355

30 184

30

3

37

22

41

7

6

28

39

10

355

0

185

0

2

45

20

10

0

7

59

29

28

11 354

30

185

30

1

55

17

10

178

30

8

48

30

7

12 354

0 186

0

1

5

13

7

0

179

0

9

29

30

35

13

353

30 186

30

0

11

5

30

30

179

30

10

32

30

21

14

353

0

187

0

0

0

0

0

0

0 180

0

10

45

30

55

15 352

30

187

30

0

0

0

0

16 359

30 180

30

11

32

30

51

b

d

f

g

b

c

d

a

c

e

h

Variants from B in C, E, P, and V: a1: P149 V149 corr.349; b1: V37 corr.38; b3, b5, b7, b9, b11, b13, b15: C30; c1: CPV23 E33; c3, c5, c7, c9, c11, c13, c15: C30; f11: CEP57 V59 corr. 57; f13: CEP39 V29 corr. 39; f16: P33; g5: P29 V19; h10: C29; h13: CEP34 V35 corr. 34; h14: CEPV51; h16: V51 corr.53.

a

30 180

M

Puncta eclipsis

e

f

g

h

Variants from B in C, E, P, and V: a15: E360 P vac.; a16: CV352; a17: CV352; a17: C351; b14: E23; b15: E0; b16: C0; b17: C30; c15: E180; c16: V188; d15: E0; d16: V0; e1: CEV10; e5: C7; e6: E6 corr. 7; e8: E4 corr.7; e15: E10; e16: V0; f2 : E39 V29 corr39; f3: E48 corr. 28 V28 corr. 48; f4: C57 E21 corr. 51 V27 corr. 57; f6: E13 corr. 7 V23 corr. 13; f15: E45; f16: V0; g15: E30; g16: V0; h2: E34 corr.27; h15: E55; h16: V0.

130

edi t i on of the tab les with comments

This table is mentioned in Chapter 35 of the canons beginning Priores astrologi (see Saby 1987, pp. 231–249). Variant readings For this table, the base manuscript used is MS B: Basel, Universitätsbibliothek, F II 7. The variant readings make it possible to easily detect scribal errors in MS B. As an example, consider the first value of the argument in the sub-table for longest distance, at the beginning of eclipse. The argument is given in the two columns as 6;37º and 173;32º. It is clear that the real value for the second argument, which is the complement in 180º to the first, should be 173;23º. This is indeed the value found in the other manuscripts we have collated, as well as in almost all manuscripts containing the Toledan Tables examined by Pedersen (2002, p. 1461). A quite different case is given by the first two complementary values of the argument in the sub-table for shortest distance, at the beginning of the eclipse: 7;1º and 172;40º. These two values do not add up to 180º. All other manuscripts collated read 7;1º, and two of them read 172;49º. The main tradition in the Toledan Tables, however, gives 7;11º and 172;49º, the sum of which is indeed 180º. It would therefore seem that the minutes in both values, 7;1º and 172;40º, are erroneous. Surprisingly, these are exactly the two values appearing in the corresponding table in the editio princeps of the Parisian Alfonsine Tables in 1483 (m1v) and in the second edition in Santritter, 1492 (k4r), an indication that both editions depended on this faulty tradition related to John of Lignères’ tables.

1 31

s ol ar eclip ses with argumen t of luna r latitude a s a rg ument

Tabula eclipsis solis ad longitudinem propinquiorem Argumentum latitudinis septentrionalis G

M

G

M

Tabula eclipsis solis ad longitudinem propinquiorem

Puncta eclipsis

Minuta casus

P

M

M

Argumentum latitudinis meridiane

S

G

M

1

7

1

172

40

0

0

0

0

1 359

2

7

0

173

0

0

17

7

56

2 359

0

3

6

30

173

30

1

9

14

11

3 358

4

6

0

174

0

2

0

18

32

4 358

5

5

30

174

30

2

53

21

37

5

6

5

0

175

0

3

45

24

2

6

7

4

30

175

30

4

37

26

8

4

0 176

0

5

28

9

3

30 176

30

6

10

3

0

177

0

11

2

30

177

12

2

0

13

1

30

G

30 180

M

Puncta eclipsis

Minuta casus

P

M

M

S

30

11

30

33

30

181

0

10

38

33

15

30

181

30

9

47

32

45

0

182

0

8

56

32

15

357

30

182

30

8

5

31

31

357

0

183

0

7

12

30

19

12

7 356

30

183

30

6

20

29

17

27

53

8 356

0 184

0

5

28

27

53

20

29

17

9

355

30 184

30

4

37

26

12

7

12

30

19

10

355

0

185

0

3

45

24

2

30

8

5

31

31

11 354

30

185

30

2

53

21

17

178

0

8

56

32

15

12 354

0 186

0

2

0

18

32

178

30

9

37

32

49

13

353

30 186

30

1

9

14

16

14

1

0

179

0

10

48

33

15

14

353

0

187

0

0

17

7

16

15

0

30

179

30

11

30

33

30

15 352

30

187

30

0

0

0

0

16

0

0 180

0

12

44

33

34

e

f

g

h

b

c

d

a

b

c

d

Variants from B in C, E, P, and V: b4, b6, b8, b10,b12, b14, b16: C30; d1: C49 V40 corr.49; f7: C17 V19 corr.37; f13: V37 corr.47; f16: V40 corr. 44; h1: V0 corr.10; h10: V10 corr.19; h12: V51 corr.15.

a

e

f

g

h

Variants from B in C, E, P, and V: b15: EP49 V30 corr. 49; c4: C181; c6: C182; c8: C183; c10: C184; c12: C185; c14: C186; d15: E11 V11 corr. 31; f2: V28 corr.38; h8: CEPV13.

132

edi t i on of the tab les with comments

25. Solar eclipses with lunar latitude as argument Tabula eclipsis solis in longitudine longiori and Tabula eclipsis solis in longitudine propinquiori (MS B, 75v) Titles in other manuscripts MS C, 86r: Tabula eclipsis solis in longitudine longiori and Tabula eclipsis solis in longitudine propinquiori MS E, 45r: Tabula eclipsis solis in longitudine longiori and Tabula eclipsis solis in longitudine propinquiori MS P, 51r: Tabula eclipsis solis in longitudine longiori and Tabula eclipsis solis in longitudine propinquiori MS V, 30v: Tabula eclipsis solis in longitudine longiori and Tabula eclipsis solis in longitudine propinquiori Description As mentioned in the description of the previous table, this one adheres to the tradition set by Ptolemy in the Handy Tables, which was followed by al-Battānī and many others, and it is also to be found in the Toledan Tables in coexistence with the Almagest tradition. In this case, one enters the table with lunar latitude, given in minutes and seconds. However, the real variable, displayed for each integer number from 0 to 12, is the number of digits of the eclipsed diameter. Table 25 is presented in two sub-tables: longest distance (ad longitudinem longiorem) and shortest distance (ad longitudinem propinquiorem). As was the case with Table 24, both have two columns: one for the digits of the eclipse (in puncta and minutes of puncta) and another for the duration of immersion, minuta casus (in minutes and seconds). The presentation is the same as that in the Toledan Tables (Pedersen 2002, p. 1474), and the entries agree, but for a few cases. This table is mentioned in Chapter 35 in the canons beginning Priores astrologi (see Saby 1987, pp. 231–249). Variant readings For this table, the base manuscript is also Basel, Universitätsbibliothek, F II 7. In the two sub-tables, there are eight cases where the entries differ from the corresponding tables in the mainstream version of the Toledan Tables and in the zij of al-Battānī (Nallino 1899–1907, 2:91), which is ultimately their source. Of these, three correspond to entries in the argument, which are intended to be presented at regular intervals, here alternating 2;42’ and 2;43’: the arguments in rows 5 and 7 in the sub-table for longest distance read 20;17’ and 14;41’, respectively, while they should read 20;10’ and 14;45’. In the sub-table for shortest distance, the argument in row 6 is 20;20’ rather than 20;28’ All manuscripts we have collated containing the tables of John of Lignères for 1322 share the same readings

133

solar ec lip ses with luna r latitude a s a rg ument

with MS B. This would indicate that these faulty entries are a specific characteristic of the set by John of Lignères. Moreover, the 1492 edition of the Parisian Alfonsine Tables includes these two sub-tables (k2v) with the same faulty entries. This is a sign of the strong link between this edition and John of Lignères’ tables, as already highlighted with regard to Table 24. John of Lignères. Solar eclipses with lunar latitude as argument (25) Base : Basel F II 7, 75 v Tabula eclipsis solis in longitudine longiori Latitudo lune Dyameter equalis M

S

P

Tabula eclipsis solis in longitudine propinquiori Latitudo lune equalis

Quantitas casus M

S

M

Eclipsis

S

Quantitas casus

M

M

S

1

31

0

0

0

0

1

34

0

P

0

0

2

28

18

1

12

39

2

31

18

1

11

16

3

25

35

2

17

30

3

28

35

2

18

25

4

22

53

3

20

25

4

25

53

3

22

2

5

20

17

4

23

33

5

23

10

4

24

50

6

17

28

5

25

36

6

20

20

5

27

9

7

14

41

6

27

36

7

17

45

6

29

0

8

12

3

7

28

34

8

15

3

7

30

30

9

9

20

8

29

33

9

12

20

8

31

56

10

6

38

9

30

17

10

9

38

9

32

37

11

3

55

10

30

45

11

6

55

10

33

56

12

1

13

11

30

59

12

4

13

11

33

44

13

0

0

12

31

0

13

1

30

12

33

48

14 a

b

c

d

e

Variants from B in C, E, P, and V: a2: V34 corr.28; e1: C39; e2: V39 corr.37; e3: V30 corr.35; e5: V33 corr.23; e8: V33 corr.34; e10: V17 corr. 32; e11: V45 corr. 31; e12: V59 corr.30.

0 a

0 b

12 c

34 d

Variants from B in C, E, P, and V: c14: V13 corr.12.

0 e

134

edi t i on of the tab les with comments

26. Lunar eclipses with argument of lunar latitude as argument Tabula eclipsis lunaris ad suam longitudinem longiorem and Tabula eclipsis lune ad suam longitudinem propinquiorem (MS B, 77r–v) Titles in other manuscripts MS C, 87v–88r: Tabula eclipsis lunaris ad suam longitudinem longiorem and Tabula eclipsis lunaris ad longitudinem suam propinquiorem MS E, 45v: Tabula eclipsis lunaris ad suam longitudinem longiorem and Tabula eclipsis lune ad suam longitudinem propinquiorem MS P, 52r: Tabula eclipsis lunaris ad suam longitudinem longiorem and Tabula eclipsis lunaris ad suam longitudinem propinquiorem MS V, 32r–v: Tabula eclipsis lunaris ad suam longitudinem longiorem in epiciclo and Tabula eclipsis lune ad suam longitudinem propinquiorem in epiciclo Description As was the case for solar eclipses (Tables 24 and 25), the two tables for lunar eclipses (Tables 26 and 27) follow different traditions, both set by Ptolemy, first in the Almagest, using the argument of lunar latitude for his eclipse tables, and later in the Handy Tables, where lunar latitude is the variable to enter the tables. As noted, the Toledan Tables included both forms of presentation, and this was also the pattern used by John of Lignères in his own tables. This table follows the Almagest tradition, and it is presented in four sub-tables, two for longest distance (ad longitudinem longiorem) and two for shortest distance (ad longitudinem propinquiorem), both for positive and negative values of the argument of lunar latitude. In the four sub-tables, there are three additional columns besides those for the argument (in degrees and minutes). One is for the digits of the eclipse (in puncta and minutes of puncta), where the solar disc is divided in 12 linear digits. We note that in this case the number of digits can exceed 12, because the diameter of the Earth’s shadow is much bigger than the lunar diameter. The second column is, again, for the duration of immersion, minuta casus (in minutes and seconds), and the third column displays the duration of half totality, here called minuta more (in minutes and seconds). The presentation and the entries in this table agree, but for scribal errors, with those in the Toledan Tables (Pedersen 2002, pp. 1467–1470). This table is mentioned in Chapters 35 and 38 of the canons beginning Priores astrologi (see Saby 1987, pp. 231–249 and pp. 253–258). Variant readings As was the case for most of the previous tables, the base manuscript in this case is Basel, Universitätsbibliothek, F II 7. It is worth noting that MS B has a second copy of two of the four sub-tables, found on ff. 94v–95r: those for the northern values of the argument.

135

lun ar ec lip ses with argument of luna r latitude a s a rg ument

John of Lignères. Lunar eclipses with argument of lunar latitude as argument (26) Base: Basel F II 7, 77r-v Tabula eclipsis lunaris ad longitudinem longiorem

1

Tabula eclipsis lunaris ad longitudinem propinquiorem

Argumentum latitudinis septentrionalis

Puncta Minuta Minuta eclipsis casus more

Argumentum latitudinis septentrionalis

Puncta Minuta Minuta eclipsis casus more

G

P

G

P

M

G

M

M

M

0

0

S

M

0

S

M

G

M

13

0 167

0

M

M

S

M

0

26

12

25

S

0

1

0

0

2

12

30 167

30

1

13

20

52

0

0

0

0

3

12

0 168

0

2

2

26

7

0

0

0

4

11

30 168

30

2

50

30

22

0

0

3

36

34

27

0

0

4

34

37

0

0

0

11

0 169

0

0

0

2

10

30 169

30

0

40

12

10

3

10

0 170

0

1

40

19

30

4

9

30 170

30

2

40

24

32

0

5

9

0

171

0

3

32

28

12

0

0

5

11

0 169

0

6

8

30

171

30

4

35

31

13

0

0

6

10

30 169

30

0

0

7

8

0 172

0

5

30

34

10

0

0

7

10

0 170

0

5

29

41

27

0

0

8

7

30 172

30

6

25

36

27

0

0

8

9

30 170

30

6

10

42

26

0

0

9

7

0 173

0

7

23

38

42

0

0

9

9

0

171

0

6

54

45

21

0

0

10

6

30 173

30

8

21

40

28

0

0

10

8

30

171

30

7

41

47

25

0

0

11

6

0 174

9

9

20

42

11

0

0

11

8

0 172

0

8

31

49

28

0

0

12

5

30 174

30

10

17

43

36

0

0

12

7

30 172

30

9

26

51

6

0

0

13

5

0 175

0

11

14

44

52

0

0

13

7

0 173

0

10

11

52

44

0

0

14

4

30 175

30

12

11

45

4

0

0

14

6

30 173

30

10

54

54

9

0

0

15

4

0 176

0

13

9

36

42

10

21

15

6

0 174

0

11

43

55

20

0

0

16

3

30 176

30

14

7

34

1

13

47

16

5

30 174

30

12

34

47

14

9

7

17

3

0 177

0

15

4

32

44

15

48

17

5

0 175

0

13

27

43

53

14

9

18

2

30 177

30

16

2

31

34

17

40

18

4

30 175

30

14

25

40

54

17

25

19

2

0 178

0

17

0

30

31

19

14

19

4

0 176

0

15

0

39

9

19

57

20

1

30 178

30

17

57

30

3

20

12

20

3

30 176

30

15

50

37

50

21

7

21

1

0 179

0

18

53

29

52

20

52

21

3

0 177

0

16

38

36

51

23

32

22

0

30 179

30

19

50

29

19

21

16

22

2

30 177

30

17

25

36

0

24

49

23

0

0 180

0

20

46

29

16

21

22

23

2

0 178

0

18

15

35

31

25

47

e

f

g

h

i

j

24

1

30 178

30

19

5

35

5

26

32

a

b

c

d

25

1

0 179

0

19

54

34

49

27

2

26

0

30 179

30

20

43

34

40

27

16

27

0

0 180

0

21

31

34

35

27

27

e

f

g

h

i

j

a

Variants from B in C, E, P, and V: f5: V32 corr.14; f6: V35 corr.32; f9: P22; g11: CE44 V43 corr 42; g12: CEP44 V42 corr.43; g13: CE42 V erased and corr.44; g14: CEP46 V 46 corr.41; g15: CE3 P4 V34 corr.36; g16: CE36; g17: CE31; g18: CE30 V30 corr.31; g19: V30 corr.31; g20: C24 EP29; h5: CEP7; h7: E2; h8: C37; h9: EP43; h18: CEP38; h22: V19 corr.29; h23: V16 corr.46; j18: V49 corr.40.

b

c

d

Variants from B in C, E, P, and V: a21: P2; f4: E15; f16: CEPV35; f21: V38 corr.30; g4: E34 V34 corr.30; g8: CEPV43; h2: P53; h4: V22 corr.23; h13: V34 corr.44; h23: C30; j19: V7 corr.57; j20 C5; j23: CEP49 V49 corr.47.

136

edi t i on of the tab les with comments

In this case, two additional columns for the argument have been added, as well as two additional rows filled in with zeros. Because of symmetry, the entries for northern latitudes and southern latitudes in the two sub-tables for longest distance should be equal. The same applies for the two sub-tables for shortest distance. However, this is not so in about 20 cases, 12 are repeated in all manuscripts examined and disagree with the entries in the Toledan Tables. The following two examples illustrate the situation. Firstly, in the sub-tables for longest distance, the entries in the column for the duration of immersion are 28;7’ (MS B adds a scribal error, 28;12’) for arguments 9;0º/171;0º and also 28;7’ for arguments 351;0º/189º, in all manuscripts consulted. However, the entries found in the Toledan Tables are 28;50’ in all cases. The other example is slightly different. It is found in the sub-tables for shortest distance, in the column for the digits of the eclipse. The entries for arguments 10;30º/169;30º and 349;30º/190;30º are, respectively, 4;34 p and 4;38 p (MS B adds a scribal error, 4;0 p) in all manuscripts consulted. Both entries should be 4;38 p, as it is indeed the case in the Toledan Tables. These examples of diverging entries, together with the other faulty entries, seem to be a repeated feature of the tables for 1322 compiled by John of Lignères. We note that the analogous sub-tables in the Parisian Alfonsine Tables printed in 1483 have 28;7’ in the first case and 4;34’ in the second, and that both agree with John of Lignères’ entries. As noted in the descriptions of previous tables, it would seem that John of Lignères’ eclipse tables were the model, not only in the format but also its contents used by Ratdolt in the editio princeps of the Parisian Alfonsine Tables (1483) and this reinforces the idea that the Tables of 1322 by John of Lignères is a set to be considered as an intermediary stage between the Toledan Tables and the Parisian Alfonsine Tables.

137

lun ar ec lip ses with argument of luna r latitude a s a rg ument

Tabula eclipsis lunaris ad suam longitudinem longiorem Argumentum latitudinis meridiane G 1 360

M

G

0 180

Tabula eclipsis lune ad suam longitudinem propinquiorem

Puncta Minuta Minuta eclipsis casus more

M

P

M

M

0 20 46 29

Argumentum latitudinis meriodionalis

S

M

S

G

16

21

22

1 360

M

Puncta Minuta Minuta eclipsis casus more

G

M

0 180

0

P

M

21

M

S

M

S

31 34 45

27

27

2 359 30 180 30

19 50 29

19

21

16

2 359 30 180 30 20 43 34 40

27

16

3 359

0

18

53 29

52 20

52

3 359

27

2

4 358 30 181 30

17

57 30

3

19

12

4 358 30 181 30

19

5

35

5 26

32

5 358

0

17

0 30

31

17

14

5 358

0

18

15

35

31

6 357 30 182 30

16

2

31

38

15

38

6 357 30 182 30

17

25

36

0 24 49

7 357

0

15

5

32 44

13 48

7 357

16

38

36

51

23

32

8 356 30 183 30

14

7 34

10 47

8 356 30 183 30

15 50

37 50

21

57

9 356

0

13

9

36 42

0

22

9 356

0

15

0

39

9

19

57

10 355 30 184 30

12

11

41

4

0

0

10 355 30 184 30

14

25 40 54

17

25

11 355

0

11

14 44

52

0

0

11 355

0

13

27 43

53

14

9

12 354 30 185 30

10

16 43

36

0

0

12 354 30 185 30

12

39 47

14

9

7

55 20

13 354

0 181 0 182 0 183 0 184 0 185

1

0 181 0 182 0 183 0 184 0 185

0

0

19 54 34 44

9 20 42

11

0

0

13 354

14 353 30 186 30

8

21 40

22

0

0

14 353 30 186 30

10 54 54

15 353

0

7

23

38 42

0

0

15 353

10

16 352 30 187 30

6

25

36

27

0

0

16 352 30 187 30

9 26

17 352

5 30 34

10

0

0

17 352

0

18 351 30 188 30

4

32

31

13

0

0

19 351

3

35 28

7

0

20 350 30 189 30

2 40 24

32

21 350

0

1 40

22 349 30 190 30 23 349

0

0

9

0

0

52 44

0

0

51

6

0

0

8

31 49 28

0

0

18 351 30 188 30

7

41 47

25

0

0

0

19 351

6 54 45

21

0

0

0

0

20 350 30 189 30

6

10 43 26

0

0

19 30

0

0

21 350

5 29

41

27

0

0

0 40

12 30

0

0

22 349 30 190 30

4

38

37

0

0

0

0

0

0

0

0

0

0

23 349

3

36 34

27

0

0

24 348 30 191 30

0

0

0

0

0

0

24 348 30 191 30

2 50

31

22

0

0

25 348

0

2

2 26

7

0

0

26 347 30 192 30

1

13 20

52

0

0

27 347

0 26

12

25

0

0

g

h

a

0 186 0 187 0 188 0 189 0 190 0 191 b

c

0

0 0

d

e

f

g

h

i

j

a Variants from B in C, E, P, and V: a24 to j24: CEPV vac.; f1: E48; f7: CEP4; f20: CE30; f21 to f22: C30; h1: EPV46; h2: V19 corr.49; h9: E0; h14: CEPV28; h22: CEV10; i4 to i9: EPV one-cell shift downwards; i11, i14, i15, i17, i19: P vac.; j12, j14, j16, j18: P vac.

0 186 0 187 0 188 0 189 0 190 0 191 0 192 0 193 b

c

0 0

11 43

25 47

0 0 0

0 d

e

11

f

i

j

Variants from B in C, E, P, and V: f1: E25: f2: E40 V43 corr. 47; f3: E49; f5: E31; f6: E0; f7: E51; f9: C5; f10: E54; f11: E53; f12: C35 E14 V25 corr.35; f13: E20; f14: E9; f15: E44; f16: E6; f17: E28; f18: E25; f19: E21; f20: E26; f21: E27; f22: E0; f23: E27; f24: E22; f25: E7; f26: E52; f27: E25; g5: E39; g13: V45; g24: CP30; h1: C35 EP25 V25 corr. 35; h3: CEPV49; i14, i16, i18, i20, i22, i24, i26: P vac. ; j11: C15; j15, j17, 19, j21, j23, j25: P vac.

138

edi t i on of the tab les with comments

27. Lunar eclipses with lunar latitude as argument Tabula eclipsis lune ad suam longitudinem longiorem and Tabula eclipsis lune ad suam longitudinem propinquiorem (MS B, 76v) Titles in other manuscripts MS C, 87r: Tabula eclipsis lune ad suam longitudinem longiorem and Tabula eclipsis lune ad longitudinem suam propinquiorem MS E, 45v: Tabula eclipsis lunaris ad longitudinem longiorem and Tabula eclipsis lune ad suam longitudinem propinquiorem MS P, 51v: Tabula eclipsis lune ad suam longitudinem longiorem and Tabula eclipsis lune ad suam longitudinem propinquiorem MS V, 31v: Tabula eclipsis lune ad longitudinem longiorem and Tabula eclipsis lune ad longitudinem propinquiorem in epiciclo Description One enters this table with lunar latitude, thus following Ptolemy’s tradition in the Handy Tables, the zij of al-Battānī and many others, including the Toledan Tables. Lunar latitude is displayed in minutes and seconds. In contrast to other tables for eclipses, we are given a second way to enter the table because the argument of lunar latitude (more specifically, the lunar longitude from the node) is listed as a column, with entries in degrees and minutes. It should also be noted that, as in Table 25, the real variable is displayed for each integer number from 0 to 21 and corresponds to the number of digits of the eclipsed diameter. As was the case for the analogous table for solar eclipses (Table 25), this table is presented in two sub-tables: longest distance (ad longitudinem longiorem) and shortest distance (ad longitudinem propinquiorem). Both sub-tables have three columns: one for the digits of the eclipse (in puncta and minutes of puncta), another for the duration of immersion, minuta casus (in minutes and seconds), and a third one displaying the duration of half totality, minuta more (in minutes and seconds). The presentation and the entries in this table agree -discounting scribal errors- with those in the Toledan Tables (Pedersen 2002, pp. 1478). Table 27 is mentioned in Chapters 35 and 38 of the canons beginning Priores astrologi (see Saby 1987, pp. 231–249 and pp. 253–258). Variant readings For this table, the base manuscript used to collate other manuscripts is Basel, Universitätsbibliothek, F II 7. These two sub-tables share the problems already pointed out with the other tables for eclipses. An additional problem here is found in the sub-table for greatest distance. The column for the longitude from the node begins with 9;11º and decreases non-regularly to 0;0º, whereas in the Toledan Tables it begins with 10;11º and decreases smoothly to 0;0º; in MS B, the entries for degrees were shifted upwards one line, discarding the first entry, 10º, and adding an extra 0º at the end of the column.

lun ar eclip ses with luna r latitude a s a rg ument

John of Lignères. Lunar eclipses with lunar latitude as argument (27) Base : Basel F II 7, 76 v Tabula eclipsis lune ad suam longitudinem longiorem Longitudo lune a nodo G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

a

M 9 8 8 7 7 6 6 5 5 4 4 4 3 3 2 2 1 1 1 0 0 0 0

b

Latitudo lune M

11 43 18 46 17 49 20 53 22 55 26 58 30 2 24 5 27 9 41 12 44 16 0

c

S 52 50 48 45 43 40 38 35 33 30 28 25 23 21 18 16 13 11 8 6 3 1 0

d

Eclipsis P

0 33 5 38 10 43 19 48 22 53 25 58 30 3 35 8 40 13 45 18 50 23 0

e

Minuta casus M

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 21

f

S 0 15 22 26 29 31 36 39 41 43 45 46 47 38 35 33 32 31 30 29 29 29 29

g

Minuta more M

0 36 39 16 45 15 20 5 52 5 44 12 30 11 14 24 5 9 27 58 41 31 20

h

S 0 0 0 0 0 0 0 0 0 0 0 0 0 10 14 17 19 20 21 22 23 23 23

i

0 0 0 0 0 0 0 0 0 0 0 0 0 27 23 5 7 39 49 39 11 28 30

Variants from B in C, E, P, and V: a12: C3; b3: C16; b8: CE52; b9: CE23; b11: CE36; b17: P37; c14: V22 corr. 21; d7: CE15 V15 corr.19; d10: C25; f4: CEPV 25; f6: CE 33; f14: CEPV48; g7: CEPV42; i12; V7 corr. 11; k1: V5; k14: C21; l10: EP38; l11: CV38 E37 P35; l12: V32 corr. 31; l15: CPEV22; m6: V57 corr. 56; m8: CEPV3; o14: CV55; p6: C41; p7: V52; r18: V14 corr. 18.

1 39

14 0

edi t i on of the tab les with comments

This feature, and a similar shift in the column for the lunar latitude in the sub-table for shortest distance, is shared by all manuscripts examined containing the tables of John of Lignères. Moreover, the latitude associated with the erroneous first entry, 9;11º, is 52;0’ in all manuscripts we examined containing John of Lignères’ set, in contrast to the latitude of 53;0’ in the Toledan Tables and the zij of al-Battānī. This table was not included in the first edition of the Parisian Alfonsine Tables.

lun ar eclip ses with luna r latitude a s a rg ument

Tabula eclipsis lune ad suam longitudinem propinquiorem Longitudo lune a nodo G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

j

M 12 11 11 10 9 9 8 8 7 7 6 5 5 4 4 3 3 2 2 1 0 0 0

k

Latitudo lune M

15 41 7 33 58 24 49 15 41 7 33 59 25 51 17 43 9 36 2 28 24 20 0

l

S 63 60 57 54 51 48 45 42 40 37 34 31 28 25 21 19 16 13 10 7 4 1 0

m

Eclipsis P

36 39 43 46 49 56 59 1 6 9 13 16 19 23 26 29 33 36 40 40 43 46 0

n

Minuta casus M

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 21

o

S 0 19 27 32 36 40 43 47 49 51 53 55 56 45 42 40 38 37 36 35 35 35 35

p

Minuta more M

0 9 20 35 53 42 59 13 25 40 39 25 29 47 15 2 27 20 37 56 35 23 20

q

S 0 0 0 0 0 0 0 0 0 0 0 0 0 12 17 20 22 24 26 27 27 28 28

r

0 0 0 0 0 0 0 0 0 0 0 0 0 35 16 32 38 18 2 12 52 13 16

Variants from B in C, E, P, and V: a12: C3; b3: C16; b8: CE52; b9: CE23; b11: CE36; b17: P37; c14: V22 corr. 21; d7: CE15 V15 corr.19; d10: C25; f4: CEPV 25; f6: CE 33; f14: CEPV48; g7: CEPV42; i12; V7 corr. 11; k1: V5; k14: C21; l10: EP38; l11: CV38 E37 P35; l12: V32 corr. 31; l15: CPEV22; m6: V57 corr. 56; m8: CEPV3; o14: CV55; p6: C41; p7: V52; r18: V14 corr. 18.

1 41

14 2

edi t i on of the tab les with comments

28. Eclipsed parts of the solar and lunar discs Tabula quantitatis tenebrarum in utraque eclipse and Tabula superficiei corporis amborum luminarium eclipsate (MS B, 76v) Titles in other manuscripts MS C, 86v: Tabula quantitatis tenebrarum eclipsis and Pars duodecima puncti equati ad solem et lunam MS E, 45r: Tabula quantitatis tenebrarum eclipsis solis et lune and Pars duodecima puncti equati ad solem et ad lunam MS P, 51r–v: Pars duodecima puncti equati ad solem et ad lunam and Tabula quantitatis tenebrarum eclipsis solis et lune MS V, 30v: Tabula quantitatis tenebrarum in utraque eclipsi and Pars duodecima puncti equati ad solem et lunam Description The eclipsed parts of the solar and the lunar discs are presented in two very similar sub-tables, which derive from two different traditions. They are both present in the Toledan Tables: a tradition perhaps dependent on al-Battānī and the common tradition. Both have three columns, one for the linear digits of the diameters of the luminaries, given as integer numbers from 1 to 12. The other two columns correspond the eclipsed part of the Sun and the Moon, in area digits, given in puncta and fractions of them, with a maximum of 12. For further details, the reader may consult Chabás and Goldstein 2012, pp. 174–175. As was the case with other tables for eclipses (Tables 24–27), John of Lignères included tables stemming from different traditions in his set. The presentation and the entries in both tables agree -save for scribal errors- with those in the Toledan Tables (Pedersen 2002, pp. 1450–1451). The table for the eclipsed parts is not mentioned in John of Lignères’ canons. Variant readings For this table, we also used MS B as the base manuscript. If anything, the variant readings in the two sub-tables show that some copyists were confused by the strong similarity between the various columns and took the entries in one to be used for another. For example, in the first sub-table, it is frequent to see that the puncta for the Sun at arguments from 9 to 11 (9;20, 10;40, and 11;50), as is the case in MSS B, E, and P, were miscopied from those for the Moon (9;10, 10;20, and 11;20). As noted by Pedersen, among the manuscripts he examined, only two share this oversight: Erfurt, Universitätsbibliothek, CA 8º 82 and Paris, Bibliothèque nationale de France, lat. 16655. This is exactly the case found in Table 1.

1 43

eclip sed parts of the sola r a nd luna r discs

John of Lignères. Eclipsed parts of the solar and lunar discs (28) Base : Basel F II 7, 76 v Tabula quantitatis tenebrarum in utraque eclipsi

Tabula superficiei corporis amborum luminarium eclipsate

Puncta Quantitas eclipsis Quantitas eclipsis dyametri solis lune solis et lune

Puncta Puncta superficiei Puncta superficiei dyametri solaris corporis lunaris corporis solis vel lune eclipsata

P

a

P

M

P

M

P

P

M

P

M

1

0

20

0

30

1

0

30

0

30

2

1

0

1

10

2

1

1

1

10

3

1

50

2

5

3

2

45

2

8

4

2

40

3

10

4

3

40

3

10

5

3

20

4

20

5

4

40

4

20

6

5

40

5

30

6

5

40

5

30

7

7

50

7

42

7

7

40

6

41

8

8

0

8

0

8

8

0

8

0

9

9

10

9

10

9

9

20

9

10

10

10

20

10

20

10

10

40

10

20

11

11

20

11

20

11

11

50

11

20

12

12

0

12

0

12

12

0

12

0

b

c

d

e

f

g

h

i

j

Variants from B in C, E, P, and V: b6: CV4; b7: CV5; b8: CV7; b9: CV8; b10: CV9; b11 : CV10; c9: CV20; c10: CV40; c11: CV50; d7: V6; e7: C42 V40 corr. 42; g1: P1; g2: P2; g3: P3; g4: P4; g5: P5; g6: P7; g7: P8; g8: P8; g9: P9; h1: CEPV20; h2: EP0; h3: E25; h7: EP50; j7: CPV45 .

14 4

edi t i on of the tab les with comments

29. Finding lunar latitude from the argument of latitude Tabula latitudinis lune in principio et in medio etiam et in fine eclipsis (MS B, 76v) Titles in other manuscripts MS C, 51v: Tabula latitudinis lune in principio et in medio et in fine eclipsis MS E, 45v: Tabula latitudinis lune in principio et in medio et in fine eclipsis MS P, 51v: Tabula latitudinis lune in principio et in medio et in fine eclipsis MS V, 31r: Tabula latitudinis lune in principio, in medio et in fine eclipsis etc Description The purpose of this table is to convert the argument of lunar latitude, given here for every degree from 0º to 13º, into lunar latitude, and it is valid in the vicinity of an eclipse. This table is an excerpt of Table 9, displaying the lunar latitude as a function of the argument of lunar latitude, with a maximum of 5;0º, for the range of values for which an eclipse is possible. This table is not mentioned in John of Lignères’ canons. Recomputation As was the case for Table 9, the entries can be recomputed by means of the following modern formula: β = arcsin (sin 5 sin ω), where β is the lunar latitude, ω is the argument of latitude, and 5º is the inclination of the lunar orb. Despite the low number of entries in this table –only thirteen– there are four differences with regard to Table 9. The entries in Table 29 for arguments 3º, 4º, 6º, and 13º are 0;15,44º, 0;20,52º, 0;31,13º, and 1;7,29º, respectively, whereas they are 0;15,40º, 0;20,53º, 0;31,19º, and 1;7,23º, respectively, in Table 9. We note, however, that for argument 13º, none of the two entries agree with computation, since arcsin (sin 5 · sin 13) = 1;7,24º. Hence, the entries displayed in Table 29 are not fully consistent with those given in Table 9, but they are with those found in the corresponding tables in the zij of al-Battānī (Nallino 1899–1907, 2: 78–83) and the Toledan Tables (Pedersen 2002, p. 1457), showing that John of Lignères used at least two different copies of the Toledan Tables for his Tables 9 and 29.

f in din g lun ar latitude from the a rg ument of latitude

Variant readings The base manuscript used for this table is MS B. The variant readings show that the entry for argument 6º (0;31,13º) agrees in all manuscripts consulted but disagrees in the seconds with the corresponding entry (0;31,19º) in Table 9 and in the Toledan Tables, indicating that this value might be a characteristic feature of John of Lignères’ Table 29. Among the many manuscripts containing the Toledan Tables examined by F. S. Pedersen, three present this specific variant in the seconds: Paris, Bibliothèque nationale de France, n.a.l. 3091 (thirteenth century), Montpellier, Bibliothèque Interuniversitaire Section Médecine, H323 (thirteenth and fourteenth centuries), and Cambridge, University Library, Ji.I.17 (fourteenth century). John of Lignères. Conversion from argument of latitude into latitude (29) Base : Basel F II 7, 76 v Tabula latitudinis lune in principio et in medio etiam et in fine eclipsis Argumentum latitudinis septentrionalis Signa

a

0 0 0 0 0 0 0 0 0 0 0 0 0 0

G

b

Signa 0 1 2 3 4 5 6 7 8 9 10 11 12 13

c

6 5 5 5 5 5 5 5 5 5 5 5 5 5

Argumentum latitudinis meridionalis

G

d

Signa 0 29 28 27 26 25 24 23 22 21 20 19 18 17

e

12 11 11 11 11 11 11 11 11 11 11 11 11 11

Variants from B in C, E, P, and V: k3: E34 P4 V40 corr. 44; k13: C13 EPV23.

G

f

Signa 0 29 28 27 26 25 24 23 22 21 20 19 18 17

g

6 6 6 6 6 6 6 6 6 6 6 6 6 6

Latitudo ipsius lune

G

h

G 0 1 2 3 4 5 6 7 8 9 10 11 12 13

i

M 0 0 0 0 0 0 0 0 0 0 0 0 1 1

j

0 5 10 15 20 26 31 36 41 46 52 57 2 7

S

k

0 13 27 44 52 7 13 31 42 52 1 9 16 29

1 45

146

edi t i on of the tab les with comments

30. Corrections Tabula equationis seu directionis portionum seu actacium (MS B, 74r) Titles in other manuscripts MS C, 84r: Tabula actacium sive directionis portionum MS E, 44v: Tabula equationis sive directionis portionum seu actacium MS P, 49r: Tabula equationis sive directionis portionum seu actacium MS V, 29r: Tabula equationis seu directionis proportionum. Tabula actacium Description The term ‘actatium’ or ‘attacium’, usually found in the title of this table, derives from ‘attacuim’, which is a transliteration from the Arabic term al-taqwīn, meaning ‘correction’ or ‘equation’, according to Nallino (1899–07, 2: 350). In this table, the argument is lunar anomaly, given at intervals of 6º. There are three columns for corrections. The first is for the minutes of proportion, and its entries are displayed in minutes and seconds, with a maximum of 60;0’. The same entries are found in an auxiliary table for eclipses in Almagest VI.8 (Toomer 1984, p. 308), to be used for the longest and shortest distances between the Moon and the Earth. In Ptolemy’s, table the argument is also displayed at intervals of 6º. This tradition was strictly followed by al-Battānī in his zij (Nallino 1899–1907, 2: 89). On the other hand, the Toledan Tables followed the tradition set by al-Khwārizmī in his table of proportions at intervals of 2º (Suter 1914, pp. 187–189). In this sense, John of Lignères’ entries for the minutes of proportion are in the tradition of al-Battānī. The two other columns are for correcting lunar parallax at intermediate positions of the Moon on its epicycle (circulus brevis) and its deferent (circulus egressus), and they derive from Ptolemy’s Handy Tables (Stahlman 1959, p. 257). We note that the column for the lunar epicycle was reproduced as a separate table (see Table 23). The three columns were presented in a single table in the zij of al-Battānī (Nallino 1899–1907, 2: 89), which was included as part of the Toledan Tables (Pedersen 2002, p. 1440). According to F. S. Pedersen, “our table is universally present in the Toledan Tables and must be native to the tradition” (Pedersen 2002, p. 1437). There is little doubt that this set was at the origin of John of Lignères’ table. This table is mentioned in Chapters 34, 35, and 38 in the canons beginning Priores astrologi (see Saby 1987, pp. 230, 243–246, and 255). Variant readings The base manuscript for this table is MS B. We note that in all manuscripts we examined, the entries in the column for the minutes of proportion for arguments 144º and 150º (53;42’ and 55;34’, respectively) differ from those given by Ptolemy and al-Battānī. However, they agree with those found in the Toledan Tables, indicating that John of Lignères also borrowed this table from the Toledan set.

corrections

John of Lignères. Corrections (30) Base : Basel F II, 74 r Tabula equationis seu directionis porcionum seu actacium Linee numeri G

a

G 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180

b

Longitudo porcionum M

360 354 348 342 336 330 324 318 312 306 300 294 288 282 276 270 264 258 252 246 240 234 228 222 216 210 204 198 192 186 180

c

Circulus Circulus brevis egressus

S 0 0 0 1 2 4 5 7 9 11 14 16 19 22 25 28 31 34 38 41 44 46 49 51 53 55 57 58 59 59 60

d

M 0 21 42 42 42 1 21 18 15 37 0 48 36 36 36 42 48 54 0 0 0 45 30 39 42 34 15 18 21 41 0

e

S 0 0 0 0 1 1 1 2 2 2 3 3 4 4 5 6 6 7 8 8 9 9 10 10 11 11 11 12 12 12 12

f

Variants from B in C, E, P, and V: c24: E3; d96: V42 corr. 48; d102: V48 corr. 53; d126: V41 corr. 45.

0 0 0 1 2 3 4 5 6 8 9 11 13 14 16 17 19 21 22 24 26 27 28 29 30 30 31 31 32 32 32

1 47

148

edi t i on of the tab les with comments

31. Tabula reflexionis tenebrarum Tabula reflexionis tenebrarum in utraque eclipsi scilicet solis et lune (MS B, 76r) Titles in other manuscripts MS C, 86v: Tabula reflexionis tenebrarum in utraque eclypsi MS E, 45r: Tabula reflexionis tenebrarum in utraque eclipsi MS P, 51r: Tabula reflexionis tenebrarum in utraque eclipsi MS V, 30v: Tabula reflexionis tenebrarum in utraque eclipsi Description The title of this table indicates that it is to be used for both eclipses: solar and lunar. More specifically, it is intended for computing the angle between the ecliptic and the great circle passing through the centres of the Sun and the Moon, or the shadow, during an eclipse. Although it is not explicitly stated, the argument is the number of digits of the solar and lunar discs, and the entries in the three columns are given in degrees. Besides the column for the argument, there are three other columns corresponding to the beginning and the end of a solar eclipse, the beginning and the end of a lunar eclipse, and the beginning and the end of totality in a lunar eclipse, here labelled c2, c3, and c4. This table goes back to Ptolemy and is found in Almagest VI.12 (Toomer 1984, p. 319), with entries given to minutes, and in the Handy Tables (Stahlman 1959, p. 256), with entries given only to degrees. The table in the Handy Tables was reproduced in the zij of al-Battānī (Nallino 1899–1907, 2: 89) and in the Toledan Tables (Pedersen 2002, p. 1455). John of Lignères followed this tradition and borrowed his table from the Toledan Tables. Table 31 is not mentioned in the canons beginning Priores astrologi. Recomputation As indicated by Pedersen (2002, p. 1454), the entries in columns c2, c3, and c4 can be recomputed by means of the following modern expressions: c2 = arcsin ((1 – d/6) · rs + rm) / (rs + rm)), c3 = arcsin ((1 – d/6) · rm + ru) / (rm + ru)), c4 = arcsin ((1 – d/6) · rm + ru) / (ru – rm)), where d is the variable found in column 1 and rs, rm, and ru are, respectively, the radii of the Sun, the Moon, and the shadow. The Ptolemaic values rs = 0;15,40º, rm = 0;16,40º, and ru = 0;43,20º (Almagest VI.11) yield good agreement. For example, for d = 12 digits, we find c2 = 1;46º (text: 2º), c3 = 26;23º (text: 26º), and c4 = 90º (text: 90º). Variant readings The base manuscript used here is again MS B. As the entry for c3 at argument for d = 15 is 19º in MS B, but 18º in all other manuscripts examined, as well as in the zij of al-Battānī and the Toledan Tables, we take it to be an isolated scribal error in MS B. Recomputation yields c3 = 17;47º.

ta bula reflexionis tenebra rum

John of Lignères. Tabula reflexionis tenebrarum (31) Base : Basel F II 7, 76r Tabula reflexionis tenebrarum in utraque eclipsi scilicet solis et lune Numerus punctorum eclipsis ex dyametro

Initium eclipsis solis et eius recessionis

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

90 67 57 49 43 37 31 26 21 16 11 6 2 0 0 0 0 0 0 0 0 0

a

b

Initium Puncta eclipsis eclipsis lune lune et finis et initium recessionis recessus

c

90 73 65 59 54 50 46 43 39 36 32 29 26 23 21 19 15 12 10 7 4 2

d

0 0 0 0 0 0 0 0 0 0 0 0 90 64 52 43 36 29 22 16 10 4

Variants from B in C, E, P, and V: b1: V74 corr.73; b2: V49 corr.57; b3: V43 corr.49; b4: V37 corr.43; b5: V31 corr.37; b6: V26 corr.31; b7: V21 corr.26; b8: V16 corr.21; b9: V11 corr.16; b10: V6 corr.11; b11: V2 corr.6; b12: V0 corr.2; c1: C60; c15: P18 V18 corr. 19.

1 49

15 0

edi t i on of the tab les with comments

32. Proportions at intervals of 2º Tabula proporcionis (MS B, 75v) Titles in other manuscripts MS C, 84v: Tabula proporcionis [added in another hand]: vere aucta per duos gradus MS E, 45v: Tabula proporcionis MS L, 45r: Tabula proporcionalis MS P, 51v: Tabula proporcionis Description This table complements Tables 24–27 for solar and lunar eclipses, and it is used to interpolate between greatest and shortest distances between the Moon and the Earth. The argument is lunar anomaly, and it is given at intervals of two degrees, from 2º to 180º. The entries are coefficients of interpolation, to be applied to the distance between the positions of the Moon and the Earth at apogee and perigee, and they are displayed in minutes and seconds. Table 32 derives from Almagest VI.8 (Toomer 1984, p. 308), where the entries are given at intervals of 6º, and it is found in many sets, including the zij of al-Battānī, also at 6º-intervals (Nallino 1899–1907, 2: 89) and the Toledan Tables, at 2º-intervals (Pedersen 2002, p. 1446). The table is mentioned in Chapters 35 and 38 in the canons beginning Priores astrologi (see Saby 1987, pp. 231–249 and pp. 253–258). Variant readings For this table, the base manuscript used is MS B. For the variant readings, we used MSS C, E, L, and P. All manuscripts examined agree with regard to some faulty entries found in the main tradition of the Toledan Tables as compared with previous analogous tables, indicating that John of Lignères’ table was taken from the Toledan Tables. There is an entry for which all manuscripts examined, MSS B, C, E, L, and P, agree, but simultaneously disagree with the Toledan Tables. For argument 46º, all five manuscripts have 8;31 minutes, whereas the entry found in the Toledan Tables is 8;32. The critical apparatus for the Toledan table provided by Pedersen indicates that in the very many manuscripts he surveyed, the entry for argument 46º is always 8;32, except in two of them, Erfurt, Universitätsbibliothek, CA 8º 82 and Paris, Bibliothèque nationale de France, lat. 16655, which give precisely 8;31. As was the case for Table 1 (sine) and Table 28 (eclipsed parts), these two manuscripts were also the only ones to share an error with the manuscripts containing the Tables of 1322. Another relevant variant reading occurs for argument 148º, where MS B gives 55;50 minutes, whereas the other manuscripts have 55;59. The Toledan Tables indeed read 59 seconds, but the number of minutes is correctly given as 54. Thus, in this case MS B presents two differences in relation to the Toledan Tables and to the original table by John of Lignères, but for different reasons. For the seconds, the number 59 can easily be confused with 50,

proportions at interva ls of 2º

1 51

John of Lignères. Proportions for correcting lunar parallax (32) Base : Basel F II 7, 75 v Tabula proportionis Gradus proportionis G 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60

G 358 356 354 352 350 348 346 344 342 340 338 336 334 332 330 328 326 324 322 320 318 316 314 312 310 308 306 304 302 300

Minuta proportionalia M

S 0 0 0 0 0 0 0 1 1 1 2 2 3 3 3 4 4 5 5 6 7 7 8 9 10 10 11 12 13 14

Gradus proportionis G

2 6 12 20 30 42 57 15 34 55 18 42 5 25 54 21 50 21 57 34 13 52 31 15 0 46 33 21 10 0

62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120

G 298 296 294 292 290 288 286 284 282 280 278 276 274 272 270 268 266 264 262 260 258 256 254 252 250 248 246 244 242 240

Minuta proportionalia M 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 38 39 40 41 42 43 44

S 55 45 41 38 36 36 36 36 36 36 36 36 36 40 42 44 46 48 50 52 54 56 58 0 0 0 0 0 0 0

Gradus proportionis G 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180

G 238 236 234 232 230 228 226 224 222 220 218 216 214 212 210 208 206 204 202 200 198 196 194 192 190 188 186 184 182 180

Minuta proportionalia M 44 45 46 47 48 49 50 51 52 52 53 54 55 55 56 57 57 58 58 59 59 59 59 59 59 59 59 59 59 60

S

a b c d e f g h i j k l Variants from B in C, E, L, and P: d36: C22; d40: C37; g98: C33; h62: CEL52; h86: CE38; h106; C50; k138: C51; k164: C58; l130: E47; l148: CL59; l170: C33; l174: C51; l172: CE43 L44.

0 0 7 46 57 30 19 6 50 32 11 48 24 50 34 8 42 15 43 8 31 50 7 21 53 12 11 56 58 0

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edi t i on of the tab les with comments

and for the minutes, all manuscripts consulted have some entries increased by one unit in the minutes with respect the Toledan Tables, between arguments 148º to 164º. This is a feature that also occurs in BnF, lat. 16655, and it is worth noting that this manuscript was owned by Peter of Limoges, and was available in Paris at the Collège de Sorbonne from 1306. Again, as was the case for Tables 1 and 28, this Parisian manuscript, or El Escorial, O.II.10, might have been at the origin of John of Lignères’ Tables of 1322.

List of manuscripts

(We have not listed the references to MSS B, C, E, P, and V in the commentaries, for they appear in almost all pages.) Basel Universitätsbibliothek, F II 7 (MS B)  11, 24–27, 29, 31, 36 Universitätsbibliothek, F II 10  19 Bernkastel-Kues Cusanusstiftsbibliothek, 210  24 Cusanusstiftsbibliothek, 211  21–22 Cusanusstiftsbibliothek, 212  24, 26–27 Cusanusstiftsbibliothek, 213  24

Bonn 24 Universitäts- und Landesbibliothek, S 498  Brussels Bibliothèque Royale, 926–40  13 Bibliothèque Royale, 1022–47  21 Bibliothèque Royale, 10117–260  18 Cologne Historisches Archiv der Stadt, W* 178 (MS K)  24, 26–27, 29, 37 Cracow Biblioteka Jagiellońska, 459  24 Biblioteka Jagiellońska, 546  24 Biblioteka Jagiellońska, 547  24 Biblioteka Jagiellońska, 549  24 Biblioteka Jagiellońska, 550  24 Biblioteka Jagiellońska, 551 (MS C)  12, 24–27, 29, 31 Biblioteka Jagiellońska, 553  24 Biblioteka Jagiellońska, 555  18 Biblioteka Jagiellońska, 557  18 Biblioteka Jagiellońska, 602  24 Biblioteka Jagiellońska, 610  24

15 4

l i s t of m an uscr ip ts

Biblioteka Jagiellońska, 613  24 Biblioteka Jagiellońska, 618  24 Biblioteka Jagiellońska, 715  21 El Escorial Biblioteca del Real Monasterio, O.II.10  44, 152 Erfurt Universitätsbibliothek, CA 2º 377 (MS E)  12, 24, 26-27, 29, 33 Universitätsbibliothek, CA 2º 384  24 Universitätsbibliothek, CA 2º 386  11 Universitätsbibliothek, CA 2º 388  20 Universitätsbibliothek, CA 2º 395  20–21 Universitätsbibliothek, CA 4º 355  18 Universitätsbibliothek, CA 4º 366  12, 18 Universitätsbibliothek, CA 8º 82  44, 142, 150 Florence Biblioteca Medicea Laurenziana, San Marco 185  24, 26–27 Frankfurt Stadt- und Universitätsbibliothek, Barth 134  20 Leipzig Universitätsbibliothek, 1484  24 London British Library, Egerton 889 (MS L)  24–27, 29, 32, 76, 80, 115–116 Lüneburg Ratsbücherei, Miscell. D 2º 11  24 Ratsbücherei, Miscell. D 2º 13  24 Madrid Biblioteca Nacional, 9288  16, 18 Biblioteca Nacional, 10002  24, 26–27, 102 Montpellier Bibliothèque Interuniversitaire Section Médecine, H 323  145 Moscow Russian State Library, F 68 N 450  24

list of ma nuscripts

Munich Bayerische Staatsbibliothek,  Clm 5640  24 Universitätsbibliothek, F 593  17 Oxford Bodleian Library, Can. Misc. 27 (MS Ox)  24–27, 29, 37, 80 Bodleian Library, Can. Misc. 499  24 Bodleian Library, Digby 168  18 Paris Bibliothèque nationale de France, lat. 7281  11–12, 16 Bibliothèque nationale de France, lat. 7282 (MS P)  24, 26–27 Bibliothèque nationale de France, lat. 7285  24 Bibliothèque nationale de France, lat. 7286C  24, 26–27, 34–35, 112 Bibliothèque nationale de France, lat. 7295  18 Bibliothèque nationale de France, lat. 7295A  10, 24, 26–27 Bibliothèque nationale de France, lat. 7290  10 Bibliothèque nationale de France, lat. 7329  11 Bibliothèque nationale de France, lat. 7378A  10 Bibliothèque nationale de France, lat. 16621  34 Bibliothèque nationale de France, lat. 16655  39, 44, 142, 150, 152 Bibliothèque nationale de France, Mélanges Colbert 60  17 Bibliothèque nationale de France, n.a.l. 3091  145 Philadelphia Free Library, Lewis E.3  17 University of Pennsylvania, LJS 174  24 Prague National Library, X A 23  24 Rome Biblioteca Casanatense, 653  12, 25 Osservatorio astronomico, III C 14  25 Rostock Universitätsbibliothek, math.-phys. 1  21, 25 Vatican Biblioteca Apostolica Vaticana, Ottob. 1826 (MS X)  25–27, 29, 38 Biblioteca Apostolica Vaticana, Pal. lat. 446  17 Biblioteca Apostolica Vaticana, Pal. lat. 1188  19 Biblioteca Apostolica Vaticana, Pal. lat. 1367  25 Biblioteca Apostolica Vaticana, Pal. lat. 1373  25

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156

l i s t of m an uscr ip ts

Biblioteca Apostolica Vaticana, Pal. lat. 1374 (MS V)  25–27, 29, 36, 101 Biblioteca Apostolica Vaticana, Pal. lat. 1375  18 Biblioteca Apostolica Vaticana, Pal. lat. 1376  25 Biblioteca Apostolica Vaticana, Pal. lat. 1390  20 Biblioteca Apostolica Vaticana, Pal. lat. 1412 (MS W)  23, 25–27, 29, 37 Biblioteca Apostolica Vaticana, Pal. lat. 1413  25 Biblioteca Apostolica Vaticana, Pal. lat. 1445  20 Biblioteca Apostolica Vaticana, Pal. lat. 1446  17 Biblioteca Apostolica Vaticana, Vat. lat. 4275  13 Biblioteca Apostolica Vaticana, Urb. lat. 1399  18, 23 Venice Biblioteca Nazionale Marciana, lat. VI, 29 (2526)  25 Wolfenbüttel Herzog August Bibliothek, 36.21 Aug.2º (2401)  25 Zürich Stadtbibliothek, 361  25 Zentralbibliothek, II.88  23

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