Jose Chabas Bergon, Bernard R. Goldstein Essays on Medieval Computational Astronomy

Essays on Medieval Computational Astronomy

Time, Astronomy, and Calendars texts and studies

Editors Charles Burnett Sacha Stern

Editorial Board Dáibhí Ó Cróinín – Benno van Dalen – Gad Freudenthal – Tony Grafton Leofranc Holford-Strevens – Bernard R. Goldstein – Alexander Jones Daryn Lehoux – Jörg Rüpke – Julio Samsó – Shlomo Sela – John Steele

volume 5

The titles published in this series are listed at brill.com/tac

Essays on Medieval Computational Astronomy By

José Chabás Bernard R. Goldstein

leiden | boston

Cover illustration: Detail of the star chart in the Alfonsine Libro de las estrellas de la ochava espera in Madrid, Real Academia de la Historia, ms 9/5707, f. 103. Courtesy Real Academia de la Historia ©. Library of Congress Cataloging-in-Publication Data Chabás, José, 1948Essays on Medieval computational astronomy / by José Chabás, Bernard R. Goldstein. pages cm. – (Time, astronomy, and calendars, ISSN 2211-632X ; volume 5) Includes bibliographical references and index. ISBN 978-90-04-28174-5 (hardback : alk. paper) – ISBN 978-90-04-28175-2 (e-books) 1. Astronomy, Medieval. 2. Astronomy–Tables. I. Goldstein, Bernard R. II. Title. III. Title: Medieval computational astronomy. QB26.C43 2015 521.01'51–dc23 2014041332

This publication has been typeset in the multilingual “Brill” typeface. With over 5,100 characters covering Latin, ipa, Greek, and Cyrillic, this typeface is especially suitable for use in the humanities. For more information, please see www.brill.com/brill-typeface. issn 2211-632X isbn 978-90-04-28174-5 (hardback) isbn 978-90-04-28175-2 (e-book) Copyright 2015 by Koninklijke Brill nv, Leiden, The Netherlands. Koninklijke Brill nv incorporates the imprints Brill, Brill Nijhoff and Hotei Publishing. All rights reserved. No part of this publication may be reproduced, translated, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission from the publisher. Authorization to photocopy items for internal or personal use is granted by Koninklijke Brill nv provided that the appropriate fees are paid directly to The Copyright Clearance Center, 222 Rosewood Drive, Suite 910, Danvers, ma 01923, usa. Fees are subject to change. This book is printed on acid-free paper.

Contents List of Figures vii Introduction

1

part 1 Conjunctions and Oppositions 1

Nicholaus de Heybech and His Table for Finding True Syzygy 9

2

Computational Astronomy: Five Centuries of Finding True Syzygy 40

3

Transmission of Computational Methods within the Alfonsine Corpus: The Case of the Tables of Nicholaus de Heybech 57

part 2 Planetary Motions 4

Ptolemy, Bianchini, and Copernicus: Tables for Planetary Latitudes 73

5

Displaced Tables in Latin: The Tables for the Seven Planets for 1340 99

6

Computing Planetary Positions: User-Friendliness and the Alfonsine Corpus 150

part 3 Sets of Tables 7

Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād 179

8

Early Alfonsine Astronomy in Paris: The Tables of John Vimond (1320) 227

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9

John of Murs’s Tables of 1321

308

10

Isaac Ibn al-Ḥadib and Flavius Mithridates: The Diffusion of an Iberian Astronomical Tradition in the Late Middle Ages 338

part 4 Other Tables 11

Ibn al-Kammād’s Star List 373

12

Astronomical Activity in Portugal in the Fourteenth Century Index 407

389

List of Figures 1.1 1.2 1.3 3.1 3.2 3.3 4.1 5.a 5.b 5.c 5.d 5.e 5.f 5.g 5.h 6.1 6.2 7.1 8.1 8.2 8.3 8.4 9.1 9.2 9.3

The mean conjunction (λ̄ s = λ̄ m) at time t takes place after the true conjunction at time t′ 11 The corresponding true conjunction (λ′s = λ′m) at time t′ takes place before the mean conjunction at time t′ 12 Graphical representations of the entries in the five columns of Nicholaus de Heybech’s Table at intervals of 6° 18 Facsimile of Nicholaus de Heybech’s table (excerpt): Vienna, Nationalbibliothek, ms 2440, f. 74v 59 Facsimile of tv 7 (excerpt): Madrid, Biblioteca Nacional, ms 3385, f. 106r 62 Facsimile of tv 8 (excerpt): Madrid, Biblioteca Nacional, ms 3385, f. 107r 63 The functions y =| cos x| and y = cos2 x in the interval (0°, 180°), where the cosine function represents c5(x) in formulas (1) and (2) 91 The solar equation displaced vertically 106 Ptolemy’s second lunar model 110 The model for Mars 120 Mars, equation of center and equatio porcionis 123 Mars, minutes of proportion 125 First station of Mars 130 Equation of anomaly near greatest distance for Mars 133 Venus, equation of anomaly as a function of the true argument of anomaly 140 Ptolemy’s model for Mars 152 The equation of center of Jupiter 170 The geometrical model underlying Ibn al-Kammād’s table for trepidation, as reconstructed 209 A geometric interpretation of Vimond’s tables for the mean motion for Venus 253 Vimond’s equation of center, col. 5, for Venus, Mars, Jupiter, and Saturn 262 Vimond’s equation of center, col. 5, for Mercury 262 Ptolemy’s model for the three superior planets and Venus 269 Correction for Saturn as a function of its mean argument of center 315 Correction for the Moon for a given “age” (20 days) 319 Correction for the Moon for a given value of mean lunar argument of anomaly (4,0°) 319

viii 12.1 12.2 12.3

list of figures Table of samt in Burgos (from Lib 1° to Ari 1°) 393 Calendaric matters (Madrid, ms 3349, f. 3v) 394 Domiciles according to al-Bīrūnī 401

Introduction During the Middle Ages and early modern times tables were a most successful and economical way to present mathematical procedures and astronomical models, facilitating computations based on them. One reason for depending on these techniques was the absence of modern mathematical notation to represent the algorithms that astronomers were using. Indeed, some major sets of tables were the direct result of the development of new astronomical approaches. In the second century ad Ptolemy composed the Almagest, a comprehensive handbook of astronomy and related mathematical procedures: in it he presented a set of observations on the basis of which he determined the parameters of his geometrical models for planetary motion. He then compiled tables which were also included in this treatise. At a later date he revised these tables, making them easier to use, in a work called the Handy Tables where, for example, Ptolemy displayed entries at 1°-intervals, rather than at 3°- and 6°intervals as he had done previously. Nicolaus Copernicus (1473–1543) provides a somewhat different example of this pattern. In 1543 his magnum opus, De revolutionibus, was published, in which he described a set of planetary models to replace those based on the Almagest, and included tables for computing planetary positions with these new models. In 1551 Erasmus Reinhold (1511– 1553) compiled the Prutenic Tables, based on the models of Copernicus’s De revolutionibus, presented in ways that facilitate computation. Most of the time, however, the compilation of complete sets of tables—or of individual tables— just reflected partial changes in the parameters and the models underlying a particular theory, or new methods to compute the positions of the celestial bodies without changing the underlying models. This is particularly true in the Middle Ages, when the Ptolemaic models were rarely challenged. Astronomical tables are a basic component of astronomy, although they are frequently neglected (or considered an unintelligible sequence of numbers) because, more often than not, it is no simple matter to establish the structure of such tables, and sometimes it is even difficult to identify the problem which a particular table addresses. Yet, to understand astronomical tables and to fully tackle the procedures used to compile them, various skills, both linguistic and mathematical, are required at the same time. The papers included in this volume give many examples where the meaning and purpose of such tables has been determined by careful analysis. As we have shown in our recent monograph, A Survey of European Astronomical Tables in the Late Middle Ages (Brill, 2012), astronomical tables are a primary source of historical information. Through their analysis it is possible to insert

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them and their compilers in an astronomical tradition, thus displaying unexpected links between authors. These analyses also allowed us to discover computational techniques, interpolation methods, and approximation procedures, as well as to identify changes in the standard parameters and the geometrical models. It should be stressed that an analyst of a medieval table should only appeal to modern mathematics very selectively and judiciously, given the risk of falling into anachronisms that may lead to incorrect results. In particular, modern mathematical functions are generally to be avoided although medieval mathematical procedures expressed verbally may legitimately be translated into algebraic notation. In recomputing a table consistency requires making use only of computational techniques, concepts, and strategies that were available at the time. For example, in computing entries in a table for finding the length of daylight on a given day, one needs to use trigonometrical tables that can be found in the same set of tables or in a previous set; indeed, this is how it was done by medieval astronomers. In other words, we have to evaluate medieval scholars according to criteria consistent with their own time, which may be different from those appropriate for other periods. The pioneering work on this subject goes back to the beginning of the twentieth century. In the years between 1899 and 1907 C.A. Nallino published an edition, translation, and commentary on the zij (i.e., a set of astronomical tables) of al-Battānī (ca. 900). This work was extended in 1956 by E.S. Kennedy in his survey of Islamic astronomical tables. Moreover, in 1962 O. Neugebauer provided an extensive commentary on the zij of al-Khwārizmī, originally composed in Baghdad in the ninth century that is only extant in a Latin version by Adelard of Bath (1075–1160), which was published by H. Suter in 1914. While the publication and analysis of Islamic astronomical tables have continued to this day, considerable attention has also been devoted to astronomical tables in Latin, Byzantine Greek, Hebrew, and the vernacular languages in Europe, including our own contributions to this field which have been built on those of our distinguished predecessors. As a result of these scholarly efforts a critical mass of studies is available which provides a good understanding of the overall framework for the transmission of astronomical ideas and procedures through tables in the period from Ptolemy to the early sixteenth century. The study of many sets of tables in a variety of languages still to be examined in European libraries (and others in different parts of the world) will surely lead to significant modifications of this framework in addition to an expansion of its contents. In particular, there are at least three major topics concerning medieval astronomy in the Latin West, where the analysis of tables is likely to play a critical role in enhancing our understanding of developments in this domain: the early stages of astronomy in Latin Europe, based on Arabic materials in the

introduction

3

Iberian Peninsula and the Maghreb; the importance of several sets of tables in Hebrew compiled in the Iberian Peninsula in the fourteenth and fifteenth centuries; and the shaping of the Parisian Alfonsine Tables in the central decades of the fourteenth century in Paris. The twelve articles presented in this collection have been jointly written by the two of us in a period covering about two decades. They are not the only ones, for our collaboration has extended to many other papers and monographs, for the most part devoted to astronomical tables. This four-handed collaboration, in the sense that all words and sentences as well as all numbers and computations have been written and checked by both of us, has proved to be fruitful and stimulating, an accomplishment that was achieved despite an ocean—a few thousand kilometers wide—separating us. The focus of attention in history of science is generally on scientists who produced new theories, such as Kepler and Newton. In the Middle Ages, however, the innovations were made in the context of a long tradition. The essays selected here address the major issues in medieval astronomy, and offer examples of several of the cleverest solutions in tabular form we have ever found. The first group of essays concerns syzygies; specifically, the determination of the time interval from mean to true syzygy, where mean syzygy refers to a conjunction or opposition of the mean Sun with the mean Moon, and true syzygy refers to a conjunction or opposition of the true Sun with the true Moon. The concept of “mean Sun” and “mean Moon” (or the mean positions of the Sun and the Moon) was used by Ptolemy in his Almagest, where the mean position of the Sun, the Moon, or a planet is the place it would reach on the ecliptic if its motion were uniform. This mean position is then corrected—using a table derived from a geometrical model—to yield its true position at a given time. Finding the time interval from mean to true syzygy is indeed a fundamental problem in computational astronomy, for it is the first step in computing the time and circumstances of a solar or lunar eclipse. Ptolemy had given an approximate solution by means of an iterative procedure without a table, and medieval astronomers devoted much effort to provide innovative, more precise, and more user-friendly solutions. One impressive approach to this problem is due to Nicholaus de Heybech (ca. 1400), who split the time from mean to true syzygy into two terms, one for the Sun and the other for the Moon, and managed to present his solution in the form of a single table in five columns that makes computation easy to do in a few steps, for it requires only addition, subtraction, and simple interpolation, rather than an iterative procedure with many steps. The second topic addressed here concerns the motion of the planets. The history of tables for computing planetary latitude (that is, the angular distance

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from a planet to the ecliptic) and using planetary equations (that is, the difference between the mean and true positions of a planet, where the Sun and the Moon are included among the planets) is surveyed in two papers, and many examples of their evolution are given. The other paper deals with the Tables for the Seven Planets for 1340, compiled by an anonymous author working within the framework of the Alfonsine Tables, who clearly had a profound understanding of planetary astronomy. He succeeded in simplifying the astronomer’s task by eliminating subtractions in the course of computing planetary positions. This was very helpful at a time when negative numbers were not yet available, for many complicated arithmetic rules were needed to express what can now be represented algebraically using negative numbers. To accomplish this, the compiler of these tables introduced what we call horizontal and vertical displacements (following the terminology of E.S. Kennedy), such that the results of the computations agree with those that depend on the standard Alfonsine Tables, although the intermediate steps are all different. In the third group we present different sets of tables by several authors, beginning with the zij by Ibn al-Kammād (Córdoba, ca. 1100). In the Iberian Peninsula astronomers were heirs to two traditions: a Greek tradition largely based on the works of Ptolemy, and an Indian tradition introduced into the Islamic world in the eighth and ninth centuries. The first is represented by the zij of al-Battānī, and the second by the zij of al-Khwārizmī. Ibn al-Kammād relied on both traditions, and we analyze each of his tables in terms of its structure and sources. We then turn to the sets of tables by John Vimond (ca. 1320) and John of Murs (ca. 1321), both of whom were active in Paris and engaged in recasting the tradition of Alfonsine astronomy, specifically, astronomical tables produced in Toledo in the thirteenth century under the patronage of Alfonso x of Castile. The Parisian version of the Alfonsine Tables is extant in many copies and it was the most influential set of tables in Latin Europe during more than two centuries. Finally, we review the tables compiled by Isaac Ibn al-Ḥadib (d. ca. 1426), who left Spain to live in Sicily in about 1396, and whose tables, which concentrate on the motions of the two luminaries, give innovative solutions. It is worth noting that the sets of tables by these four authors were all available at the time in Latin manuscripts, even though the work of Ibn al-Kammād was originally written in Arabic and that of Isaac Ibn al-Ḥadib in Hebrew. Finally, of the two papers listed under “Other Tables”, one is entirely devoted to a list of 30 stars that depends on Ptolemy’s star catalogue. It was compiled by Ibn al-Kammād, and it was surprisingly successful, for it is preserved in texts in Arabic, Latin, and Hebrew. The other paper describes the contents of a manuscript containing a mixture of tables of diverse origins and intended

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for different purposes, some of which are strictly astrological. The tables, none of them for planetary motion, reflect astronomical activity in Portugal during the fourteenth century. Among the tables in this manuscript is one for the daily “lunar progress” in an “astrological month” and another is for “astrological months”, where 13 such months constitute a tropical year; this astrological “Moon” (a fictitious celestial body) advances 390° in each astrological month. We have found that these astrological tables appear in many medieval and early modern sets of astronomical tables. For an explanation and historical account of these astrological tables, see our article, “Planetary velocities and the astrological month”, Journal for the History of Astronomy, 44 (2013), 465–478. Despite their great variety, underlying the tables reviewed in this collection of essays are models for the Sun, the Moon, and the planets, which follow essentially the Ptolemaic tradition in an ingenious and original way. Some of them also provide elegant and intelligent solutions to standard astronomical problems, within a general tendency to offer the practitioner user-friendly tables that facilitate his task of computing planetary positions and celestial events. We thank the editors and publishers of the journals in which the chapters of this book first appeared as independent essays for graciously granting us permission to reprint them: bibliographic data for the original publication are given in the first footnote of each chapter. Since it was decided to reset type for these essays, we have taken the opportunity to make some minor changes in them.

part 1 Conjunctions and Oppositions

∵

chapter 1

Nicholaus de Heybech and His Table for Finding True Syzygy* Introduction The calculation of eclipses was a major task for medieval astronomers, and the first step in this procedure was the determination of the time from mean to true syzygy (where syzygy means either the conjunction or the opposition of the Sun with the Moon). The basic approach given in Ptolemy’s Almagest was refined by subsequent astronomers in various ways. In the Latin West of the late Middle Ages the corpus of Alfonsine astronomy held a dominant position. Of special interest in this tradition is the solution to this problem in Chap. 22 of John of Saxony’s canons for the Alfonsine Tables (ca. 1327)1 as well as the solution attributed to Nicholaus de Heybech of Erfurt (ca. 1400).2 The text and table of Nicholaus de Heybech (see Appendix 1) have not been discussed previously.3 lt will be shown that the method of Nicholaus de Heybech is much

* Historia Mathematica 19 (1992), 265–289. 1 Editio princeps, Ratdolt [1483]; cf. Poulle [1984, 17]. 2 In a short note, Thorndike [1948] collected what little is known about Nicholaus de Heybech of Erfurt. His principal text is the canon and table to find the time of true conjunction and opposition of the Sun and the Moon. The date of composition of this text is uncertain but some of the manuscripts of it were copied in the 1440s. Thorndike adds that most of the manuscripts are of the fifteenth century but one manuscript, dated 1394, was in the Library of Grenville Kane, Tuxedo Park, New York (the present location of that manuscript is in Princeton). In addition to listing manuscripts of the text which will be discussed below, Thorndike reports that ms Köln W* 178 (fol. 29v) contains a table of mean conjunctions for the years from 1384 to 1504 computed for the meridian of Paris and ascribed to a certain “Nicholaus,” suggesting that it is the same author. In the manuscript we find the author’s name in the phrase given in the heading as “composita per Nycolaum de Er*” where the asterisk represents here a balloon-shaped symbol. However, it is by no means certain that this author is to be identified with our Nicholaus. Moreover, Hartmann [1919, 12–13] refers to a certain “Nicholaus de Heybech” who was registered as a student at Erfurt University in 1421, and he suggests that this might be our Nicholaus. According to Thorndike, in 1392 Nicholaus de Heybech completed a copy of Gerard of Cremona’s Theorica Planetarum, preserved in ms Cues 213. From this meager information we conclude that Nicholaus de Heybech was active circa 1400. 3 A translation of the text of Nicholaus de Heybech appears in Appendix ii.

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simpler than that of John of Saxony. In the same period another way to solve this problem by means of tables, introduced by Levi ben Gerson (d. 1344) and followed by Jacob ben David Bonjorn (fl. ca. 1361), has recently received extensive treatment.4 Levi ben Gerson’s method requires the use of a set of four tables and calls for many more computations than the method of Nicholaus de Heybech. The Alfonsine tradition is identified with Alfonso x, king of Castile (Spain), who reigned from 1252 to 1284. He sponsored much scientific activity, and a number of astronomical works were written in, or translated into, Castilian at that time. The Alfonsine Tables are a special case because they are only extant in a Latin version produced in Paris in the 1320s, and derivatives from it. lt has even been argued that there never was a Spanish version prior to the Parisian one [see Poulle 1988]. The manuscript tradition of these tables is very complex and has not been adequately examined. For this reason, it has been customary to identify the Alfonsine Tables with those published by E. Ratdolt in 1483 despite the difficulty in deciding which tables were intended to be so designated by the Parisian group in the 1320s, let alone by the astronomers serving under Alfonso x. One of the characteristic features of the Parisian version (and the edition of 1483) is the division of the circle into “physical” signs of 60° rather than “natural” signs of 30° that were prevalent in ancient and medieval astronomy [Poulle 1988, 100]. We refer to the entire manuscript tradition associated with these tables as the corpus of Alfonsine astronomy. We shall first discuss Ptolemy’s method and then the method presented by John of Saxony, neither of which is reduced to specific tables for this purpose. Then we turn to the method of Nicholaus de Heybech, who successfully presented his solution to this problem in a single table. We begin by introducing some definitions and notation. Let λs, λ̄ s, λm, and λ̄ m be the true and mean longitudes of the Sun and the Moon, respectively, at a mean syzygy that occurs at a given time t. The solar correction (cs) and the lunar correction (cm) are defined as cs(κ̄) = λs – λ̄ s and cm(ᾱ) = λm – λ̄ m

4 On Levi ben Gerson’s method, see Goldstein [1974, 136–144, 229–241]; on Jacob ben David Bonjorn see Chabás [1989, 26–39, and 1991].

nicholaus de heybech and his table for finding true syzygy

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figure 1.1 The mean conjunction (λ̄ s = λ̄ m) at time t takes place after the true conjunction at time t′ (see Figure 1.2)

where κ̄ is the mean solar anomaly and ᾱ is the mean lunar anomaly. Note that for 0° ≤ κ̄ ≤ 180°, cs ≤ 0°; and that in the simple lunar model, for 0° ≤ ᾱ ≤ 180°, cm ≤ 0°. However, in Ptolemy’s complete lunar model cm depends on the mean elongation η̄ = λ̄ m – λ̄ s, as well as on ᾱ. At mean syzygy 2η̄ = 0°, whereas at true syzygy 2η = 0, where η = λm – λs.5 We add a prime symbol to all variables related to the true syzygy taking place at a time t′, corresponding to a mean syzygy that takes place at time t. Let Δt = t′ – t be the interval between the two events. When the true syzygy comes after mean syzygy, Δt is positive and λm < λs. Figure 1.1 illustrates a mean conjunction where λm > λs, and Δt is negative; Figure 1.2 illustrates the corresponding true conjunction. From the definition of Δt it follows that (1)

5 For a discussion of Ptolemy’s solar and lunar models see, for example, Pedersen [1974, 122– 202] and Neugebauer [1969, 191–198].

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figure 1.2 The corresponding true conjunction (λ′ s = λ′ m) at time t′ takes place before the mean conjunction at time t′ (shown in Figure 1.1)

where the functions vm(t) and vs(t) are the time dependent velocities in longitude of the Moon and the Sun. Since λ̄ m = λ̄ s at mean conjunction, Eq. (1) can be written as (2) The difficulties in assigning proper values to these functions in the calculation of Δt gave rise to a variety of approaches in the determination of the times of true syzygies. The functions vm(t) and vs(t) are not tabulated in Ptolemy’s Almagest. For the calculation of Δt Ptolemy depended on the approximation vs(t)/vm(t) ≈ 1/13.6 Then Eq. (1) becomes (3)

6 Almagest vi.4; Toomer [1984, 281]; Neugebauer [1975, 122]; Pedersen [1974, 221–226].

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According to Ptolemy the next step is to compute vm(t) as a function of the lunar anomaly by means of a relation which, expressed in modern notation, is (4) vm(t) = 0;32,56 + 0;32,40 Δc, where 0;32,56 °/h is the hourly mean lunar velocity in longitude, 0;32,40 °/h is the hourly mean lunar velocity in anomaly, and Δc = cm(ᾱ) – cm(ᾱ – 1) is the difference in the lunar correction corresponding to one degree of mean anomaly at the time of mean syzygy. However, since Ptolemy’s table for the lunar correction (Almagest iv.10) is arranged for intervals in the argument of 6° in the upper part and 3° in the lower part, the previous expression is stated as Δc = (cm(ᾱ) – cm(ᾱ – d))/d, where d is 6° or 3°, as appropriate. At time t + Δt the true positions of the Sun and the Moon are to be computed anew. If there is still a sensible difference between them, this procedure is to be iterated.

The Alfonsine Method according to the Canons of John of Saxony7 If the time t at which a mean syzygy takes place is known, the values for all the required quantities may be found and the time difference Δt from mean to true syzygy may be derived by means of a successive approximation procedure involving two principal steps. 1. First, it is necessary to compute a time interval τ which yields the time of true syzygy, correct to the nearest hour. To do so, John of Saxony appeals to Ptolemy’s complete lunar model in which Ptolemy introduces an equation of center q, a function of η̄ . However, since q(η̄ ) = 0° at mean syzygy, the canons introduce an “equated” lunar anomaly α in order to determine a consistent value for vm. The relation proposed is (5) where the mean lunar anomaly ᾱ and the elongation η are to be replaced by their values at mean syzygy. In the text, the elongation η = λm – λs, is called “longitudo lune” when η > 0° and “longitudo solis” when η < 0°. The text offers no justification for Eq. (5) but clearly its purpose is to account for the change

7 Poulle [1984, 80–87].

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in lunar velocity in the time interval from mean syzygy to true syzygy. The lunar velocity will now be treated as a function of α rather than of ᾱ, where α corresponds to the midpoint of the time interval from mean syzygy to true syzygy. To derive Eq. (5), let t be half the time interval from mean syzygy to true syzygy and α = ᾱ – v̄α · t, where v̄α, the hourly mean lunar velocity in anomaly, is 0;32,40 °/h. In Eq. (5), the coefficient 13/24 (= 0;32,30) seems to represent an approximation of v̄α. We further assume that the velocity in elongation from mean syzygy to true syzygy can be approximated by its mean value, v̄η = 0;30,29 °/h. It follows that the time, 2t, from mean syzygy to true syzygy equals η/½, or 2η; hence t = η. Note that in Eq. (5) η is therefore measured in hours. The value of vm corresponding to α is intended to represent the velocity of the Moon at the midpoint of the time interval from mean syzygy to true syzygy. The text assumes that the velocity of the Sun remains the same in this time interval. According to the canons, the time interval τ is then computed by means of the relation (6) where δ = ±0;0,1 < (|Int(η)| – 1) and where Int(η) is the integer part of the elongation at mean syzygy (in degrees). When δ = 0°, Eq. (6) has the same structure as Eq. (1); that is, the difference in lunar and solar longitudes is divided by the difference in their velocities. Note, however, that in Eq. (6) lunar velocity is a function of the equated anomaly defined in Eq. (5) rather than the mean anomaly. In the text the difference between the lunar and solar velocities is called “superatio” and the denominator of Eq. (6) is called “superatio equata.” The solar velocity in the time from mean conjunction to true conjunction is essentially constant; therefore, the term δ is intended to modify the lunar velocity which changes noticeably in that same time interval. This term serves the same function in the Toledan Tables and Azarquiel’s almanach, and John of Saxony apparently followed the same tradition.8 This adjustment to lunar

8 Toomer [1968, 85] and Millás [1943–1950, 233] display a table of corrections to the hourly lunar velocity corresponding to δ; the argument varies from 1° to 7° and the entries from 0″to 6″. In al­Battānī’s zīj a similar table is found [Nallino 1907, 88], but here the argument varies from 1° to 7° and the corresponding entries from 1″to 7″. The tabulated function is thus 0;0,1 |Int(η)|. It would seem that for al-Battānī this is a correction to the hourly velocity of the Moon to account for the change in lunar velocity in the time interval from mean to true

nicholaus de heybech and his table for finding true syzygy

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velocity is no longer needed because Eq. (5) satisfies the same purpose. Indeed, δ has no sensible effect on the value of τ because its maximum of 0;0,6° is very small compared with the least difference in lunar and solar velocities (0;27,45° according to the Toledan Tables, or 0;27,47° according to the edition of 1483 of the Alfonsine Tables).9 The text presents rules for the algebraic sign of δ, but they are incomplete. The text just considers the case where the longitude of the Moon is greater than that of the Sun, i.e., η > 0°; this implies that true conjunction precedes mean conjunction (Δt < 0), and that the lunar anomaly in that interval is always smaller than its value at mean conjunction. Under this circumstance, the text gives the correct rule for the algebraic sign of δ: if α ≤ 180°, then δ ≥ 0°, and vice versa. Unfortunately, the text uses the expression “longitudo”10 instead of “longitudo lune” (η > 0°) and omits any discussion of the case where η < 0° (“longitudo solis”). 2. The second step requires the computation of a time interval τ* to yield the time, correct to a minute of an hour, for the true syzygy: τ* + τ = Δt. In order to do so, the canons propose a “differential” method that involves calculating and comparing all the quantities at two different times 24 min apart (i.e., 1/60 of a day): t + τ and t + τ + 0;1 d. For a given time dependent variable x, let dx = x(t + τ + 0;1) – x(t + τ) be its variation in a minute (i.e., a sixtieth) of a day. According to the canons, the time interval τ* is determined as (7)

9

10

syzygy. Al-Battānī’s maximum argument of 7° reflects the maximum value of η which for him is 7;0,10°. The maximum entry in this table of 7″is an “average” (where the range is from 0″to 11″) for the effect on lunar velocity in degrees per hour corresponding to a change in lunar anomaly of about 3¾° (corresponding to half the time for the Moon to reach the point of true conjunction). Toomer [1968, 82]; Ratdolt [1483, fol. g6r–g7r]. In his canons, John of Saxony mentions a table by John of Lignères for solar and lunar velocity, but it is not clear which set of entries is intended [cf. Poulle 1984, 82]. Ratdolt [1483, fol. b1r, line 5]; cf. Poulle [1984, 82, line 57].

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where η* is the elongation at time t + τ. The denominator of Eq. (7) can also be written as dλm – dλs = λm(t + τ + 0;1) – λs(t + τ + 0;1) – λm(t + τ) + λs(t + τ) = η(t + τ + 0;1) – η(t + τ) = dη. Since the mean elongation is no longer 0°, the lunar longitude has to be computed according to Ptolemy’s complete lunar model rather than the simple lunar model, both of which are represented in the Alfonsine Tables.11 The variation in elongation in a minute of a day, dη, is very nearly the “instantaneous” velocity in elongation at the time of true syzygy. Analogously, dλm represents the “instantaneous” velocity in lunar longitude in a minute of a day. To illustrate the method described in the canons to the Alfonsine Tables by John of Saxony, we show in Table i some basic magnitudes (in degrees) involved in the computation of the true conjunction corresponding to the mean conjunction of 20 July 1327, 3;58,10 h after noon.12 True conjunction is estimated to take place at time t + τ + τ* = 3;58,10 h – 8;43,30 h + 0;9,49 h = 19;24,29 h (19 July 1327) in Toledo.

A Nicholaus among All the Johns The Latin tradition of the Alfonsine Tables was mainly developed by the group of astronomers who may be called “the Johns” (i.e., John of Murs, John of Lignères, and John of Saxony), and also by such lesser known figures as John of Genoa and John of Montfort who computed tables for lunar velocities.13

11

12

13

Ptolemy’s table for his complete lunar model appears in Almagest v.8 [Toomer 1984, 238]. The corresponding table (with modified parameters) appears in the Alfonsine Tables [Ratdolt 1483, fol. e4r–e6v; cf. Poulle 1984, 148–153]. Poulle [1984, 214–218] presents a worked example according to John of Saxony’s canons for this conjunction. We have recomputed all the magnitudes involved (using the tables, as he did, in the 1483 edition of the Alfonsine Tables) and our results differ somewhat from his. For example, we compute the equation for the “centrum lune” according to John of Saxony and the appropriate tables in [Poulle 1984] at time t + τ as –1;18,46° whereas he found it to be –1;13,47°. Despite the discrepancies, the resulting time of true conjunction is essentially the same in both computations. Nothing is known about John of Montfort except that, ca. 1332, he produced a table for

nicholaus de heybech and his table for finding true syzygy table i

A worked example for the mean conjunction of 20 July 1327 according to the instructions by John of Saxony

t = 3;58,10 h t+τ (mean conj.) (τ = –8;43,30 h) λ̄ s λs κ̄ λ̄ m λm ᾱ α 2η̄ η

17

2, 4;53,30 2, 3;40,35 35;25, 4 2, 4;53,30 2, 8;26,14 3,42;26, 7 3,42;26, 7 0 4;45,39

2, 4;32, 0 2, 3;19,45 35; 3,34 2, 0; 6, 5 2, 3;13,15 3,37;41, 8 3,36;22,22 5,51; 8,10 –0; 5,29

t + τ + τ* t + τ + 0;1 d (τ* = +0;9,49 h) 2, 4;32,59 2, 3;20,42 35; 4,34 2, 0;19,16 2, 3;28,37 3,37;54,12 3,36;39, 6 5,51;32,33 0; 7,55

2, 4;32,24,13 2, 3;20, 8, 2 35; 3,57 2, 0;11,28,50 2, 3;20, 7,57 3,37;46,28,42 3,36;29,12, 2 5,51;18, 8,55 –0; 0, 0, 5

Despite the differences of their tables in detail, all the Johns required two kinds of tables for the determination of the time of true syzygy: tables of correction and tables of velocity for each of the luminaries. However, some manuscripts within the Alfonsine corpus contain a single table to solve this problem where all entries are given in units of time. The author of that table, according to its heading, is Nicholaus de Heybech of Erfurt, and this table is usually accompanied by a short canon explaining its use. This canon adds that, after the time of true syzygy is determined, a correction is required to take into account the equation of time.14 Both the canon and the table survive in

14

solar and lunar velocities. John of Genoa was the author of a set of canons and tables for eclipses (also dated 1332), and a computation for the solar eclipse of March 1337 according to the Alfonsine Tables [cf. Thorndike and Kibre 1963, Cols. 51, 61, and 1690]. The tables for lunar velocity by John of Montfort and by John of Genoa are analyzed in Goldstein [1992]. Nicholaus de Heybech refers to a table for the equation of time in the canon (see Appendix i [18]). His allusion to a table for the equation of time is simply copied from John of Saxony’s canon (Chap. 22). This table appears in the 1483 edition of the Alfonsine Tables (fol. k1r–k2r) and has the heading “Tabula elevationum signorum in circulo directo.” It is arranged for signs of 30°, and each entry for the equation of time lies under each sign, as it is described both by John of Saxony and Nicholaus de Heybech in Appendix 1: [18]. Note that this table was taken from al-Battānī [cf. Nallino 1907, 61–64] and already appeared in the Toledan Tables [cf. Toomer 1968, 34–35], and that Poulle [1984, 6, 222] excluded it from the Alfonsine Tables.

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figure 1.3 Graphical representations of the entries in the five columns of Nicholaus de Heybech’s Table at intervals of 6°

several Latin manuscripts, often together with the Alfonsine Tables and Jacob ben David Bonjorn’s tables for syzygies. Nicholaus de Heybech’s canon and table are transcribed in Appendix i, based on the following manuscripts: B D P P′

Basel, Universitätsbibliothek f.ii.7, fol. 36r–37v (table and canon); Dijon, bm 447, fol. 62r–v (canon); Paris, bn Lat. 7287, fol. 72r–73r (table), fol. 86va–87ra (canon); Paris, bn Lat. 7290a, fol. 103r–104r (table).

Thorndike and Kibre [1963, 1390, 1478, 1562] listed the first three as well as three other manuscripts: Bern 454, Cues 211, and Vat. Pal. 1376. Most of the information seems to have been taken from E. Zinner [1925]. This table seems to have had a wide diffusion, for it is extant in quite a number of manuscripts in addition to those already mentioned: Cracow, Jagiellonian Library, mss 609 (table and canon), 610 (table and canon), 613 (canon), 1852, and 1865; Princeton, Library of Grenville Kane, ms 51; Vienna, Nationalbibliothek 2440 (table and canon) and 5227 (table and canon). The argument of the table is arranged at 1° intervals from 0° to 180° using signs of 60°. Column i has the heading “equatio solis” and the entries are displayed in hours and minutes; column ii gives the “diversitas equationis solis” in hours and minutes; column iii lists the “minuta proportionalia”; column iv displays the “equatio lune” in hours and minutes; and column v gives the “diversitas equationis lune” in minutes of an hour. Figure 1.3 illustrates the five columns at 6° intervals.

nicholaus de heybech and his table for finding true syzygy

19

The structure of the table suggests that the time interval between mean and true syzygy given by Eq. (2) is here distributed into two terms, each of which accounts for the role of one luminary: (8) The calculation of the first, or solar term (Δts), involves columns i, ii, and iii, and that of the second, or lunar term (Δtm), involves columns iii, iv, and v. The strategy consists in treating each term separately, and within each term computing a set of minimum and maximum values and then to use an interpolation scheme for intermediate values. The solar term can be approximated as (9) where v̄s is the mean solar velocity (°/h) and c3(ᾱ) is a coefficient of interpolation ranging from 0 to 1, depending on the mean lunar anomaly, ᾱ, and tabulated in col. iii in minutes. When ᾱ = 0°, the Moon reaches its minimum velocity, min(vm), and c3(ᾱ) = 0. Hence Eq. (9) reduces to (10) The values computed by means of Eq. (10) correspond to those tabulated in Col. i. Analogously, when ᾱ = 180°, the Moon reaches its maximum velocity, max(vm), and c3(ᾱ) = 1. In this case, Eq. (9) reduces to (11) The values computed by means of the term in brackets in Eq. (9) correspond to those tabulated in Col. ii. Let c1 represent an entry in Col. i and c2 an entry in Col. ii of Nicholaus de Heybech’s table: c1 and c2 are both functions of the mean solar anomaly, κ̄ . Then (12) Δts = c1(κ̄) – c2(κ̄) · c3(ᾱ). Similarly, the lunar term in Eq. (8) can be approximated as (13)

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where c3(κ̄) is a coefficient of interpolation ranging from 0 to 1, depending on the mean solar anomaly, κ̄. Column iii serves as the interpolation scheme for both the Sun and the Moon. When κ̄ = 0°, the Sun is at its minimum velocity, min(vs), and c3(κ̄) = 0. Hence Eq. (13) reduces to (14) The values computed by means of Eq. (14) correspond to those tabulated in Col. iv. Analogously, when κ̄ = 180°, the Sun is at its maximum velocity, max(vs), and κ̄ = 1. In this case, Eq. (13) reduces to (15) The values computed by means of the term in brackets in Eq. (13) correspond to those tabulated in Col. v. Let c4 represent an entry in Col. iv and c5 an entry in Col. v of Nicholaus de Heybech’s table: c4 and c5 are both functions of the mean lunar anomaly ᾱ. Then (16) Δtm = c4(ᾱ) – c5(ᾱ) · c3(κ̄). Finally, from Eqs. (12) and (16) we find that the time from mean to true syzygy is (17) In this equation the algebraic signs of c1(κ̄) and c2(κ̄) are the same as the corresponding sign of c3(κ̄), and those of c4(ᾱ) and c5(ᾱ) are the same as that of cm(ᾱ). In the canon by Nicholaus de Heybech we are first told (sentences [5]–[8]) to compute the six quantities in Eq. (17); note that the sign convention for c4(ᾱ) and c5(ᾱ) differs from the one we have used. The rule for computing the second term is found in sentence [9], and sentence [10] tells us to compute the difference between the first and the second terms, that is, the solar term, Δts. In sentence [11] we are told to compute the fourth term, and sentence [12] tells us to compute the lunar term, that is, Δtm (according to our convention). In sentences [13] and [14] we are given the rules for adding the solar and lunar terms. For the example given above (the mean conjunction of 20 July 1327), we find c1 = –2;41 h c4 = +6;24 h

c2 = –0;35 h c5 = +0;02 h

c3(κ̄) = 0;06, c3(ᾱ) = 0;53,

nicholaus de heybech and his table for finding true syzygy

21

using as arguments κ̄ = 35;25,4° and ᾱ = 3,42;26,7°. The resulting time interval Δt is –8;34 h, whereas the value according to John of Saxony’s method is –8;33,41 h. It is noteworthy that with the table of Nicholaus de Heybech essentially the same result is obtained with much less work; we also note that his entire procedure is in units of time restricted to two sexagesimal places. In contrast, John of Saxony’s procedure involves the computation of many intermediate quantities, most of which are not in units of time, and he does not specify the degree of precision of these quantities needed to assure the minutes of time in the ultimate result. Equation (17) incorporates the instructions in the canon of Nicholaus de Heybech’s table, and we have seen that, at least for the example presented above, the entries are reasonable. As was frequently the custom among medieval astronomers, Nicholaus de Heybech does not describe his method for computing the entries in his table. We now present our reconstruction of his method for computing the table. To recompute the entries in column i, we need to fix a set of values in Eq. (10). For the solar correction we use the values in the 1483 edition of the Alfonsine Tables; we consider min(vm) = 0;29,37,13 °/h and v̄s = 0;2,27,51 °/h. The first parameter is the minimum lunar velocity found in John of Genoa’s table, arranged at intervals of 6° and given to 3 sexagesimal places.15 The value for the mean solar velocity v̄s is derived from the table for mean solar motion in the editio princeps of the Alfonsine Tables. To recompute the entries in Col. ii, we also have to fix the value for the maximum lunar velocity in Eq. (11): max(vm) = 0;36,58,54 °/h, which is found in the same table by John de Genoa mentioned above.16 The values in Col. ii are computed as the difference between those in Col. i and those found by means of Eq. (11) with the same argument. These parameters yield the entries in Col. i and Col. ii quite closely (see Table ii). It should be stressed that, restricting our attention to attested parameters, we obtain best agreement with the entries in column ii using John of Genoa’s velocity table which contains an isolated

15

16

The Alfonsine corpus has many variant tables for lunar velocity (see Goldstein [1992]) and there is no reason to assume that Nicholaus de Heybech would have used the version in the editio princeps of the Alfonsine Tables in which the minimum value is 0;30,21°/h. We use this value for the lunar velocity in John of Genoa’s table which appears in many manuscripts, and hence it is textually secure. Nevertheless, it may have been miscomputed according to an analysis of the entire table: see Goldstein [1992]. Other values for the maximum lunar velocity in the Alfonsine corpus are: 0;36,4°/h [cf. Goldstein 1980], 0;36,25°/h [Ratdolt 1483, fol. g7r] and 0;36,53°/h, ascribed to John of Lignères [cf. Goldstein 1992].

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error for the maximum entry, 0;36,58,54 °/h (instead of 0;36,53,20 °/h which is consistent with the rest of the table). The fact that this value is so much greater than any other strongly supports the claim that Nicholaus de Heybech indeed depended on this table by John of Genoa. As we shall see, the entries in column iv in Nicholaus de Heybech’s table also agree with recomputation based on this table by John of Genoa. The recomputation of columns i and ii of Nicholaus de Heybech’s table is only displayed here for values of the argument in multiples of 6°, for it is likely that he derived the other entries by distributing them uniformly between the values calculated at 6° intervals. For instance, in Col. i after finding 28 min for κ̄ = 6°, one way to proceed would be to distribute 28 into six parts according to the sequence 5–5–5–5–4–4 (leading to the values 5, 10, 15, 20, 24, and 28, that actually appear in Nicholaus de Heybech’s table). After finding 57 min for κ̄ = 12°, one can distribute 57 – 28 = 29 into 6 parts according to the sequence 5–5–5–5–5–4 (leading to the values in Nicholaus de Heybech’s table: 33, 38, 43, 48, 53, and 57). These sequences generate smooth curves within each interval of 6°, but they are no longer smooth when considered as a whole. The results are shown in Table ii, where c is the recomputed value and t–c is its difference from the value given in the text. We now argue that the coefficient c3, tabulated in column iii (“minuta proportionalia”), has been recomputed according to the formula (18) where, in a simple eccentric model, d = 60 + e (the distance of the luminary at apogee), d = 60 – e (its distance at perigee), d(κ) is its distance for a true anomaly, κ, and e is the eccentricity.17 In a simple eccentric model, this interpolation coefficient is a function of κ and e, but, as is shown in Table iii, c3 does not depend strongly on the eccentricity, for similar results are obtained using 2;16 (the solar eccentricity corresponding to the maximum solar correction in the Alfonsine Tables: 2;10°) or 5;10 (the lunar eccentricity corresponding to the maximum lunar correction in these tables: 4;56°). This justifies the use of the same interpolation coefficient in both the solar and lunar terms of Δt (Eqs. 9 and 13). Note also that while the entries in Col. iii have been computed with the true anomaly as argument, the instructions by Nicholaus de Heybech indicate

17

The argument for the distance is here the true anomaly in an eccentric model, whereas in the Almagest vi.8 [cf. Toomer 1984, 308] Ptolemy presents a similar interpolation table based on the mean anomaly in an eccentric model [cf. Toomer 1984, 654].

nicholaus de heybech and his table for finding true syzygy table ii

A recomputation of columns i and ii of Nicholaus de Heybech’s table

Arg. °

i h

c h

t–c min

ii h

c h

t–c min

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

0;28 0;57 1;25 1;53 2;19 2;43 3; 6 3;27 3;47 4; 4 4;18 4;29 4;38 4;44 4;47 4;47 4;43 4;37 4;27 4;14 3;58 3;39 3;18 2;55 2;29 2; 1 1;32 1; 3 0;32 0; 0

0;28,28 0;56,54 1;25, 4 1;52,50 2;18,58 2;43,25 3; 5,58 3;27,39 3;47,21 4; 4,40 4;18,55 4;29,38 4;37,57 4;43,48 4;47, 7 4;46,54 4;43, 2 4;36,51 4;27, 1 4;13,58 3;57,53 3;38,37 3;18, 0 2;54,13 2;28,17 2; 0,14 1;31,52 1; 2,34 0;31,38 0

0 0 0 0 0 0 0 –1 0 –1 –1 –1 0 0 0 0 0 0 0 0 0 0 0 +1 +1 +1 0 0 0 0

0; 6 0;12 0;18 0;24 0;30 0;35 0;40 0;44 0;48 0;51 0;54 0;57 0;59 1; 1 1; 1 1; 1 1; 0 0;59 0;57 0;54 0;50 0;46 0;42 0;38 0;32 0;26 0;20 0;14 0; 7 0; 0

0; 6, 4 0;12, 9 0;18, 8 0;24, 4 0;29,38 0;34,50 0;39,40 0;44,17 0;48,29 0;52,11 0;55,13 0;57,30 0;59,17 1; 0,31 1; 1,14 1; 1,11 1; 0,22 0;59, 3 0;56,57 0;54, 9 0;50,44 0;46,37 0;42,13 0;37, 9 0;31,37 0;25,39 0;19,35 0;13,21 0; 6,45 0

0 0 0 0 0 0 0 0 0 –1 –1 –1 0 0 0 0 0 0 0 0 –1 –1 0 +1 0 0 0 +1 0 0

23

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table iii

A recomputation of column iii of Nicholaus de Heybech’s table

Arg. iii t–c for e = 2;16 t–c for e = 5;10 ° min min min 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

1 2 4 7 11 15 20 25 31 36 42 47 51 54 56 58 60 60

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

0 0 0 –1 –1 –1 –1 –1 0 –1 +1 +1 +1 0 0 0 0 0

that the argument for Col. iii is the mean anomaly. Indeed, these instructions do not suggest that the true anomalies need be computed. In any event, the differences between entries computed with true, rather that mean, anomaly are small (less than 3 min in the worst case), and so there is little practical effect on the final result. For our recomputation of Cols. iv and v, we have taken the values of the lunar correction cm(ᾱ) from the 1483 edition of the Alfonsine Tables, vm(ᾱ) from John of Genoa’s lunar velocity table, and from his solar velocity table the following values for the maximum and minimum velocities: max(vs) = 0;2,33,40 °/h and min(vs) = 0;2,22,30 °/h.18

18

See, for example, Paris BnF Lat. 7282, fol. 129r–v [cf. Poulle 1984, 210].

nicholaus de heybech and his table for finding true syzygy

25

These parameters are used to evaluate the time given by Eq. (14) which corresponds to Col. iv and the difference between it and that resulting from Eq. (15) with the same argument. It would seem that Nicholaus de Heybech increased the argument of lunar anomaly by 6° in calculating the lunar velocity in between mean syzygy and true syzygy. Hence Eq. (14) becomes (19) and Eq. (15) is modified analogously: (20) Again, the recomputation is only displayed for the values of the argument in multiples of 6°. The results are shown in Table iv. The recomputed value according to Eq. (19) is c, and t – c is its difference from the value given in the text. The shift of 6° significantly diminishes the discrepancy between the text and the recomputed values, with the result that a value in the text is always within 1 min of time of the recomputed value. We note again that John of Genoa’s table for lunar velocity is arranged at intervals of 6° so that a shift of 6° is a shift of a line in the table. Column v is recomputed as the difference between the time found in column iv and the time found by means of Eq. (20) with the same argument. Table v displays Col. v together with our recomputed values (the differences between the times found by means of Eq. (19) and Eq. (20)). Since the entries in Col. v are small and have a very limited range, the rounding procedure plays a more significant role than in other columns. For this reason we have not displayed the differences between text and recomputation.

Conclusion Our principal goal in studying the canon and the table of Nicholaus de Heybech of Erfurt was to understand their use in finding the time from mean syzygy to true syzygy, as well as to reconstruct his table. It turned out that it works very well for its intended purpose and is quite easy to use. When we review some of the different approaches prior to that of Heybech, we see that Ptolemy had already solved the problem by means of an iterative procedure without recourse to any tables. On the other hand, John of Saxony’s method took advantage of solar and lunar velocity tables, while also introducing what we have called a differential method for computing an “instan-

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table iv

ᾱ ° 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 recomputation of column iv of Nicholaus de Heybech’s table

cm(ᾱ) °

ᾱ+6 °

vm(ᾱ+6) °/h

iv h

c h

t–c min

0;28,28 0;56,41 1;24,27 1;51,27 2;17,29 2;42,21 3; 5,46 3;27,30 3;47,20 4; 5, 4 4;20,27 4;33,18 4;43,28 4;50,41 4;54,54 4;55,56 4;53,38 4;48,10 4;39,15 4;27, 0 4;11,23 3;52,47 3;31, 3 3; 6,35 2;39,35 2;10,26 1;39,27 1; 7, 6 0;33,47 0

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 –

0;29,42, 6 0;29,46,17 0;29,51,51 0;29,59,31 0;30, 8,34 0;30,19,43 0;30,32,15 0;30,46,53 0;31, 2,55 0;31,18,55 0;31,37,45 0;31,57,15 0;32,16,45 0;32,39,45 0;33, 8,31 0;33,30,36 0;33,49,21 0;34,10,19 0;34,35,24 0;34,58,23 0;35,18,25 0;35,38,47 0;35,57,36 0;36,13,37 0;36,28,15 0;36,38, 0 0;36,46,22 0;36,53,15 0;36,58,54 –

1; 3 2; 5 3; 5 4; 3 4;59 5;50 6;38 7;21 7;59 8;31 8;58 9;18 9;32 9;40 9;40 9;34 9;23 9; 7 8;44 8;15 7;41 7; 3 6;20 5;33 4;42 3;49 2;54 1;57 0;59 0; 0

1; 2,56 2; 4,59 3; 5,35 4; 3,47 4;59, 5 5;50,48 6;38,24 7;21,10 7;58,48 8;31,22 8;57,36 9;17,53 9;32,19 9;39,25 9;38,36 9;33,37 9;23,34 9; 6,58 8;43, 7 8;14,16 7;40,39 7; 2, 9 6;19, 9 5;32,32 4;42,22 3;49,41 2;54,25 1;57,17 0;58,53 0

0 0 –1 –1 0 –1 0 0 0 0 0 0 0 +1 +1 0 –1 0 +1 +1 0 +1 +1 0 0 –1 0 0 0 0

nicholaus de heybech and his table for finding true syzygy table v

ᾱ ° 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 recomputation of column v of Nicholaus de Heybech’s table

v min

c Eq.(19) Eq.(20) min h h

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

0;26 = 1; 2,56 – 1; 2,30 0;50 = 2; 4,59 – 2; 4, 9 1;15 = 3; 5,35 – 3; 4,20 1;39 = 4; 3,47 – 4; 2, 8 2; 1 = 4;59, 5 – 4;57, 4 2;20 = 5;50,48 – 5;48,28 2;38 = 6;38,24 – 6;35,46 2;53 = 7;21,10 – 7;18,17 3; 6 = 7;58,48 – 7;55,42 3;17 = 8;31,22 – 8;28, 5 3;25 = 8;57,36 – 8;54,11 3;29 = 9;17,53 – 9;14,24 3;34 = 9;32,19 – 9;28,45 3;34 = 9;39,25 – 9;35,51 3;30 = 9;38,36 – 9;35, 6 3;20 = 9;33,37 – 9;30,17 3;20 = 9;23,34 – 9;20,14 3;12 = 9; 6,58 – 9; 3,46 3; 1 = 8;43, 7 – 8;40, 6 2;50 = 8;14,16 – 8;11,26 2;35 = 7;40,39 – 7;38, 1 2;22 = 7; 2, 9 – 6;59,47 2; 6 = 6;19, 9 – 6;17, 3 1;50 = 5;32,32 – 5;30,42 1;33 = 4;42,22 – 4;40,49 1;15 = 3;49,41 – 3;48,26 0;57 = 2;54,25 – 2;53,28 0;38 = 1;57,17 – 1;56,39 0;19 = 0;58,53 – 0;58,34 0 =0 –0

27

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taneous” velocity in elongation at a time very close to true syzygy. To be sure, a modern definition of a differential would require the evaluation of a function at two moments separated by an infinitesimal time interval. A value for the velocity very close to that obtained by the modern definition can be computed by taking a sufficiently small time interval, and John of Saxony set that interval as 0;1 days. Less than a century after John of Saxony presented his method, Nicholaus de Heybech offered a much simpler alternative without sacrificing the accuracy of the previous methods. First, instead of requiring many steps of computation, some of which called for tables while other did not, Heybech presented his solution in the form of a single table whose entries are given in time and whose arguments are the mean positions of the Sun and the Moon at mean syzygy, the initial values for all methods of computation. Second, the use of the table involves very simple arithmetic operations in contrast to the complexity of John of Saxony’s method. Nicholaus de Heybech has produced a “user-friendly” table for which he deserves much credit. In this respect he is following Ptolemy who, for example, established tables for finding planetary positions that replaced complex trigonometric computations. Both John of Saxony and Nicholaus de Heybech showed real insight into the ways Ptolemy’s models work and, on this basis, they were able to facilitate the computation of certain astronomical phenomena. This example shows that astronomers in the Middle Ages could make significant contributions without introducing any modification of the models for the motions of the celestial bodies. Finally, we wish to emphasize that it is inappropriate to define the corpus of Alfonsine Tables by the tables as they appear in the editio princeps, or by the canons of John of Saxony, or even to assign either of them any privileged status. Indeed, it has long been known that there is much more in the vast number of manuscripts within this corpus, but little attention has been paid to their contents. In this case study we have indicated the significance of Nicholaus de Heybech’s table and its dependence on a lunar velocity table by John of Genoa, neither of which is included in the restricted definition of the Alfonsine Tables. It is reasonable to expect that similar studies of other parts of this unexplored corpus would reveal hitherto unsuspected riches.

nicholaus de heybech and his table for finding true syzygy

29

Appendix i a Text in Latin The text presented here is a transcription of Dijon bm 447, fol. 62r–v (D). Variant readings from other manuscripts are noted: Basel f.11.7, fol. 37v (B) and Paris BnF Lat. 7287, fol. 86va–87ra (P). Sentence numbers and punctuation have been added by the editors. Canon super tabulas magistri nicholay de heybech [1] Tempus uere coniunctionis et oppositionis solis et lune per tabulas a magistro nicholao de heybech de erfordia compositas inuenire. [2] Scias tempus medie coniunctionis si uolueris coniunctionem ueram, uel tempus medie oppositionis si uolueris oppositionem ueram. [3] Scias eciam argumentum solis et argumentum lune ad idem tempus. [4] Quibus scitis et habitis intra primo cum argumento solis in tabulam equationis temporis uere coniunctionis et oppositionis luminarium. [5] Et accipe equationem solis et scribe super eam m si argumentum solis sit minus tribus signis phisicis, uel scribe a si sit plus tribus signis. [6] Accipe eciam ibidem diuersitatem equationis solis atque minuta proportionalia et scribe super ea solis. [7] Quibus habitis et seruatis intra postea cum argumento lune in easdem tabulas in lineis numeri et accipe equationem lune in directo existentem et scribe super am a si argumentum sit minus tribus signis, uel m si sit plus. [8] Accipe eciam ibidem diuersitatem equationis lune et minuta proportionalia et scribe super ea lune. [9] Istis habitis et seruatis accipe partem proportionalem de diuersitate equationis solis secundum proportionem minutorum lune ad 60. [10] Quam partem proportionalem minue ab equatione solis prius seruata et habebis eam bene equatam. [11] Accipe similiter partem proportionalem de diuersitate equationis lune secundum proportionem minutorum proportionalium solis ad 60. [12] Quam partem proportionalem adde super equationem lune prius inuentam et habebis eam bene equatam. [13] Postea uide si super ambas equationes, scilicet solis et lune, scriptum sita; tunc adde eas simul cum tempore medie coniunctionis prius inuento et seruato; sed si super utramque equationem scriptum fuerit m, tunc minue eas similiter ab eodem tempore. [14] Si uero super unam scriptum fuerit a et super aliam m, tunc minue minorem equationem a maiori et residuum adde cum tempore predicto si super maiorem scriptum fuerit a, uel minue si super maiorem scriptum fuerit m. [15] Et sic habebis ueram coniunctionem solis et lune si operatus es ad coniunctionem, et similiter ueram oppositionem si operatus es ad eam reperiendam, et hoc diebus non equatis, ad meridianum loci ad quem predicta inquirebas. [16] Et

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ad istud tempus debent queri loca planetarum. [17] Si gradum ascendentem scire uolueris, oportet equare dies. [18] Intra ergo cum uero loco solis in tabulam equationis dierum cum suis noctibus et inuenies in directo graduum equationem dierum, scilicet sub signo illo in quo est sol. [19] Et si inueneris ibi gradus et minuta accipe pro quolibet gradu quatuor minuta hore et pro quolibet minuto quatuor 2a hore, que adde cum tempore predicto quodlibet ad suum genus et proueniet tempus uere coniunctionis uel oppositionis diebus equatis. [20] Et cum illo debet queri ascendens et reliqua que pertinent ad figuram. Title: canon … heybech] om. B; post tabulas add. P coniunctionis solis et lune. [1] per tabulas … compositas] om. B. [3] eciam] igitur P; idem] iddem B. [5] sit] om. P. [7] in easdem] easdem B D P. [9] post minutorum add. B proportionalium. [13] inuento et] om. B; similiter] simul B. [14] equationem] om. P; maiorem 1 ipsam B et add. maiorem s. l.; maiorem 2] ipsam B. [17] si] si uero B. [18] ergo] igitur B; graduum] graduum ascendentem P; equationem] equationum B. [20] post figuram add. D et sic est finis secundum uasseun, add. P Et sic finis scripti Maruanti per me Mertinj anno domini 1447 12 die Julij.

b

The Table of Nicholaus de Heybech of Erfurt for Finding the Time of True Syzygy The table presented here is a transcription of Basel f.ll.7, fol. 36r–37r (B). Variant readings from two other manuscripts are noted: Paris BnF Lat. 7287, fol. 72r–73r (P) and BnF Lat. 7290a, fol. 103r–104r (P′). Column iii is found before Col. ii in P and P′. mss P and P′ share 28 errors (23 entries of Col. iv are shifted upward one place); P exhibits 5 additional errors and P′ just one. Therefore, we suggest that P was copied from P′. Title B

P

lncipit tabula equationum temporis uere coniunctionis et oppositionis solis et lune ordinata per Nycholaum dictum de Heybech de Erfordia (f. 36r) Tabula equationum uere coniunctionis et oppositionis solis et lune (signs are used for the last four words) secundum magistrum Nycholaum de Heybech de Erfordia (f. 72v)

nicholaus de heybech and his table for finding true syzygy

P′

Tabula equationum uere coniunctionis et oppositionis solis et lune secundum magistrum Nycholaum de Heybech de Erfordia (f. 103r) Column Headings

Argument i ii iii iv v

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

31

Linee numeri Equatio solis Diversitas equationis solis Minuta proportionalia Equatio lune Diversitas equationis lune

Argument °/s ° 1/5 2/5 3/5 4/5 5/5 6/5 7/5 8/5 9/5 10/5 11/5 12/5 13/5 14/5 15/5 16/5 17/5 18/5 19/5 20/5 21/5 22/5 23/5 24/5

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

h

i min

h

ii min

iii min

h

iv min

v min

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

5 10 15 20 24 28 33 38 43 48 53 57 2 7 12 17 21 25 30 35 40 44 49 53

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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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

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

11 22 33 43 53 3 14 24 34 45 55 5 15 25 35 45 55 5 14 24 34 43 53 3

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

P, P′: 1

32

chapter 1 (cont.)

s

Argument °/s °

h

i min

h

ii min

iii min

h

iv min

v min

0 0 0 0 0 0

25/5 26/5 27/5 28/5 29/5 30/5

35 34 33 32 31 30

1 2 2 2 2 2

58 3 7 11 15 19

0 0 0 0 0 0

25 26 27 28 29 30

3 4 4 4 4 4

4 4 4 4 4 4

12 22 31 41 50 59

2 2 2 2 2 2

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

31/5 32/5 33/5 34/5 35/5 36/5 37/5 38/5 39/5 40/5 41/5 42/5 43/5 44/5 45/5 46/5 47/5 48/5 49/5 50/5 51/5 52/5 53/5 54/5 55/5 56/5 57/5 58/5

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

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 3 3 3

23 27 31 35 39 43 47 51 55 59 3 6 10 14 17 20 24 27 31 35 38 41 44 47 50 53 56 59

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

31 32 33 34 35 35 36 37 38 39 40 40 41 42 42 43 44 44 45 46 46 47 48 48 49 49 50 50

5 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 11 11 11 12 12 12 13 13 14

5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8

8 17 26 34 42 50 58 6 14 22 30 38 46 53 0 7 14 21 28 35 41 47 53 59 5 11 16 21

2 2 2 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

nicholaus de heybech and his table for finding true syzygy

Argument s °/s °

i h min

ii h min

iii min

iv h min

v min

0 1

59/5 0/5

1 0

4 4

2 4

0 0

51 51

14 15

8 8

26 31

3 3

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

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

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

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

6 9 11 14 16 18 20 22 24 26 28 29 31 32 34 35 37 38 39 40 41 42 43 44 45 45 46 46 47 47

0 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

52 52 52 53 53 54 54 55 56 56 57 57 57 58 58 58 59 59 59 59 0 0 0 1 1 1 1 1 1 1

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

8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

36 41 46 50 54 58 2 6 9 12 15 18 21 24 26 28 30 32 34 36 37 38 39 40 40 40 40 40 40 40

3 3 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

1 1

31/4 32/4

29 28

4 4

47 47

1 1

1 1

32 32

9 9

39 38

4 4

P: 55 P: 56

P: 24

33

34

chapter 1 (cont.)

s

Argument °/s °

h

i min

h

ii min

iii min

h

iv min

v min

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

33/4 34/4 35/4 36/4 37/4 38/4 39/4 40/4 41/4 42/4 43/4 44/4 45/4 46/4 47/4 48/4 49/4 50/4 51/4 52/4 53/4 54/4 55/4 56/4 57/4 58/4 59/4 0/4

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

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

47 47 47 47 46 46 45 45 44 43 42 41 40 39 38 37 36 34 32 30 29 27 25 23 21 19 17 14

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

1 1 1 1 1 1 1 0 0 0 0 0 59 59 59 59 58 58 58 57 57 57 56 56 55 55 54 54

33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 42 42 43 43 44 44 45 45 46 46 47

9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8

37 36 35 34 33 31 29 27 25 23 21 19 16 13 10 7 4 0 56 52 48 44 40 35 30 25 20 15

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

P, P′: 16

2 2 2 2 2 2

1/3 2/3 3/3 4/3 5/3 6/3

59 58 57 56 55 54

4 4 4 4 4 3

12 10 7 4 1 58

0 0 0 0 0 0

53 52 52 51 50 50

47 48 48 49 49 49

8 8 7 7 7 7

10 4 59 53 47 41

3 3 3 3 3 3

P: 4 58, P′: 3 58 P, P′: 55

nicholaus de heybech and his table for finding true syzygy

Argument s °/s °

i h min

ii h min

iii min

iv h min

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

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

53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30

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

55 52 49 46 43 39 36 33 29 25 22 18 15 11 7 3 59 55 51 47 43 38 34 29

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

49 48 48 47 46 46 45 45 44 43 43 42 41 41 40 39 39 38 37 36 35 34 33 32

50 50 50 51 51 51 52 52 52 53 53 53 54 54 54 54 55 55 55 55 56 56 56 56

7 7 7 7 7 7 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 4 4

35 29 23 17 10 3 56 49 42 35 28 20 13 5 57 49 41 33 25 17 9 0 51 42

3 3 3 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 2

31/3 32/3 33/3 34/3 35/3 36/3 37/3 38/3 39/3 40/3 41/3

29 28 27 26 25 24 23 22 21 20 19

2 2 2 2 2 2 1 1 1 1 1

24 19 15 10 6 1 57 52 47 42 37

0 0 0 0 0 0 0 0 0 0 0

31 30 29 28 27 26 25 24 23 22 21

56 57 57 57 57 57 58 58 58 58 58

4 4 4 4 3 3 3 3 3 3 3

34 25 16 7 58 49 40 31 22 13 3

2 2 2 2 2 2 2 1 1 1 1

35

v min P, P′: 52 P, P′: 49 P, P′: 46 P, P′: 43 P, P′: 39 P, P′: 36 P, P′: 33 P, P′: 29 P, P′: 25, P: 0 47 P, P′: 22 P, P′: 18 P, P′: 15 P, P′: 11, P′: 6 10 P, P′: 7 P, P′: 3 P, P′: 2 59 P, P′: 55 P, P′: 51 P, P′: 47 P, P′: 43 P, P′: 38 P, P′: 34

P: 42

36

chapter 1 (cont.)

s 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3

Argument °/s ° 42/3 43/3 44/3 45/3 46/3 47/3 48/3 49/3 50/3 51/3 52/3 53/3 54/3 55/3 56/3 57/3 58/3 59/3 0/3

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

h

i min

h

ii min

iii min

h

iv min

v min

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

32 28 23 18 13 8 3 58 53 48 43 38 32 27 22 17 12 6 0

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

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

58 59 59 59 59 59 59 59 60 60 60 60 60 60 60 60 60 60 60

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

54 44 35 25 16 6 57 47 38 28 19 9 59 50 40 30 20 10 0

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

P, P′: 0 P, P′: 0

Appendix ii Translation The sentence numbers added to the Latin text have been retained in the translation; words in square brackets have been added by the editors. Sentences [16]–[20] are virtually identical with John of Saxony’s canon to the Alfonsine Tables, Chap. 22 [cf. Poulle 1984, 86, lines 127–138]. Canon for the Tables of Master Nicholaus de Heybech [1] To find the time of true conjunction and opposition of the Sun and the Moon by means of the tables compiled by Master Nicholaus de Heybech of Erfurt, [2] you must know the time of mean conjunction if you seek [the time

nicholaus de heybech and his table for finding true syzygy

37

of] true conjunction, or the time of mean opposition if you seek [the time of] true opposition. [3] You must also know the solar argument and the lunar argument at that time; [4] and when these are known and obtained, first enter the table for the equation of the time of true conjunction and opposition of the luminaries with the solar argument. [5] Take the solar correction [in Col. i] and write above it “m” if the solar argument is less than 3 physical signs [i.e., 180°], or write “a” if it is greater than 3 [physical] signs. [6] Take also in the same place the difference of the solar correction [in Col. ii] as well as the minutes of proportion [in Col. iii], and write “Sun” above them; [7] and when these are obtained and recorded, then enter the same tables in the column [labeled] “argument” [Lat.: linee numeri] with the lunar argument, and take the lunar correction opposite it [in Col. iv], and write above it “a” if the [lunar] argument is less than 3 [physical] signs, or “m” if it is more. [8] Take also in the same place the difference of the lunar correction [in Col. v] and the minutes of proportion [in Col. iii], and write “Moon” above them; [9] and when these are obtained and recorded, take the proportional part from the difference of the solar correction according to the ratio of the minutes of the Moon to 60. [10] Subtract this proportional part from the solar correction which was recorded previously, and you will have it well corrected. [11] Similarly, take the proportional part from the difference of the lunar correction according to the ratio of the minutes of proportion of the Sun to 60. [12] Add this proportional part to the lunar correction previously found, and you will have it well corrected. [13] Then, see if there is an “a” written above both corrections, the solar and the lunar: then add together with them the time of mean conjunction previously found and recorded; but if there has been an “m” written above both of them, then subtract them from the same time in the same way. [14] However, if there is an “a” written above one of them and an “m” above the other, then subtract the smaller correction from the larger and add the remainder together with the predicted time if there has been an “a” written above the larger, or subtract [it] if there has been an “m” written above the larger one. [15] You will thus obtain for the meridian of the place for which you were seeking the predictions the true conjunction of the Sun and the Moon, if you were working to find the [true] conjunction, and similarly, the true opposition, if you were working to find it, but without the equation of time [Lat.: diebus non equatis]. [16] And the positions of the planets must be sought for that particular time. [17] If you wish to know the degree of the ascendant, it is necessary to equate the days. [18] Therefore, enter the table for the equation of time [lit.: the equation of the days with their nights] with the true solar position, and you will find the equation of time opposite [the number] of degrees, that is, beneath the sign in which the Sun is. [19] If you find degrees and minutes there, take four minutes of an

38

chapter 1

hour for each degree, and four seconds of an hour for each minute; and add these together with the predicted time, each to its own rank, and there results the time of true conjunction or opposition with the equation of time. [20] With that [time] you must seek the ascendant and the other [magnitudes] relating to the figure [of the sky for the horoscope].

Acknowledgments We are grateful to José Luis Mancha (Sevilla) for establishing the Latin text of the canon given in Appendix i, and we are indebted to him for many helpful remarks. Alan C. Bowen (Princeton) read a draft translation of this canon, and many of his suggestions have been incorporated in Appendix ii. We have also benefited from comments by Emmanuel Poulle (Paris) on a previous draft of this paper.

References Alfonsine Tables. 1483. Tabulae astronomicae Alfontii regis castelle. Venice: E. Ratdolt. Chabás, J. 1989. Análisis del contenido astronómico de las tablas de Jacob ben David Bonjorn. Barcelona: Ph.D. dissertation, University of Barcelona (u. of b. Microfilm 814). Chabás, J. 1991. The astronomical tables of Jacob ben David Bonjom. Archive for History of Exact Sciences 42, 279–314. Goldstein, B.R. 1974. The Astronomical Tables of Levi ben Gerson. Hamden, ct: Archon Books (Transactions of the Connecticut Academy of Arts and Sciences, Vol. 45). Goldstein, B.R. 1980. Solar and lunar velocities in the Alfonsine Tables. Historia Mathematica 7, 134–140. Goldstein, B.R. 1992. Lunar velocity in the Ptolemaic tradition. In In the Prime of lnvention: Essays in the History of Mathematics and the Exact Sciences, P. Harman & A.E. Shapiro, Eds. Cambridge: Cambridge Univ. Press, pp. 3–17. Hartmann, J. 1919. Die astronomischen Instrumente des Kardinals Nikolaus Cusanus. Abhandlungen der königlichen Gessellschaft der Wissenschaften zu Göttingen, Neue Folge, 10 (6). Millás, J.M. 1943–1950. Estudios sobre Azarquiel. Madrid/Granada: Consejo Superior de Investigaciones Científicas. Nallino, C.A. 1907. Al-Battānī sive Albatenii Opus Astronomicum, ii. Milan: Reale Osservatorio di Brera in Milano. Neugebauer, O. 1969. The Exact Sciences in Antiquity. New York: Dover.

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Neugebauer, O. 1975. A History of Ancient Mathematical Astronomy. Berlin/New York: Springer­ Verlag. Pedersen, O. 1974. A Survey of the Almagest. Odense: Odense University Press. Poulle, E. 1984. Les tables alphonsines avec les canons de Jean de Saxe. Paris: Centre national de la recherche scientifique. Poulle, E. 1988. The Alfonsine Tables and Alfonso x of Castille. Journal for the History of Astronomy 19, 97–113. Ratdolt, E. 1483. See Alfonsine Tables. Thorndike, L. 1948. Nicholaus de Heybech of Erfurt. Isis 39, 59–60. Thorndike, L., and Kibre, P. 1963. A Catalogue of Incipits of Medieval Scientific Writings in Latin. Cambridge, ma: Medieval Academy of America. Toomer, G.J., 1968. A survey of the Toledan Tables. Osiris 15, 5–174. Toomer, G.J. 1984. Ptolemy’s Almagest. New York: Springer-Verlag. Zinner, E. 1925. Verzeichnis der astronomischen Handschriften des deutschen Kulturgebietes. Munich: C.H. Beck.

chapter 2

Computational Astronomy: Five Centuries of Finding True Syzygy* 1

Introduction

Medieval astronomers were aware of the difficulties and enormous effort needed to follow the rules for determining astronomically significant events and, from Ptolemy on, it was generally understood that numerical tables could be constructed that represented the underlying geometrical models adequately. Indeed, tables were already the result of many computations that relieved the user of much work and thus reduced the possibility of making mistakes.1 Moreover, the use of a table could be described more simply than the corresponding rules for calculating the same quantity. In short, tables were the most successful and economical way to present complex mathematical procedures in the Middle Ages. Many solutions to the same problem were generated, usually without justification or reference to previous treatments of it. For this reason, the task of the historian is to provide the astronomical significance of the quantities present in the mathematical relationships, as well as to indicate the dependence of later scholars on the works of their predecessors. In this paper we explore the specific case of finding the time from mean to true syzygy (conjunction or opposition of the Sun and the Moon).2 The procedures introduced for this purpose rely entirely on arithmetic, and do not involve any observations. Rather, they illustrate the methods and approaches of medieval practitioners in computational astronomy, and yield insights into

* Journal for the History of Astronomy 28 (1997), 93–105. 1 See, for example, B.R. Goldstein, “Descriptions of astronomical instruments in Hebrew”, in From deferent to equant: A volume of studies in the history of science in the ancient and medieval Near East in honor of E.S. Kennedy, ed. by D.A. King and G. Saliba (New York Academy of Sciences, New York, 1987), 105–141 (espec. p. 128, where Ibn al-Ḥadib’s view is cited). 2 In addition to the authors whose works are discussed here, we know of a few other treatments of this question. The most important is probably that of John of Murs noted by E. Poulle, “John of Murs”, in Dictionary of scientific biography (New York, 1970–1980), vii, 128–133 (espec. p. 130), and we have also located tables for this purpose in Hebrew astronomical manuscripts: Moses Farissol Botarel (c. 1481) in Munich, ms heb. 343, ff. 96b–97a, and Isaac ben Elia Kohen of Syracuse (c. 1491) in London, ms British Library Or. 2806, ff. 43a–44b.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_004

computational astronomy: five centuries of finding true syzygy

41

the ways that various traditions in the Middle Ages affected early modern astronomy. As far as we can tell, solutions to this problem that make use of double argument tables first appeared in Spain, and later in other parts of Europe. To be sure, medieval examples of double argument tables have been noted previously, but they pertain to the calculation of lunar and planetary equations (beginning with Ibn Yūnus, d. 1009),3 and for problems concerning the time of day.4 We will focus our attention on tabular solutions, while also noting procedures that were not reduced to tables. Simply stated, the problem is: given the time at which a mean syzygy takes place (easily solved using available tables), to determine Δt, the time from mean syzygy, t, to true syzygy, t′, where Δt > 0 indicates that true syzygy takes place after mean syzygy. There are four variables to be taken into consideration: λs and λm, the true longitudes of the Sun and the Moon at mean syzygy; and vs and vm, the velocities in longitude of the Sun and the Moon during the time from mean to true syzygy. The most serious difficulty is that the lunar velocity cannot properly be considered constant in this time interval, and there was no simple way to approximate its ‘average’ value. In Almagest vi.4 Ptolemy (c. 150) presented an approximate solution (without reducing it to a table) which may be expressed in modern notation as (1)

Δt = –13η/12vm(t),

where η is the true elongation at mean syzygy (η = λm – λs for conjunction, and η = λm – λs + 180° for opposition), together with a rule for computing vm(t), the velocity of the Moon at the time of mean syzygy. This method makes the simplifying assumptions that the solar and lunar velocities are constant over the relevant time interval, and that the ratio of lunar to solar velocity is 13 to 1. Though these approximations are crude, they can be refined by

3 C. Jensen, “The lunar theories of al-Baghdadi”, Archive for history of exact sciences, viii (1972), 321–328; D.A. King, “A double argument table for the lunar equation in Ibn Yūnus”, Centaurus, xviii (1974), 129–146; D.A. King, “Some early Islamic tables for determining lunar crescent visibility”, in King and Saliba (eds), op. cit. (ref. 1), 185–225; J.D. North, “The Alfonsine Tables in England”, in Prismata, ed. by Y. Maeyama and W.G. Saltzer (Wiesbaden, 1977), 269–301; G. Saliba, “The double argument lunar tables of Cyriacus”, Journal for the history of astronomy, vii (1976), 41–46; M. Tichenor, “Late medieval two-argument tables for planetary longitudes”, Journal of Near Eastern studies, xxvi (1967), 126–128, reprinted in E.S. Kennedy, Studies in the Islamic exact sciences (Beirut, 1983), 122–124. 4 See, for example, B.R. Goldstein, “A medieval table for reckoning time from solar altitude”, Scripta mathematica, xxvii (1964), 61–66, reprinted in Kennedy, Studies (ref. 3), 293–298.

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introducing an iteration and checking procedure. After computing a first value for Δt, λm and λs may be recomputed from the solar and lunar correction tables for t + Δt. If they are not equal, the procedure in Equation (1) is repeated, using the new value for η. The process converges rapidly, and it is rarely, if ever, necessary to perform more than two iterations to achieve equality of solar and lunar longitudes, to the precision of minutes of are. For most medieval astronomers this approach (with or without refinement) was quite satisfactory: it was adopted by al-Battānī;5 it appears in the canons to the tables of al-Khwārizmī;6 and it is found in the principal astronomical tables composed in medieval Spain, namely, the Toledan Tables and the Alfonsine Tables (as attested in Chapter xxx of the Castilian canons).7

2

The Method of Ibn al-Kammād

The earliest solution to the syzygy problem that we have found in the form of a table appears in a work of Ibn al-Kammād (c. 1116) called al-Muqtabis.8 This is one of his three known works (and the only one extant), preserved uniquely in a Latin manuscript: Biblioteca Nacional de Madrid, ms 10023. The text was translated into Latin in 1260 in Palermo by John of Dumpno. Ibn al-Kammād’s solution consists of a double argument table (f. 52r): its entries, Δt(vm(t) – vs(t), η), are given in hours and minutes, and are functions of the difference between the hourly velocities of the Moon and the Sun (in minutes and seconds per hour, from 0;27,30°/h to 0;33,30°/h at intervals of 0;0,30°/h) and the elongation, η (in degrees and minutes, from 0;30° to 12;0° at intervals of 0;30°). An excerpt of this table is reproduced in Table 1. The entries were calculated by means of the formula 5 O. Neugebauer, A history of ancient mathematical astronomy (New York and Berlin, 1975), 123, points out that Ptolemy does not indicate whether iteration is necessary, whereas in works of the Byzantine period it is explicitly stated that the procedure is to be iterated until no elongation remains. See also C.A. Nallino, Al-Battānī sive Albatenii opus astronomicum (2 vols, Milan, 1903–1907), i, 94. 6 O. Neugebauer, The astronomical tables of al-Khwārizmī (Copenhagen, 1962), 63. 7 On zijes (sets of astronomical tables), see E.S. Kennedy, A survey of Islamic astronomical tables, Transactions of the American Philosophical Society (Philadelphia, 1956), xlvi/2; see also F.S. Pedersen, “Canones Azarchelis: Some versions, and a text”, Cahiers de l’Institut du Moyen­Age grec et latin, liv (1987), 129–218 (espec. p. 182); M. Rico y Sinobas, Libros del saber de astronomía del Rey Alfonso x de Castilla (5 vols, Madrid, 1866), iv, 150–151. 8 J. Chabás and B.R. Goldstein, “Andalusian astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for history of exact sciences, xlviii (1994), 1–41.

computational astronomy: five centuries of finding true syzygy table 1

43

Ibn al-Kammād (Madrid, ms 10023, f. 52r)

[η]

27;30

28; 0

0;30 1; 0 1;30 … 6; 0 … 11;30 12; 0

1; 6 2;11 3;17 … 13; 5 … 25; 6 26;11

1; 4 2; 8 4;12 … 12;51 … 24;39 25;43

[vm – vs] … 33; 0 0;54 1;49 2;43 … 10;45 * … 20;36 21;30

33;30 0;53 1;47 2;41 … 10;45 … 20;46 ** 21;50 **

* Read 10;54. ** These values seem to belong to the column labelled 33;0, and the values there to this column.

(2) Δt = –η/[vm(t) – vs(t)]. This solution gives rather crude results (see Section 6, below), because the relative velocity of the luminaries is assumed to be constant during the time interval from mean to true syzygy, but its simple presentation made it attractive, and led to its adoption (with minor modifications) by a number of subsequent astronomers. – The astronomical work of Juan Gil de Burgos (c. 1350), as preserved in Hebrew in London, Jews College, ms heb. 135, contains a table (ff. 92b–93b) with the same range, intervals, and precision for the horizontal argument (vm – vs) as that of Ibn al-Kammād. The vertical argument (η) has the identical range and the same accuracy, but the interval is different, for here it is 0;6° (rather than 0;30°). The entries in this table are more accurately computed from Equation (2) than those in Ibn al-Kammād’s table. – The Tables of Barcelona, extant in Catalan, Hebrew, and Latin manuscripts, and probably completed in 1381,9 depend on the same solution as that

9 J.M. Millás, Las tablas astronómicas del Rey Don Pedro el Ceremonioso (Madrid and Barcelona, 1962); J. Chabás, “Astronomía Andalusí en Cataluña: Las Tablas de Barcelona”, in From Baghdad to Barcelona: Studies in the Islamic exact sciences in honour of Prof Juan Vernet, ed. by

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table 2

Yosef Ibn Waqār (Munich, ms heb. 230, f. 78b)

[vm]

[η] 0;10 0;20 … 1; 0 …

12; 0 12;10 12;20 … 13; 0 … 14;50

0;21 0;21 0;21 … 0;20 … 0;17

0;42 0;42 0;42 … 0;40 … 0;34

2;11 2; 9 2; 7 … 2; 0 … 1;43

6; 0 13; 6 12;54 12;40 … 12; 0 … 10;12

given by lbn al-Kammād, although the presentation is somewhat different. The horizontal argument has a minimum value of 0;28,0°/h (rather than 0;27,30°/h) and the same upper limit; the same interval (0;0,30°/h) is also used here. The range and interval of the vertical argument differ from those in the tables of Ibn al-Kammād and Juan Gil: the elongation is given from 0;10° to 6;40° at intervals of 0;10°. – Yosef lbn Waqār of Seville (mid-fourteenth century) presents a slightly different solution in the form of tables, uniquely preserved in Munich, ms heb. 230, f. 78b, where the headings are written in Arabic in Hebrew characters. He treats the effect of the velocities of each luminary in Equation (2) separately in two double argument tables. In the first table (Munich, ms heb. 230, f. 78b top) only the lunar velocity varies while it is assumed that the solar velocity is constant. In this table the elongation, η (in degrees and minutes, from 0;10 ° to 1;0°, at intervals of 0;10°, and from 1° to 6° at intervals of 1°), is the horizontal argument (rather than vertical, as before), and the lunar velocity is the vertical argument (in degrees and minutes per day, from 12;0°/d to 14;50°/d, at intervals of 0; 10°/d). Table 2 displays an excerpt of this table. The entries, Δt(η, vm), can be recomputed by means of a formula analogous to Equation (2): (3) Δt = –24η / (vm – k),

J. Casulleras and J. Samsó (Barcelona, 1996), 477–525; J. Chabás, L’astronomia de Jacob ben David Bonjorn (Barcelona, 1992), 23.

computational astronomy: five centuries of finding true syzygy table 3

45

Yosef Ibn Waqār (Munich, ms heb. 230, f. 78b)

[vs]

[Δt] 0;10 0;20 … 1; 0 …

0;57 0;58 0;59 1; 0 1; 1 1; 2

0;24 0;25 0;25 0;25 0;26 0;26

0;47 0;49 0;49 0;50 0;51 0;52

2;22 2;25 2;27 2;30 2;32 2;35

6; 0 14;12 14;30 14;42 15; 0 15;12 15;30

where k = 1°/d, and represents the constant solar velocity. Ibn Waqār’s second double argument table is for determining the position of the Sun at true syzygy (see the excerpt in Table 3): the vertical argument is vs, the solar velocity (in degrees and minutes per day, from 0;57°/d to 1;2°/d, at intervals of 0;1°/d). The horizontal argument (not specified in the ms) is Δt, the time from mean to true syzygy found in the previous table, and it is given in hours, from 0;10h to 1h, at intervals of 0;10h, and from 1h to 6h, at intervals of 1h. The entries, Δλ(Δt, vs), represent the arc-distance (in minutes and seconds) that the Sun travels during that time, and we have recomputed them by means of the formula (4) Δλ = (60/24)vs ·Δt. In the canons to Ibn Waqār’s tables (Munich, ms heb. 230, ff. 5a–5b [Hebrew] and ff. 14a–14b [Arabic in Hebrew characters]) we are told that the true positions of the Sun and Moon should be computed for noon of day 14 or day 29 of the month (rather than for mean syzygy), and then the table should be used, thus indicating that the table gives time from noon to true syzygy. Despite this statement in the canons, the entries in the table display the time from mean to true syzygy, rather than the time from noon to true syzygy (for an example, see Section 6, below). – Two centuries later, the method presented by Ibn al-Kammād appears, without ascribing it to any author, in the printed version of the Alfonsine Tables edited by P. Du Hamel in Paris (1553). In this case we have exactly the same

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pattern as in Ibn al-Kammād, but for minor details (p. 155): the entries, Δt(vm – vs, η), are given here to seconds; the relative difference in velocity is given from 0;22°/h to 0;34°/h at intervals of 0;1°/h; the elongation ranges from 1° to 8° at intervals of 1°. This table also includes columns for the differences in the entries between two consecutive values of the elongation, to assist in the cumbersome task of interpolation. The entries are computed by means of Equation (2). On pp. 156–157 there is another version of the same table: its entries are sixtieths of the corresponding previous ones.

3

New Approaches to an Old Problem

John of Saxony (c. 1330), one of the Parisian astronomers who adapted the Alfonsine Tables and composed canons to it, offered a more sophisticated solution using Ptolemy’s lunar models.10 He made allowances for the variation in the lunar velocity in the time interval between mean and true syzygy, and introduced a method of successive approximations to Δt, first to the nearest hour, and then to the nearest minute of an hour.11 This yields an improvement in accuracy, but involves a lot of computation that most practitioners in the late Middle Ages were probably not prepared to follow. Unfortunately, this solution cannot be displayed in tabular form. Another attempt to give more accurate solutions was successfully developed by Levi ben Gerson (1288–1344) who lived in southern France and wrote in Hebrew. He depended on his own lunar models rather than Ptolemy’s, but remained within the framework defined by Ptolemy. He presented his original solution in the form of four tables, and all of them contain additive coefficients to avoid calculations with negative terms.12 The equation of time is also included in the tables, so that all coefficients add up to 24; 17h. Two different

10 11 12

E. Poulle, Les tables alphonsines avec les canons de Jean de Saxe (Paris, 1984), 80ff. J. Chabás and B.R. Goldstein, “Nicholaus de Heybech and his table for finding true syzygy”, Historia mathematica, xix (1992), 265–289. B.R. Goldstein, The astronomical tables of Levi ben Gerson (New Haven, 1974), 136–146. For examples of medieval planetary correction tables where negative terms are eliminated by adding a constant. see H. Salam and E.S. Kennedy, “Solar and lunar tables in early Islamic astronomy”, Journal of the American Oriental Society, lxxxvii (1968), 492– 497; E.S. Kennedy, “The astronomical tables of Ibn al-Aclam”, Journal for the history of Arabic science, i (1977), 13–23 (espec. p. 14); and B.R. Goldstein, “The survival of Arabic astronomy in Hebrew”, Journal for the history of Arabic science, iii (1979), 31–39 (espec. p. 37).

computational astronomy: five centuries of finding true syzygy

47

sets of Levi’s tables for this purpose are known: they share the same structure, but they are based on different parameters. The time from mean to true syzygy is dependent on the motions of both luminaries in a way that renders it difficult to treat them separately. Nevertheless, Isaac Ibn al­Ḥadib, an astronomer of Spanish origin who settled in Sicily at the end of the fourteenth century,13 managed to treat the effect of each luminary separately. In his astronomical tables, entitled Oraḥ selulah, he mentioned the works of Bonfils (Tarascon, c. 1365), Bonjorn (Perpignan, c. 1361), Ibn alKammād, and Ibn al­Raqqām (Granada, d. 1315),14 and gave two corrections to be applied to the time of mean syzygy in order to find that of true syzygy: one due to the Sun and the other to the Moon, each one in separate tables (Paris, ms heb. 1086, ff. 7a–9b). The first (solar) correction has a maximum of 3;54,22h for a value of the mean argument of the Sun of 91°; the second (lunar) correction reaches a maximum of 9;42,6h when the true argument of the Moon is 96°. Ibn al-Ḥadib’s tables also give separate corrections for the positions of the luminaries between mean and true syzygy, making use of Ptolemy’s values: 0;32,56°/h for the mean hourly lunar velocity in longitude, 0;32,40°/h for the mean hourly lunar velocity in anomaly, and 0;2,28°/h for the mean hourly velocity of the Sun. Within the Spanish astronomical tradition, Abraham Zacut (d. after 1515) presented yet another approach to this problem.15 In his Almanach perpetuum printed in Leiria (Portugal) in 1496 there is a double argument table “for the equation of syzygies”. Its title is indeed surprising (pp. 65v–66v): Tabula ad verificandum horam aspectuum vel coniuntionis. The corresponding title for the same table in Zacut’s Ha-ḥibbur ha-gadol is “Table for correcting the time of conjunction and opposition and quarters of the month and all aspects of the Moon with all the planets” (Lyon, ms heb. 14, f. 142r). In this table the vertical argument is the elongation (arcus distantie), η, between the Sun and the Moon. The values taken for the elongation are 0;5°, 0;10°, 0;20°, …, 1°, and thereafter for each half degree to 13°, for a total of 31 values. The horizontal argument goes from 10;36° to 16° at intervals of 0;12°, and represents the daily increment of elongation, Δη. This table contains 868 entries, yielding not the time from

13

14 15

B.R. Goldstein, “Scientific traditions in late medieval Jewish communities”, in Les Juifs au regard de l’ histoire: Mélanges en l’ honneur de M. Bernhard Blumenkranz, ed. by G. Dahan (Paris, 1985), 235–247. On Ibn al-Raqqām, see J. Carandell, Risāla fi ʿilm al-ẓilāl de Muḥammad Ibn al-Raqqām al-Andalusī (Barcelona, 1988). On Zacut, see F. Cantera Burgos, “El judío salmantino Abraham Zacut”, Revista de la Academia de Ciencias de Madrid, xxvii (1931), 63–398; and F. Cantera Burgos, Abraham Zacut (Madrid, 1935).

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mean to true syzygy but the time, t, counted from noon, at which the true syzygy occurs in Salamanca. The 842 non-zero entries in this table can be recomputed by means of the following equation: (5) t = 24η / Δη, where η and Δη are deduced from the values of the true longitudes of the Sun and the Moon, at Salamanca for noon, found in other tables of the Almanach perpetuum (Tables 2 and 7 for the daily positions of the Sun and the Moon). The results given by Zacut differ by more than ±1′ in only 26 cases, thus indicating that the original table was satisfactorily calculated.

4

An Accurate and Easy Solution

Nicholaus de Heybech of Erfurt (c. 1400) seems to have been the first astronomer to reach a solution that met the criteria of improved accuracy and a ‘user-friendly’ presentation in tabular form (see Table 4 for an excerpt).16 To do this, he introduced two terms for the time from mean to true syzygy, to be added algebraically, one for the Sun and one for the Moon, and treated them separately. Each term can be considered as a combination of three functions, some of which depend only on the mean solar anomaly, κ̄, and some only on the mean lunar anomaly, ᾱ. In calculating the entries in his table, Nicholaus de Heybech made use of Ptolemy’s second lunar model for computing the underlying lunar velocities, in contrast to his predecessors who generally depended on Ptolemy’s simple lunar model.17 The result is a single table in 5 columns and 180 rows that gives results as good as those of his predecessors (see Section 6, below), but with much less work. The rule for computing Δt from the entries in the table is as follows: (6) Δt = Δts – Δtm = [c1(κ̄) – c2(κ̄) ·c3(ᾱ)] – [c4(ᾱ) – c5(ᾱ) · c3(κ̄)], where c1 …, c5 represent entries in columns 1, …, 5 of Heybech’s table. 16 17

Heybech’s entire table is published in Chabás and Goldstein, op. cit. (ref. 11), together with an explanation of the way it was computed; see Essay 1. For the use of Ptolemy’s second lunar model in computing lunar velocities, see B.R. Goldstein, “Lunar velocity in the Ptolemaic tradition”, in The investigation of difficult things: Essays on Newton and the history of the exact sciences, ed. by P.M. Harman and A.E. Shapiro (Cambridge, 1992), 3–17.

computational astronomy: five centuries of finding true syzygy table 4

49

Nicholaus de Heybech. Note that here ‘s’ represents signs of 60°. The columns are labelled: Argument: “linee numeri”; i: “equatio solis”; ii: “diversitas equationis solis”; iii: “minuta proportionalia”; iv: “equatio lune”; and v: “diversitas equationis lune”.

Argument s °/s °

i h min

ii h min

iii min

iv h min

v min

0 0 … 1 … 2 … 3

1/5 59 2/5 58

0 5 0 10

0 0

1 2

0 0

0 11 0 22

0 0

0/5

0

4

4

0 51

15

8 31

3

0/4

0

4 14

0 54

47

8 15

3

0/3

0

0

0

60

0

0

0

0

0

Nicholaus de Heybech’s method for finding the time from mean to true syzygy requires a few, rather simple, computations. Not long after him, John of Gmunden (d. 1442), lecturer on astronomy at the University of Vienna, fully understanding Heybech’s approach, presented it in an even simpler way by means of a double argument table, and it is preserved in a holograph manuscript (ms Vin. 5151). The table is entitled Tabula ostendens distantiam vere coniunctionis et oppositionis a media (ff. 119v–122r): the horizontal argument is κ̄, the mean solar anomaly (in degrees, from 0s 0° to 11s 24°, at intervals of 6°), and its vertical argument, ᾱ, is the mean lunar anomaly (in degrees, from 0s 0° to 6s 0°, at intervals of 6°). The table is preceded by a short canon (ff. 117v–119r), at the end of which we read: Iste canon editus et scriptus est Wienne per magistrum Johannem de Gmunden die 20 mensis Maius anno domini 1440 currente. The table, an excerpt of which appears in Table 5, displays 1,800 entries, Δt(κ̄, ᾱ), given in hours and minutes, for 0° ≤ ᾱ ≤ 180°. These entries can also be used for the other values of ᾱ because the following symmetry relation holds: (7) Δt(κ̄, ᾱ) = –Δt(360 – κ̄, 360 – ᾱ). The maximum value, Δt = 14;0h, corresponds to κ̄ = 264° and 270°, and ᾱ = 84° and, by Equation (7), the minimum value, Δt = –14;0h, corrresponds to κ̄ = 90° and 96°, and ᾱ = 276°.

50 table 5

chapter 2 John of Gmunden (ms Vin. 5151, ff. 119v–122r). We have added a minus sign where the text reads “m(inue)” and nothing where it reads “a(dde)”.

3s 0°

[κ̄ ] … 6s 0° …

–0;28 0;35 1;37

–4;47 –3;44 –2;41

0; 0 1; 3 2; 5

4;47 5;49 6;51

0;57 1;59 3; 1

0;28 1;31 2;33

9;40 9;40

9;15 9;15

5;23 5;27

9;44 9;44

14; 0 13;56

10;31 10;31

10; 5 10; 5

0;59 0; 0

0;37 –0;22

–2;48 –3;46

0;59 0; 0

5;44 3;46

1;44 0;45

1;21 0;22

[ᾱ]

0s 0°

0s 6°

0s 0° 0s 6° 0s 12° … 2s 24° 3s 0° … 5s 24° 6s 0°

0; 0 1; 3 2; 5

…

9s 0° … 11s 18° 11s 24°

It is easy to derive this table from that of Nicholaus de Heybech, but note that John of Gmunden uses signs of 30° whereas Nicholaus de Heybech uses signs of 60°. For example, the maximum entry Δt(270°, 84°) (see Table 5) can be derived directly from the entries in Nicholaus de Heybech’s table: the result is 13;57h, which differs from the entry in John of Gmunden’s table by 3 minutes. In other cases we found even closer agreement. In a fifteenth-century manuscript in Rome, ms Casanatense 1673, we have found another copy of John of Gmunden’s double argument table (ff. 89v–92r). The table is entitled Tabule distantie vere coniunctionis vel oppositionis luminarium a media: Composita Erfordie Duringie, and shares all the characteristics of Gmunden’s except for two: signs of 60° rather than 30° are used, and the differences between consecutive entries, whether in the same row or column, are shown. There is no text in the manuscript explaining the use of this table, and just after it there is an interpolation table (f. 92v) to be used in connection with our table, with the indication that it was drawn “per dominum nicolaum de Reichenbach”. This is the name of the copyist of the canons to the tables of Iohannes Bianchinus, also found in this miscellaneous manuscript. The double argument table bears no date, but in the manuscript it is found between an almanac for the planets beginning in 1456 and a table for mean syzygies the radices of which, as explicitly stated, are taken for the year 1452 (completed) and the meridian of Vienna. The same presentation as that given by John of Gmunden is also found in an earlier work on eclipses, known as The book of six wings, that was written

computational astronomy: five centuries of finding true syzygy table 6

[κ̄ ]

51

Immanuel ben Jacob Bonfils. Excerpt from the double argument table for the time from mean to true syzygy in Wing 2 after subtracting 24; 16h.

0°

[ᾱ] 90° 180°

0° 0; 0 9;54 0; 0 90° –4;10 6; 6 –3;24 180° 0; 0 9;57 0; 0 270° 4;10 13;45 3;24

270° –9;54 –13;45 –9;57 –6; 6

by Immanuel ben Jacob Bonfils of Tarascon (southern France) in the midfourteenth century. Originally written in Hebrew, this work was translated into Latin in 1406 and into Greek in 1435.18 The table that he called “Wing 2” is for determining Δλ, the longitude from mean to true syzygy, and Δt, the time from mean to true syzygy. Actually, the tabulated entries are Δλ + 3° and Δt + 24; 16h (which will be called Δt* in what follows). The additive coefficient to Δt consists of 0;16h, to account for the maximum equation of time, and 24h, to avoid calculations with negative terms.19 In Bonfils’s double argument table for finding the time to true syzygies, the horizontal argument is ᾱ, the mean lunar anomaly (in degrees, from 0s 0° to 11s 24°, at intervals of 6°), and the vertical argument is κ̄, the mean solar anomaly (in degrees, from 0s 0° to 12s 0°, at intervals of 6°). The entries Δt*(κ̄, ᾱ) are given in hours and minutes. The table for Δt* is similar to that of John of Gmunden except for interchanging the positions of the arguments, and the fact that it has twice as many entries, 3,600. Table 6 displays an excerpt of Bonfils’s table, after subtracting 24;16h. Although very close, the entries do not coincide with those given by John of 18

19

P.C. Solon, “The ‘Hexapterygon’ of Michael Chrysokokkes”, Ph.D. dissertation, Brown University, 1968; P.C. Solon, “The six wings of Immanuel Bonfils and Michael Chrysokokkes”, Centaurus, xv (1970), 1–20. This device is reminiscent of a similar one used by Levi ben Gerson for the same purpose relating to the same problem. Bonfils was aware of the work of his predecessor in southern France, and probably depended on him here: cf. B.R. Goldstein, The astronomy of Levi ben Gerson (1288–1344) (New York and Berlin, 1985), 9; see also ref. 12, above.

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table 7

[κ̄ ]

Recomputation of some entries in Bonfils’s table using the values given by al-Battānī

0°

[ᾱ] 90° 180°

270°

0° 0; 0 9;51 0; 0 –9;51 90° –4;10 6; 4 –3;26 –13;38 180° 0; 0 9;54 0; 0 –9;54 270° 4;10 13;38 3;26 –6; 4

Gmunden or Nicholaus de Heybech. Indeed, those two astronomers follow the Alfonsine tradition, whereas Bonfils’s table can be derived from al-Battānī’s tables, very nearly. Table 7 shows the recomputed values of some entries in Bonfils’s table using the values given by al-Battānī for the solar and lunar equations and velocities,20 and not taking into account any correction for the equation of time. It has not previously been noted in the scholarly literature that John of Gmunden’s approach was retained by early modern astronomers. Georg Peurbach (1423–1461), one of the earliest advocates of humanism at the University of Vienna, is the author of the Tabulae eclypsium, an extensive set of astronomical tables first printed in 1514 in Vienna, edited by Georg Tannstetter, and bound together with a work of Peurbach’s associate and student, Regiomontanus: Tabula primi mobilis Joannis de Monteregio. In Peurbach’s work there is a forty-eight-page table entitled Tabula distantie vere coniunctionis aut oppositionis a media (ff. a3v–d3r) and, in fact, it is a variant of John of Gmunden’s table for finding the time from mean to true syzygy. In Peurbach’s double argument table, the mean solar anomaly ranges from 0s 0° to 11s 30°, and the mean lunar anomaly from 0s 0° to 6s 0°, as was the case for John of Gmunden. But Peurbach’s table is much more expanded, for the intervals are 2° for κ̄ and 1° for ᾱ, yielding an impressive total number of entries of 32,400. This table, together with the rest of Peurbach’s Tabulae eclypsium, was later printed in 1553 in Basel in a volume entitled Luminarium atque planetarum motuum tabulae octoginta quinque, omnium ex his quae Alphonsum sequuntur quam faciles together with tables by Bianchini, Prugnerus, and Peurbach.

20

Nallino, op. cit. (ref. 5), ii, 80, 88.

computational astronomy: five centuries of finding true syzygy

53

By comparing the entries common to all three tables, namely, those by Nicholaus de Heybech, John of Gmunden, and Georg Peurbach, given at intervals of 1°, 6°, and 1° of mean lunar anomaly, respectively, it is clear that Peurbach’s table does not derive directly from Heybech’s, but from Gmunden’s, and that Peurbach used an interpolation scheme to fill in the intermediary values. This puts Peurbach within a medieval tradition for computing the time from mean to true syzygy rather than going back to classical sources, as might be expected of a Renaissance scholar.

5

Calculations in the Sixteenth Century

Copernicus seems to have had a method of his own to derive the time from mean to true syzygy;21 it is described in De revolutionibus iv.29. His procedure depends on Ptolemy’s second lunar model for computing lunar velocities at syzygy, as did Nicholaus de Heybech’s procedure, and considers the increment in lunar anomaly in the interval from mean to true syzygy (without introducing any new tables), in a way that is reminiscent of John of Saxony. Although Copernicus probably knew the solutions given by his immediate predecessors, it is not clear that he ever used them. According to Swerdlow and Neugebauer,22 Copernicus had copied parts of Peurbach’s Tabulae eclypsium and appended them to a bound volume containing the 1492 edition of the Alfonsine Tables and the 1490 edition of Regiomontanus’s Tabulae directionum. This handwritten quire of astronomical tables known as the “Uppsala Notebook” seems to come from Copernicus’s studies at Cracow University (1491–1495?).23 Moreover, the Prutenic tables by Erasmus Reinhold (1511–1553), based on the Copernican models and first published in Tübingen in 1551, do not contain any table for the time from mean to true syzygy. In December 1590, the Danish astronomer, Tycho Brahe, observed a lunar eclipse, and displayed the times computed for eclipse middle (which should be very close to true syzygy) according to five procedures without indicating any of the intermediate values required by each procedure. The five procedures are labelled: Alfonso, Peurbach, Copernicus, Maestlin, and Brahe’s

21 22 23

N.M. Swerdlow and O. Neugebauer, Mathematical astronomy in Copernicus’s De revolutionibus (New York and Berlin, 1984), 276. Swerdlow and Neugebauer, op. cit. (ref. 21), 272. P. Czartoryski, “The Library of Copernicus”, Studia Copernicana, xvi (1978), 355–396 (espec. p. 366).

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own.24 Brahe’s times computed according to the Alfonsine Tables and those of Peurbach differ from our recomputations, and the five values he presents for the time for eclipse middle vary from each other by as much as 2 hours. However, as can be seen from Table 8, below, the different methods we have used yield results that are much closer together. Hence, we suspect that Brahe made some mistakes in his calculations. Moreover, it has been claimed that Brahe intended to test computed times against observed times, but the text is too terse to draw any conclusions.25

6

A Test of the Methods

To compute the time of true syzygy that was necessary for determining in advance the circumstances of an eclipse, medieval astronomers had to go through a complicated procedure. They had to use different tables for the mean motions, equations, velocities, etc., of the two luminaries, and they had to follow instructions, whenever available, to perform these calculations step by step. Throughout the Middle Ages these tables and instructions evolved in different ways, although they remained within the framework established by Ptolemy. But they were adapted to different models and parameters, and thus yielded different numerical results for a given problem. We know of no case prior to the sixteenth century where the various methods for finding time from mean to true syzygy proposed by the astronomers considered above were tested against an observation. Nevertheless, we think it useful to compare these methods, and so we have derived Δt for the same syzygy according to the procedures of each author. To do this, we have used the same data in all cases, rather than depending on the intermediate data that result from each author’s way of deriving them. As our test case, we have taken a syzygy, the mean conjunction of 20 July 1327, occurring at 3;58,10h after noon in Toledo.26 The basic magnitudes for that event, as calculated according to the instructions in the canons to the Alfonsine Tables by John of Saxony, are: κ̄ = 35;25,4°, ᾱ = 222;26,7°, and η = 4;45,39°. Our results, using the different methods and tables, are displayed in Table 8. 24 25

26

Tychonis Brahe Opera omnia, ed. by J.L.E. Dreyer (15 vols, Copenhagen, 1913–1929), xii, 20–25. V.E. Thoren, “Tycho Brahe’s discovery of the variation”, Centaurus, xii (1967), 151–166 (espec. p. 158). Thoren also claimed that this observation led Brahe to the discovery of the lunar variation, but we do not believe there is enough evidence to support his view. Poulle, op. cit. (ref. 10), 214 ff.; Chabás and Goldstein, op. cit. (ref. 11), 271.

computational astronomy: five centuries of finding true syzygy table 8

55

A comparison of the various methods

Table or method

Date

Δt hours

Ptolemya lbn al-Kammād Yosef lbn Waqār John of Saxonyb Levi ben Gersonc Immanuel Bonfils Nicholaus de Heybechd John of Gmunden

c. 150 c. 1116 mid-14th cent. c. 1330 c. 1340 c. 1365 c. 1400 1440

–8;40 –8;47 –8;47 –8;34 –8;33 –8;44 –8;34 –8;33

a. With iteration; without iteration the value obtained is –8;51,30h (see ref. 5). b. For details, see Goldstein and Chabás, op. cit. (ref. 11), 271. c. With Levi’s second set of tables, the value obtained is –8;6h. d. For details, see Goldstein and Chabás, op. cit. (ref. 11), 274.

7

Conclusion

The most important point is that observations were not the driving force for innovations in the treatment of the time interval from mean to true syzygy. Rather, computational methods were devised whose goal was to simplify computations and to be ‘user-friendly’ without sacrificing accuracy. We have seen that a great deal of ingenuity went into these changes in procedure, and that they were most successfully presented in the form of tables. The first step in transforming the rules for finding the time from mean to true syzygy was a rather simple table, and the earliest example of it is found in the tables of Ibn alKammād, a Spanish Muslim astronomer. Other revisions of the rules—usually in the form of tables—were made in Spain, notably by Ibn Waqār, and then astronomers in France (and later in Germany) added new devices of their own while ultimately depending on their Spanish predecessors. So far, we have not found any examples of this type of table among astronomers from the eastern Islamic countries. Moreover, as is the case for other scientific matters in Spain, there is evidence to be gleaned from traditions in Arabic, Catalan, Castilian, Hebrew, and Latin, reflecting the ‘multicultural’ setting of the Iberian Peninsula. We have also seen that these medieval discussions continued to be copied,

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printed, and cited in the sixteenth century, i.e., they were still a part of the living astronomical tradition in the early modern period.

Acknowledgement We thank R. Mercier for allowing us to consult his notes on Bonfils’s table “Wing 2” from which we have greatly benefited.

chapter 3

Transmission of Computational Methods within the Alfonsine Corpus: The Case of the Tables of Nicholaus de Heybech* By the end of the fourteenth century the Alfonsine tradition, which originated in Castile a little more than 100 years earlier, had become the main computational tool for European astronomers. A great variety of astronomical tables, often accompanied by texts, followed this tradition, using the same models to describe the motions of the celestial bodies and the same underlying parameters, but differing in presentation. This “Alfonsine corpus”, as we have recently named it, dominated the scene of Western astronomy for several centuries.1 Within this corpus are the Parisian Alfonsine Tables of which hundreds of copies in manuscript are extant as well as two editions that appeared before 1500. Of particular interest are various methods and tables for finding the time from mean syzygy (i.e., conjunction or opposition of the Sun and the Moon) to true syzygy, starting with the method described in the canons by John of Saxony (1327).2 A specific approach to this problem, summarized below, was introduced by an otherwise almost unknown Nicholaus de Heybech of Erfurt (c. 1400).3 Heybech’s table was modified as it was transmitted from Erfurt to Poland, then to Salamanca, and finally to Jerusalem, and it is a remarkable example of the variety within the Alfonsine corpus that did not involve any

* Journal for the History of Astronomy 39 (2008), 345–355. 1 See José Chabás and Bernard R. Goldstein, The Alfonsine Tables of Toledo (Dordrecht and Boston, 2003). 2 José Chabás and Bernard R. Goldstein, “Computational astronomy: Five centuries of finding true syzygy”, Journal for the history of astronomy, xxviii (1997), 93–105. 3 José Chabás and Bernard R. Goldstein, “Nicholaus de Heybech and his table for finding true syzygy”, Historia mathematica, xix (1992), 265–289. We have seen a dozen manuscripts of Nicholaus de Heybech’s tables: Basel, Universitätsbibliothek, f.ii.7; Dijon, Bibliothèque Municipale, 447; Paris, Bibliothèque nationale de France, lat. 7287 and lat. 7290a; Cues, 211; Vienna, Nationalbibliothek, 2440; Cracow, Biblioteka Jagiellońska, 609, 610, 613, 1852, and 1865 (twice); and Princeton, University Library, Grenville Kane Collection 51. Several authors have mentioned other manuscripts containing the same material: Bern, 454; Vatican, Pal. lat. 1376; Vienna, Nationalbibliothek, 5245; and Munich, Clm 14111 and 26666.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_005

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changes in the theory or the parameters. Moreover, in his computations of the entries in his table, Heybech used a table for lunar velocity that first appears in Paris in the 1330s; this is part of the Alfonsine corpus and differs significantly from the tables for lunar velocity in the traditions of al-Battānī and of al-Khwārizmī.4 We thus offer an illustration of the transmission of a computational technique within the Alfonsine corpus that kept evolving during its long journey through a considerable part of Europe and beyond. Heybech’s method was presented in the form of a single table in 5 columns and consisted in taking the time interval between mean syzygy and true syzygy as the difference between two independent terms, one for the Sun and one for the Moon. Each term is calculated separately, and both require the computation of a set of minimum and maximum values and the use of an interpolation scheme for intermediate values. In Heybech’s table, this scheme is a list of interpolation coefficients, ranging from 0 to 1 (column iii, headed minuta proportionalia, where 60 minutes = 1), depending on the mean lunar anomaly, when computing the solar term, and on the mean solar anomaly, in the case of the lunar term. Besides column iii, the computation of the solar term requires columns i (headed equatio solis) and ii (headed diversitas equationis solis), both given in hours and minutes. As for the lunar term, besides column iii, its computation requires columns iv (headed equatio lune) and v (headed diversitas equationis lune), the former given in hours and minutes, and the latter in minutes of an hour. The solar term (Δts) can be obtained by means of the expression, Δts = c1(κ̄) – c2(κ̄) · c3(ᾱ), where κ̄ is the mean solar anomaly and ᾱ the mean lunar anomaly, and c1, c2, and c3 represent entries in columns i, ii, and iii, respectively. The entries in col. i depend on κ̄ and assume that ᾱ = 0°; those in col. ii also depend on κ̄ and represent the differences between the values for ᾱ = 0° and ᾱ = 180°, for a given κ̄; and the entries in col. iii, ranging from 0 to 1, are for interpolation for other values of ᾱ between 0° and 180°.

4 Bernard R. Goldstein, “Lunar velocity in the Ptolemaic tradition”, in The investigation of difficult things: Essays on Newton and the history of the exact sciences, ed. by P.M. Harman and A.E. Shapiro (Cambridge, 1992), 3–17; idem, “Lunar velocity in the Middle Ages: A comparative study”, in From Baghdad to Barcelona: Studies in the Islamic exact sciences in honour of Prof. Juan Vernet, ed. by J. Casulleras and J. Samsó (2 vols, Barcelona, 1996), i, 181–194.

transmission of computational methods

figure 3.1 Facsimile of Nicholaus de Heybech’s table (excerpt): Vienna, Nationalbibliothek, ms 2440, f. 74v

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Similarly, the lunar term (Δtm) can be obtained by means of the expression, Δtm = c4 (ᾱ) – c5 (ᾱ) · c3 (κ̄), where c4 and c5 represent entries in columns iv and v, respectively. The entries in col. iv depend on ᾱ and assume that κ̄ = 0°, and the entries in col. v also depend on ᾱ and represent the differences between the values for κ̄ = 0° and κ̄ = 180°, for a given ᾱ. Again, the entries in col. iii are for interpolation. Thus, according to Heybech’s table, the time from mean syzygy to true syzygy is given by (1)

Δt = Δts – Δtm = c1(κ̄) – c2(κ̄) · c3(ᾱ) – c4(ᾱ) + c5(ᾱ) · c3(κ̄).

Heybech’s method for determining Δt was appreciated by many medieval astronomers, for it simplified their computations without sacrificing accuracy.5 In the manuscripts that preserve this table there is usually a short canon explaining its use, but not the method for computing the entries. In all cases the tables are identical, but for copyist’s errors. Madrid, Biblioteca Nacional, ms 3385, containing a set of tables in Latin which we call the Tabule Verificate for Salamanca (ff. 104r–113r), has much the same material as in Heybech’s table, but with a different presentation.6 In these tables, fully part of the Alfonsine corpus, the epoch is 1 Jan. 1461. The name of the author of the Tabule Verificate (henceforth tv) is unfortunately not known, although we have identified Nicholaus Polonius as the most likely candidate among the few astronomers working in the Castilian city of Salamanca. Polonius was a Polish scholar who came to Salamanca no later than 1460 and held the newly established chair in astronomy/ astrology at the university there until 1464. It is reasonably clear that he brought the Tabule Resolute (a form of the Parisian Alfonsine Tables) with him from Poland.7 The

5 See Chabás and Goldstein, “Computational astronomy” (ref. 2). The conventions for the algebraic signs in Eq. 1 are not well described in Heybech’s canons, whereas the versions in tv and Zacut are unambiguous because the headings tell the user when to add and when to subtract. 6 José Chabás and Bernard R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the transition from manuscript to print (Philadelphia, 2000), especially pp. 23−36. 7 On this set of tables, see Jerzy Dobrzycki, “The Tabulae Resolutae”, in De astronomia Alphonsis Regis, ed. by M. Comes, R. Puig, and J. Samsó (Barcelona, 1987), 71–77; José Chabás, “Astronomy at Salamanca in the mid-fifteenth century: The Tabulae Resolutae”, Journal for the history of astronomy, xxix (1999), 167–175.

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tables we have labelled tv 7 (f. 106r–v) and tv 8 (f. 107r–v) in a previous publication represent Heybech’s columns for the lunar correction and the solar correction, respectively, but they are organized in a different way: column 2 in tv 7 displays c4(ᾱ), that is, column iv in Heybech’s table; column 4 in tv 7 is equivalent to 1 – c3(ᾱ), that is, the complement in 1 to column iii in Heybech’s table; column 2 in tv 8 represents c1(κ̄) – c2(κ̄), that is, the difference between columns i and ii in Heybech’s table. However, there is no longer a column equivalent to Heybech’s column v. The rest of the columns in tv 7 and tv 8 contain the arguments and line-by-line differences of the entries of other columns. With the entries in tv 7 and tv 8, (2) Δt = [c1(κ̄) – c2(κ̄)] – c4(ᾱ) + [1 – c3(ᾱ)] · c2(κ̄). This expression is equivalent to (3) Δt = c1 (κ̄) – c2 (κ̄) · c3 (ᾱ) – c4 (ᾱ), and it agrees with the first three terms in Eq. (1). The suppression of the fourth term, c5 (ᾱ) · c3 (κ̄), is indeed an acceptable approximation because in Eq. (1) its contribution is at most 0;04h, a small amount compared with the maximum values of the first, second, and third terms (4;47h, 1;01h, and 9;40h, respectively). It should be noted that the changes introduced in these two tables by the unknown author of the Tabule Verificate are not mere variations in the positions of the columns, but imply a different approach from that in Heybech’s table, among other things because attention shifts from lunar apogee (ᾱ = 0°), which is assumed for col. i in Heybech’s table, to lunar perigee (ᾱ = 180°), which is assumed for col. 2 in tv 8. The next step in the transformation of this table took place in 1513 in Jerusalem where Abraham Zacut (1452–1515) had recently arrived.8 Zacut’s best known astronomical work was composed in Hebrew in Salamanca (1478), and entitled ha-Ḥibbur ha-Gadol (The great composition). It was later published (with a number of modifications) in Latin and Castilian in Leiria, Portugal (1496), and entitled Almanach perpetuum. This work, in turn, was later trans-

8 For biographical details, see Chabás and Goldstein, Abraham Zacut (ref. 6), 6–15.

62

figure 3.2 Facsimile of tv 7 (excerpt): Madrid, Biblioteca Nacional, ms 3385, f. 106r

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transmission of computational methods

figure 3.3 Facsimile of tv 8 (excerpt): Madrid, Biblioteca Nacional, ms 3385, f. 107r

63

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lated into Arabic and diffused in the Islamic world.9 Zacut’s tables are mainly based on the Parisian Alfonsine Tables, but some depend on astronomical traditions in Hebrew that began in Provence in the fourteenth century. For his method of finding the time from mean to true syzygy in the Almanach perpetuum Zacut depended on this Hebrew tradition that derived from the work of Levi ben Gerson (1288–1344) and was transmitted to him through Jacob ben David Bonjorn’s tables (c. 1360), and it was quite distinct from the Alfonsine tradition. But in his new set of tables of 1513 for Jerusalem Zacut included tables for finding Δt that represent a modified version of Heybech’s table. Ironically, in 1478 Zacut used the Christian calendar for mean motions, whereas in 1513 he used the Jewish calendar for this purpose. Zacut’s tables of 1513 in Hebrew for Jerusalem are extant only in fragments:10 we consulted New York, Jewish Theological Seminary of America [jtsa], ms 2574 (not dated), 15 folios (containing only Zacut’s canons and tables).11 This manuscript contains three tables for computing Δt by a method that is similar to the one in the Tabule Verificate. Table 1 (f. 8b) is for the solar correction when the Moon is at perigee on its epicycle. The argument is the solar longitude, that is, the solar anomaly increased by 90° (under the assumption that the solar apogee is at 90°, which is an adequate value at the time), and it is given in degrees at intervals of 1°. The entries are displayed in hours and minutes. They derive from Heybech’s table, and correspond to the difference between the entries in columns i and ii, c1(κ̄) – c2(κ̄), as was the case in the Tabule Verificate for Salamanca (tv 8, col. 2). Note however that, contrary to the Tabule Verificate, Zacut’s table for the solar correction is presented as a single table, not as a column of a table, as in tv 8. Table 2 (f. 9a) is for the lunar correction when the Moon is at perigee on its epicycle. The argument is the mean lunar anomaly and it is given in degrees at intervals of 1°. The entries are displayed in hours and minutes. The entries

9

10 11

For Zacut’s tables in the Islamic world, see Julio Samsó, “Abraham Zacut and Joseph Vizinho’s Almanach perpetuum in Arabic”, Centaurus, xlvi (2004), 82–97; idem, “In pursuit of Zacut’s Almanach perpetuum in the eastern Islamic world”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, xv (2002–2003), 67–93. See Bernard R. Goldstein, “The Hebrew astronomical tradition: New sources”, Isis, lxxii (1981), 237–251, p. 248. Another fragment containing Zacut’s tables of 1513 is extant in NewYork, jtsa, ms 2567. There is no hint in Zacut’s canons that he was aware of Heybech or that he had direct access to his table.

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transmission of computational methods table 1

3s

An excerpt of the table for the first correction (New York, jtsa, ms 2574, f. 8b). Table for the correction of the Sun when the Moon is at the perigee of its epicycle, in hours and minutes.

4s

5s

subtract 1 0; 4h 1;53h 3;16h 2 0; 8 1;57 3;18 3 0;12 2; 0 3;19 4 0;16 2; 4 3;21 5 0;19 2; 6 3;22 … 10 0;38 2;22 3;30 … 15 0;56 2;36 3;37 … 20 1;15 2;49 3;42 … 25 1;32 3; 1 3;46 … 30 1;50 3;14 3;47 add

6s

7s

8s

3;47h 3;47 3;47 3;47 3;47

3;18h 3;16 3;14 3;12 3;10

1;54h 1;50 1;47 1;43 1;39

3;46

2;58

1;20

3;42

2;44

1; 1

3;37

2;30

0;42

3;31

2;14

0;21

3;22

1;57

0; 0

in this table also derive from Heybech’s table, and correspond to its column iv, c4(ᾱ), as was the case in the Tabule Verificate for Salamanca (tv 7, col. 2). Note again that Zacut’s table for the lunar correction is presented as a single table, not as a column of a table, as in the Tabule Verificate. Table 3 (f. 9b) is a double argument table where Zacut combined both solar and lunar components. The rows were computed for values of the lunar anomaly at intervals of 10°, ranging from 0s 0° to 6s 0°, whereas the columns were computed for values of the solar longitude at intervals of 15°, beginning with 3s 0° (as in Table 1). The entries in this double argument table are given in minutes of time. We note that for lunar anomaly, the maximum correction takes place for argument 0°, and vanishes for argument 180°. This should indeed be so, for this correction is to be added to, or subtracted from, the value found in Table 1 when the Moon is at its epicyclic perigee; thus, the correction at argument 180°

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table 2

0 1 2 3 4 … 10 … 15 … 20 … 25

An excerpt of the table for the second correction (New York, jtsa, ms 2574, f. 9a). Table for the correction of lunar anomaly, in hours and minutes.

0s

1s

2s

3s

4s

5s

0; 0h 0;11 0;22 0;33 0;43

4;59h 5; 8 5;17 5;26 5;35

add 8;32h 8;37 8;42 8;42* 8;51

9;42h 9;42 9;42 9;42 9;42

8;15h 8; 9 7;58 7;55 7;53

4;43h 4;34 4;25 4;16 4; 7

1;44

6;23

9;12

9;31

7;16

3;13

2;36

7; 1

9;27

9;17

6;42

2;25

3;25

7;34

9;36

9; 2

6; 4

1;39

4;13

8; 4

9;42

8;40

5;24

0;50

4;59

8;32

9;42 subtract

8;15

4;43

0; 0

… 30

* With jtsa, ms 2567, f. 65b, read: 8;46.

(i.e., lunar epicyclic perigee) is 0. The maximum effect of solar anomaly should be at 90° from the solar apogee and this is represented by the column for 6s 0°. Hence, this third correction is to improve the first correction displayed in Table 1, where the only variable considered was solar longitude. Table 3 then takes into account the effect of the change in lunar anomaly in the time interval due to the solar motion (where the heading is the solar longitude), and represents the term [1 – c3(ᾱ)] · c2(κ̄) in Eq. (2). This is certainly the case: 1 – c3(ᾱ) appears as the column for a solar longitude of 6s 0°, i.e., when the Sun is 90° ahead of apogee and the correction reaches its minimum; it is the complement in 1 of Heybech’s column iii, as in tv 7, column 4. Moreover, c2(κ̄) appears as the row for a lunar anomaly of 0s 0°, and it is column ii in Heybech’s table, as in tv 8, column 4. The product of the entries in this row and this column generates the rest of the table. For example, consider the entries in the row for 2s 0°: 0, 12, 23, 32, …, 47, …, 24, 12, 0. Each one is found

67

transmission of computational methods table 3

Solar long. Lunar anom. 0s 0° 0s 10° 0s 20° 1s 0° 1s 10° 1s 20° 2s 0° … 3s 0° … 5s 20° 6s 0°

An excerpt of the table for the third correction (New York, jtsa, ms 2574, f. 9b). Table for the correction of all values for the lunar anomaly to be added to its value at the perigee of its epicycle: a double argument table [luaḥ meḥubberet; lit.: a combined table]

3s 0° 3s 15° 4s 0° 4s 15° …

6s 0°

… 8s 0° 8s 15° 9s 0°

0 0 0 0 0 0 0

15 15 14 14 13 13 12

29 29 28 28 27 25 23

41 40 40 39 37 35 32

subtract 60 59 58 56 44* 51 47

0

8

16

23

31

17

9

0

0 0

0 0

0 0

1 0

1 0 add

1 0

1 0

0 0

31 30 29 29 28 26 24

16 15 15 15 14 13 12

0 0 0 0 0 0 0

* With jtsa, ms 2567, f. 66a, read: 54.

by multiplying the corresponding entry for a lunar anomaly of 0s 0° by 47 (the entry corresponding to a solar longitude of 6s 0°). So, for 2s 0° of lunar anomaly and 3s 0° of solar longitude: 0; 0 · 0;47 = 0; 0 (entry: 0 min); 3s 15° of solar longitude: 0;15 · 0;47 = 0;11,45 = 0;12 (entry: 12 min); 4s 0° of solar longitude: 0;29 · 0;47 = 0;22,43 = 0;23 (entry: 23 min); 4s 15° of solar longitude: 0;41 · 0;47 = 0;32, 7 = 0;32 (entry: 32 min); … 6s 0° of solar longitude: 0;60 · 0;47 = 0;47 (entry: 47 min); … 8s 0° of solar longitude: 0;31 · 0;47 = 0;24,17 = 0;24 (entry: 24 min); 8s 15° of solar longitude: 0;16 · 0;47 = 0;12,32 = 0;13 (entry: 12 min); 9s 0° of solar longitude: 0; 0 · 0;47 = 0; 0 (entry: 0 min).

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Thus, Zacut replaces two columns in two different tables in the Tabule Verificate by one double argument table where the multiplication is already done, thus facilitating the task of the computer; all that is left is linear interpolation in both the horizontal and vertical directions in the table. The use of a double argument table is consistent with Zacut’s preference for this kind of table, which appears already in his Ḥibbur, and this distinguishes him from the tradition, always within the Alfonsine corpus, represented by Heybech and the Tabule Verificate. On the other hand, the entries in these three tables are not identical with those in the Tabule Verificate, but both sets are internally consistent and differ very slightly. Chapter 3 of the canons for these tables (jtsa, ms 2574, ff. 12b–13a) concerns the time interval from mean to true syzygy, with instructions on how to find this interval from the three tables, but nothing is said about the origin of this table or the way its entries were computed. Zacut adds a worked example (f. 12b) for finding true conjunction for Tishri 5274am [= 30 Aug. 1513]: mean conjunction took place 3(d) 18;6,30h after noon.12 According to the text, the Sun’s position was then 5s 16;29° and the lunar anomaly was 8s 12;47°. The first correction with solar longitude 5s 16;29° as argument is 3;38,30h to be subtracted. The result is 14;28h [= 18;6,30h – 3;38,30h]. With argument 8s 12° for the lunar anomaly, the second correction is stated to be about 9;10h [ms: 10, written as a word] to be subtracted. The result is then given in the text as 5;22h, although it should be 5;18h [= 14;28h – 9;10h]. The third correction, with the two arguments, is 0;22h to be subtracted. Thus, the final result, as given in the text, is 5;0h. This result can be checked using the tables themselves. In Table 1 the entry for 5s 16° is 3;38h; in Table 2 the entry for 8s 12° is 9;8h and for 8s 13° it is 9;11h. So 9;10h, the value given in the text for the second correction, agrees with computation using the table. In Table 3, with 5s 15° (rounded from 5s 16° for the solar longitude) and 8s 10° (rounded from 8s 12° for the lunar anomaly) as arguments, the entry is 0;22h. Hence, the total correction is –13;10,30h (= –3;38,30h – 9;10h – 0;22h),13 and the result should be: 18;6,30h – 13;10,30h = 4;56h (text: 5;0h). It is most likely that for this refinement of a technique in the Alfonsine corpus Zacut depended on tables in Latin he had seen in Salamanca many years before he arrived in Jerusalem. Zacut then transmitted his new method for finding Δt in Hebrew, thus contributing to the enlargement of the Alfonsine

12 13

3(d) means weekday 3, i.e., Tuesday. And indeed 30 Aug. 1513 (jdn 2273923) was a Tuesday. The absolute value of this amount for the total correction is close to its maximum: see Richard L. Kremer, “Wenzel Faber’s tables for finding true syzygy”, Centaurus, xlv (2003), 305–329, p. 314 (table 2).

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corpus, a body of astronomical material with no theoretical changes or modifications in the models or the parameters that depended, directly or indirectly, on the Alfonsine Tables compiled in Toledo in the 1270s and diffused from Paris throughout Europe and beyond in the 1320s.

Epilogue There is one known instance of a later text that depended on Zacut’s tables of 1513 for finding Δt: a Geniza fragment in Hebrew, ms A 697-1, at the John Rylands University Library, Manchester, England.14 In this brief fragment of an anonymous calendrical text for 5557am (= 1796–1797),15 the goal is to compute the times of true opposition (full-moon) for each month in the year 5557 am as a function of true solar longitude and mean lunar anomaly at mean opposition. On the first line Zacut is credited for the method of determining true oppositions, and the text includes values computed from his first two correction tables (but there is no evidence of his third table). For example, the solar longitude in this text for opposition in Tishri 5557 is 6s 23;47° whose correction is given as 3;33h, and the lunar anomaly at that time is 3s 20;26° whose correction is given as 9;2h. These are exactly the values for these arguments in Zacut’s Tables 1 and 2, where the arc minutes of the arguments have been ignored. 14 15

We are grateful to Y. Tzvi Langermann for bringing this manuscript to our attention. The date given in the text is not easy to read but 5557am is confirmed by recomputing the astronomical data with Zacut’s tables for 1513.

part 2 Planetary Motions

∵

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Ptolemy, Bianchini, and Copernicus: Tables for Planetary Latitudes* The study of planetary theory in Ptolemaic astronomy has concentrated on the models and tables for planetary longitudes, and considerably less attention has been paid to Ptolemy’s models and tables for planetary latitudes. There are good grounds for this imbalance both in medieval sources and in the modern secondary literature, but it is not our goal here to examine the reasons for this. Rather, we wish to focus on some special features of tables for planetary latitude, particularly those of Giovanni Bianchini (d. ca. 1469) that are extant in many manuscript copies,1 in a printed edition of 1526, and in a copy in the hand of Copernicus.2 In Almagest xiii Ptolemy gives a full treatment of planetary latitudes and, in the case of the inferior planets, he refers to three components which we will call inclination (declinatio), slant (reflexio), and deviation (deviatio).3 However, Ptolemy’s tables for the latitudes of Venus and Mercury in Almagest xiii.5 display only the first two of these components,4 whereas Bianchini has columns for all three of them. Our plan is first to give a brief survey of the history of tables for planetary latitude, particularly those that include, for Venus and Mercury, columns for the deviation. Then we will describe Bianchini’s tables for planetary latitude in detail. Finally, we will discuss Copernicus’s copy of Bianchini’s tables for planetary latitude.

* Archive for History of Exact Sciences 58 (2004), 453–473, communicated by N. Swerdlow. 1 According to the catalogues we consulted, Bianchini’s tables survive in a large number of manuscript copies in many libraries: Naples, Nuremberg, Milan, Paris, Rome, Venice, Vatican, among others (see, e.g., Boffito 1908; Thorndike 1950 and 1953; Zinner 1990). Of special interest is Nuremberg, Stadtbibliothek, ms Cent v 57, copied in Vienna in 1460 by Regiomontanus. 2 Uppsala, University Library, ms Copernicana 4, ff. 276v–281r. Since it is most likely that Copernicus saw a manuscript of these tables while he was a student in Cracow between 1491 and 1495, we will refer to the catalogue of the scientific manuscripts there: Rosińska 1984a. 3 In Swerdlow and Neugebauer 1984, p. 523 et passim, deviatio is translated “deflection”. 4 O. Pedersen 1974, pp. 355–386; Neugebauer 1975, pp. 216–226; Riddell 1978; Toomer 1984, pp. 632–634; and Swerdlow 2005.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_006

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Tables for Planetary Latitude Prior to Bianchini

In the Handy Tables, composed after the Almagest, Ptolemy made some changes in the theory of the planetary latitudes and introduced new tables with a different presentation. The latitude tables in the Handy Tables had little influence on subsequent astronomers, although they were the source for parameters in the corresponding tables in the Mumtaḥan zij (9th century).5 In the Iberian peninsula this tradition is represented by Ibn al-Kammād (ca. 1110) whose zij (extant in a unique Latin manuscript) provides the only previously known example of a set of astronomical tables where the planetary latitudes follow the Handy Tables for the inferior planets.6 In the medieval astronomical literature there is yet another tradition, not based on Ptolemaic models, where the tables for the planetary latitudes differ substantially from those in the Almagest and the Handy Tables. This tradition whose roots lie in Indian astronomy appears in the zij of al-Khwārizmī (9th century) and, later on, in tables headed Tabula bipertalis numeri and Tabula quadripertalis numeri, in the Toledan Tables (available in Latin in the 12th century, but manuscript copies of it only begin to proliferate in the 13th century).7

5 For the Handy Tables, see Stahlman 1959, pp. 143–155, 325–334; Neugebauer 1975, pp. 1006– 1016; and Swerdlow 2005. For the Mumtaḥan zij of Yaḥyā ibn Abī Manṣūr, see Kennedy 1956, pp. 145–147, 173; and Vernet 1956. Kennedy 1956, p. 146, indicates that these tables for the latitudes of Venus and Mercury have the same parameters as the corresponding tables in the Handy Tables, but their structure is much more primitive. The extremal values for the latitudes of Venus and Mercury in the Handy Tables are reported by al-Battānī, but he does not identify his source: see Nallino 1903–1907, 1:116. 6 For the zij of Ibn al-Kammād, see Chabás and Goldstein 1994, pp. 31–32; and Madrid, Biblioteca Nacional, ms 10023, f. 45r–v. We have identified another copy of this table in the zij of Juan Gil of Burgos (14th century) that survives in a single Hebrew manuscript: see f. 139a of what was formerly London, Beth Din, ms 135 [olim, London, Jews College, ms 135; film no. 4796 at the Institute for Microfilmed Hebrew Manuscripts, Hebrew University (Jerusalem)]. The same table is also found among the Tables of Barcelona (14th century): see Chabás 1996, p. 505. 7 For the zij of al-Khwārizmī, see Suter 1914, pp. 138–167; Neugebauer 1962, pp. 101–103. For the Toledan Tables, see Toomer 1968, pp. 69–70; Richter-Bernburg 1987; and F.S. Pedersen 2002, pp. 15, 1309–1321. On the rule for computing planetary latitudes in the zij of al-Khwārizmī see Kennedy and Ukashah 1969, espec. p. 89. A variant of this tradition for treating planetary latitudes is found in the zij of Ibn ʿAzzūz (Fez, 14th century) where the entries in one column for each planet are the same as in the zij of al-Khwārizmī and those in the other are the reciprocals of the corresponding entries in the earlier zij: see Samsó 1999, pp. 114; and Samsó 1997, p. 92.

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While these two traditions were represented in some texts, the mainstream tradition found in most medieval sets of astronomical tables dealing with planetary latitudes certainly derives from the Almagest and it was transmitted to the West primarily via the zij of al-Battānī (ca. 900) and the Toledan Tables.8 As was the case in the Almagest, neither of these tables displays columns for the deviation for Venus and Mercury. Interest in the third component of latitude is found in Maghribi sources, notably in a text written by an anonymous author in Tunisia ca. 1280, a reworking of the canons to the zij of Ibn Isḥāq alTūnisī (ca. 1222), based on Ibn al-Kammād. Chapter 18 of this text (uniquely extant in Hyderabad, Andra Pradesh State Library, ms 298) presents a worked example for the latitude of Venus where the deviation is considered. Further references to this method for reckoning the latitudes of the inferior planets with three components are found in the zij of Ibn al-Bannāʾ (d. 1321) which, in turn, depends on the zij of Ibn Isḥāq.9 Written at much the same time, the canons to the Castilian Alfonsine Tables include explanations for using tables to compute the latitudes of the planets. These canons were composed no later than 1272 and they are preserved in a unique manuscript now in Madrid; the original tables, which are not extant, had an epoch of January 1, 1252.10 For the inferior planets, Chap. 22 specifically mentions columns for deviation, here called “third latitude”, for Venus (22:12) and for Mercury (22:28), as follows: [11] Mas quando quisieres saber la latitud de Venus. entra con su çentro en las tablas de su latitud y toma la que fuere en su derecho de los minutos proporçionales de la declinaçion e de los minutos proporçionales del declinamiento. et escrive cada uno dellos a su parte y escrive sobre cada uno lo que hallares en somo de la regla donde lo tomas de alto o de baxo. [12] E toma otrosi lo que fuere en aquel derecho del çentro en la regla de la latitud tercera et guardala otrosi et la parte que le hallares escripta de suso es siempre septentrional. (…)

8 9

10

For the zij of al-Battānī, see Nallino 1903–1907, 1:115–116, 2:140–141. For the Toledan Tables, see Toomer 1968, pp. 71–72; and F.S. Pedersen 2002, pp. 1322–1326. For the zij of Ibn Isḥāq al-Tūnisī, see Mestres 1999, pp. 56–59. For the zij of Ibn al-Bannāʾ, see Vernet 1952, pp. 96–100. For planetary latitude tables in the Islamic East, see van Dalen 1999, espec. p. 323. Madrid, Biblioteca Nacional, ms 3306; see Chabás and Goldstein 2003a.

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[26] Mas quando quisieres saber la latitud de Mercurio entra con su çentro en las tablas de su ladeza et toma lo que fuere en su derecho de los menudos proporçionales de la su declinaçion y de los menudos proporçionales del desviamiento. [27] E escrivelos con sus titulos segun lo diximos en Venus. [28] E toma otrosi con el çentro su ladesa terçera e escrivela con su parte e siempre la hallaras meredional. [11] But when you wish to know the latitude of Venus, enter the tables of its latitude with the center and take the minutes of proportion for inclination and the minutes of proportion for slant which are opposite it; and write them down separately and note for each of them what you will find at the top or at the bottom of the corresponding column. [12] And also take what is in the column for the third latitude which is in opposite it and keep it, and what is written is always northern. (…) [26] But when you wish to know the latitude of Mercury enter the tables of its latitude with the center and take the minutes of proportion for inclination and the minutes of proportion for slant which are opposite it. [27] And write them down with their headings, as we said for Venus. [28] And also take with its center [as argument what is in the column for] the third latitude and write it down with the rest, and it is always southern. The instructions in the text seem to refer to tables in the style of Almagest xiii.5. However, in contrast to Ptolemy’s tables, the columns for the minutes of proportion for the inclination and for the slant are not the same. The Castilian Alfonsine Tables arrived in Paris in the early 14th century and they began to spread in a modified form, in Latin, throughout Europe. Among the Parisian astronomers, John Vimond, seems to be the first to have constructed tables for the latitudes of each planet including, for Venus and Mercury, a column for the deviation. His latitude tables are also in the style of the Almagest with arguments at 12-intervals from 0s 12° to 12s 0°.11 The inclusion of columns for the deviation is exceedingly rare in the West and, to the best of our knowledge, Vimond’s tables are the earliest to display them. As a matter

11

Paris, Bibliothèque nationale de France, ms lat. 7286c, ff. 1r–8v; see Chabás and Goldstein 2003b, and Chabás and Goldstein 2004.

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of fact, the third component for Venus and Mercury is not tabulated in the manuscripts of the Parisian Alfonsine Tables we have seen that date from the 14th century. The Parisian Alfonsine Tables gave rise to a variety of tables that preserve the basic parameters of Alfonsine astronomy but differ in presentation. In particular, the latitude tables were sometimes recast in the form of double argument tables such that the entries display the latitude of the planet as a function of both its anomaly and its center. Double argument tables for planetary latitudes are found in several sets of tables: in the 14th century the Oxford Tables of 1348 ascribed to William Batecombe; and in the 15th century the tables of John of Gmunden (Vienna), ha-Ḥibbur ha-gadol by Abraham Zacut (Salamanca), and the Almanach Perpetuum (based on Zacut’s tables and printed in 1496 in Leiria, Portugal).12 We have spot checked corresponding entries in both the Oxford Tables and in the Almanach Perpetuum and it is clear that these entries all derive from Ptolemy’s tables (or a minor variant of them). We can also say that they were computed independently, based on small divergences between corresponding entries. In particular, the entries for Venus and Mercury take into account the deviation: when the anomaly and the center are 0°, the inclination and the slant are both 0° and the corresponding entries in the table only account for the deviation. In the Almanach Perpetuum and in the Oxford Tables, these entries are not 0°, but +0;10° (Venus) and –0;45° (Mercury), which are the standard extremal values for the deviation (see below). The deviation is also embedded in the rest of the entries but some recomputations are required to reveal it. Double argument tables for latitude were certainly an advance over single argument tables such as those in Almagest xiii.5. The increase in size offered more possibilities to the computer (Ptolemy’s table has 675 entries for all 5 planets, whereas the Oxford Tables have 8,220 and those by Zacut 8,680, of which 5,611 are for the inferior planets). But, above all, the latitude tables with double arguments gained in “user-friendliness”, for with them the computer could often find the latitude he sought (or, at least, a first approximation to it) simply by inspecting the table. In this way he could avoid the tedious and risky computations using the table where the entries in each column are a function of a single argument, looking up many entries (7 in the case of an inferior

12

For the Oxford Tables of 1348, see North 1977; for the tables of John of Gmunden, see Porres 2003; for ha-Ḥibbur ha-gadol and the Almanach Perpetuum, as well as a survey of such double argument tables, see Chabás and Goldstein 2000, pp. 137–143. According to Tichenor 1967, p. 128, al-Kāshī (d. 1429) also had double argument tables for planetary latitudes. On al-Kāshī, see now Kennedy 1998.

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planet), each with its proper sign, and combining them (paying attention to the algebraic signs) according to complicated rules. John of Murs, who was also active in Paris in the early 14th century, compiled various sets of tables, and among them are tables for the latitudes of Venus and Mercury. For both planets we are given double argument tables for the first two components of latitude, whereas the deviation is presented in a separate column headed, “3a latitudo”.13 A contemporary of John de Lignères, also working in Paris, John of Saxony, wrote canons to the Parisian Alfonsine Tables in 1327 in which planetary latitudes are not even mentioned. His canons were frequently copied in the 14th and 15th centuries, and published together with the editio princeps of the Alfonsine Tables (Ratdolt 1483). Despite their absence from these canons, this edition of the Alfonsine Tables has tables for the planetary latitudes but the deviation for Venus and Mercury is not taken into account. The second edition of the Alfonsine Tables, edited by J.L. Santritter in 1492 with a new set of canons, also has tables for the planetary latitudes which are the same as those in the editio princeps (i.e., in both cases, the arguments are given at intervals of 6°, and the entries are the same), differing only in presentation.14 In his canons, chapters 25 and 26 (for Venus and Mercury, respectively), Santritter discussed planetary latitudes with instructions for computing the deviation (que proveniet ex deviatione deferentis ab ecliptica) for the inferior planets. It turns out that these instructions for computing the deviation were copied from the canons by John of Lignères (1322), Priores astrologi motus corporum, chapters 22 and 23, almost verbatim.15

13

14

15

Lisbon, Biblioteca de Ajuda, ms 52-xii-35, ff. 63r–64r. The entries for the deviation for Venus, given to minutes, are based on a maximum of +0;10°, as explained in the text of the Almagest. But those for Mercury have an extremal value of –0;23°, rather than the value in the Almagest of –0;45° (see below). See Ratdolt (ed.) 1483, f. h1v; and Santritter (ed.) 1492, ff. e5r, f1r, f5r, g1r, and g5r. Rather than keeping the components of latitude for all planets in columns in a single table, Santritter collected columns for different phenomena (such as latitude, unequal daily motion, and retrograde motion) in a separate table for each planet. See Saby 1987, pp. 209, 211. For Venus, John of Lignères says: “Postea accipe de minutis proportionalibus in altero locorum servatis 6am partem que erit latitudo Veneris tertia examinata que provenit ex deviatione deferentis ab ecliptica; et est semper hec tertia latitudo septentrionalis”, while Santritter (f. c2r) has: “Postea accipe de minutis proportionalibus in altero supra loco servatis sextam partem, que erit latitudo Veneris tertio examinata que proveniet ex deviatione deferentis ab ecliptica, est quem [read: que] semper ista latitudo septentrionalis.”

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Planetary Latitude in the Tables of Bianchini

Bianchini was a prominent astronomer, active in Ferrara, whose precise dates are not known.16 One of his earliest works, written in 1442, concerns the construction and use of a surveying instrument, and it includes an explanation of the use of the decimal point.17 In the 1460s he corresponded with Regiomontanus on problems in astronomy in which Regiomontanus indicated difficulties in Ptolemy’s treatment of various phenomena.18 Bianchini’s tables were published twice in Venice (ed. S. Beuilaqua 1495; and ed. L. Gaurico 1526),19 and both editions include planetary latitudes along with planetary longitudes in large double argument tables where the anomaly is given in days and the center is given in degrees. The number of entries for the latitudes of all 5 planets is 10,584, far greater than in the other double argument tables described above. As we found in the Oxford Tables and the Almanach Perpetuum, Bianchini’s entries for the planetary latitudes are based on Ptolemy’s tables in Almagest xiii.5 (with slightly different parameters), and take into account the deviation for the inferior planets. We now turn our attention to Bianchini’s set of auxiliary tables for computing planetary latitudes in the ed. of 1526 that was not included in the editio princeps of 1495, despite the fact that these tables appear in manuscript copies. This set consists of two tables, one for the superior planets and one for the inferior planets. These two tables have not been studied previously which is somewhat surprising since they appear in a manuscript copied by Copernicus that was printed by Curtze in 1875 and by Prowe in 1884.20 However, this copy was not identified with the tables of Bianchini until 1984 by Rosińska.21 A brief description of Bianchini’s tables, with notes on the manuscripts in Cracow, was also published by Rosińska, but her goal was to catalogue the tables in these manuscripts, not to analyze their mathematical structure.22 Bianchini’s first table is for the superior planets and has 10 columns (see Table 1). The description that follows is based on Cracow, Biblioteka Jagiel-

16 17 18 19 20

21 22

Federici Vescovini 1968; Zinner 1990, p. 37. Rosińska 1996, p. 57. Swerdlow 1990. There was a third edition, edited by N. Pruckner (Basel, 1553), but we have not seen it. Curtze 1875, pp. 230–238; Prowe 1883–1884, 2:231–240. The transcription of Copernicus’s canon for the latitudes of Venus and Mercury appears in Curtze on p. 238, and in Prowe on pp. 239–240. Cf. Swerdlow and Neugebauer 1984, p. 524, n. 44. Rosińska 1984b. Rosińska 1984a, pp. 485–486.

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lońska, ms 555 (ff. 237–240), a manuscript dating from about 1453 and copied in Perugia by a scholar from Cracow:23 col. 1: argument at 1°-intervals, from 1° to 360°; col. 2: minutes of proportion for Saturn in minutes and seconds; col. 3: northern latitude for Saturn in degrees and minutes, from 2;2° (for argument 360°, or 0°) to 3;4° (for argument 180°); col. 4: southern latitude for Saturn in degrees and minutes, from 2;1° (for argument 0°) to 3;5° (for argument 180°); col. 5: minutes of proportion for Jupiter in minutes and seconds; col. 6: northern latitude for Jupiter in degrees and minutes, from 1;6° (for argument 0°) to 2;5° (for argument 180°); col. 7: southern latitude for Jupiter in degrees and minutes, from 1;4° (for argument 0°) to 2;8° (for argument 180°); col. 8: minutes of proportion for Mars in minutes and seconds; col. 9: northern latitude for Mars in degrees and minutes, from 0;5° (for argument 0°) to 4;21° (for argument 180°); col. 10: southern latitude for Mars in degrees and minutes, from 0;2° (for argument 0°) to 7;7° (for argument 180°). Note that the northern or southern latitudes are functions of the true anomaly, whereas the minutes of proportion are functions of the true center. Table 1 displays an excerpt at 10°-intervals from 0° to 180° of Bianchini’s table for the superior planets. According to Rosińska, this table is found in ten manuscripts in Cracow.24 In the printed edition of 1526, Table 1 is separated into three tables: the first is for Saturn (324r–326v) and displays, in this order, columns 1, 2, 3, and 4; the second is for Jupiter (327r–329v) and displays, in this order, columns 1, 5, 6, and 7; the third is for Mars (330r–332v) and displays, in this order, columns 1, 8, 9, and 10. The entries in columns 3, 4, 6, 7, 9, and 10 (for the northern and southern latitudes of the three planets) follow very closely the corresponding values in Almagest xiii.5. The three columns for the minutes of proportion (columns 2, 5, and 8) in fact are the same (but for shifts: see below), and they are based on the column in Almagest xiii.5 for the minutes of proportion that applies to the latitude for all planets.25 This particular column in Almagest xiii.5 is usually

23 24 25

Rosińska 1984b, p. 644. Henceforth bj, ms 555. Rosińska 1984a, p. 485. See Toomer 1984, pp. 632–634.

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ptolemy, bianchini, and copernicus table 1

Bianchini’s table for the planetary latitudes of the superior planets

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

0* 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

38;24 30 0 20 24 10 24 0 0 10 24 20 24 30 0 38 24 45 44 52 0 56 12 58 56 60 0 58 56 56 12 52 0 45 44 38 24

2; 2 2 4 2 6 2 8 2 11 2 13 2 16 2 20 2 25 2 30 2 35 2 40 2 45 2 49 2 54 2 57 3 0 3 1 3 4

2; 1 2 3 2 4 2 6 2 8 2 11 2 15 2 20 2 25 2 30 2 35 2 40 2 45 2 50 2 54 2 58 3 1 3 3 3 5

56;12 58 56 60 0 58 54 56 12 52 0 45 44 38 24 30 0 20 24 10 24 0 0 10 24 20 24 30 0 38 24 45 44 52 0 56 12

1; 6 1 8 1 8 1 10 1 12 1 13 1 16 1 20 1 25 1 30 1 35 1 40 1 45 1 49 1 53 1 58 2 1 2 3 2 5

1; 4 1 6 1 6 1 8 1 10 1 12 1 16 1 20 1 25 1 30 1 35 1 40 1 45 1 50 1 55 2 0 2 4 2 6 2 8

60; 0 58 56 56 12 52 0 45 44 38 24 30 0 20 24 10 24 0 0 10 24 20 24 30 0 38 24 45 44 52 0 56 12 58 56 60 0

0; 5 0 9 0 12 0 14 0 17 0 22 0 28 0 35 0 42 0 52 1 3 1 17 1 34 1 56 2 22 2 55 3 30 4 6 4 21

0; 2 0 4 0 6 0 8 0 11 0 16 0 22 0 28 0 38 0 49 1 2 1 16 1 37 2 3 2 41 3 29 4 39 6 0 7 7

* The entries for 0° appear in the table as entries for 360°.

referred to as c5(ω), for it is the fifth column in that table. Note that ω is the “argument of latitude” counted from the northern limit on the deferent (i.e., 90° from the nodes where the deferent crosses the ecliptic) and that no algebraic signs are associated with the coefficients c5(ω). In this paper c5(ω) will be called p(x), and Bianchini uses it for various purposes of interpolation. We note that p(x) ≈ |cos x|.26 In the instructions to compute the planetary latitudes for the superior planets (Almagest xiii.6), Ptolemy indicates that the northern limits on the deferent differ in each case from the apogees of the superior planets by +50° (Saturn), −20° (Jupiter), and 0° (Mars). For example, according to Ptolemy, the 26

Neugebauer 1975, p. 219. The entries in Ptolemy’s col. 5 are very nearly those of a cosine function; the divergence from the values of the cosine reaches a maximum of 0;0,11 (where cos 0° is taken to be 1) at about 42°.

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ascending node for Saturn is at longitude 90°, i.e., the northern limit (n) is at longitude 180°, and the apogee (a) of Saturn’s deferent is 230° (rounded from 233°).27 Hence a − n = 50°. Bianchini’s table takes these differences into account: the entries in column c2 for Saturn are shifted –50° with respect to the corresponding entries in the Almagest; column c5 for Jupiter +20°; and column c8 for Mars has no shift. Thus, if p(x) is an entry in the table in the Almagest for argument x, and ci(x) an entry in col. i in Bianchini’s table, then p(x) = c2(x − 50) = c5(x + 20) = c8(x). Bianchini’s inclusion of these shifts makes his table more “user-friendly” than Ptolemy’s since there is one less step for the user who can enter the column for interpolation with the true center in all cases. In other words, there is no longer any need to compute an argument of latitude. The second table is for the inferior planets and, as given in Cracow, Biblioteka Jagiellońska, ms 555 (ff. 241–244), it has 11 columns (see Table 2): col. 1: argument at 1°-intervals, from 1° to 360°; col. 2: inclination (declinatio) for Venus in degrees and minutes with extremal values 1;3° (for argument 360°, or 0°) and 7;22° (for argument 180°); col. 3: slant (reflexio) for Venus in degrees and minutes with extremal value 2;30° (for arguments 131°–139° and 221°–229°); col. 4: minutes of proportion for the inclination of Venus in minutes and seconds; col. 5: minutes of proportion for the slant of Venus in minutes and seconds; col. 6: deviation for Venus in minutes and seconds, ranging from +0;10,0° (for arguments 0° and 180°) to 0;0° (for arguments 90° and 270°); col. 7: inclination (declinatio) for Mercury in degrees and minutes with extremal values 1;46° (for argument 0°) and 4;4° (for argument 180°); col. 8: slant (reflexio) for Mercury in degrees and minutes with extremal value 2;30° (for arguments 112°–117° and 243°–247°); col. 9: minutes of proportion for the inclination of Mercury in minutes and seconds;

27

Neugebauer 1975, p. 208.

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col. 10: minutes of proportion for the slant of Mercury in minutes and seconds; col. 11: deviation for Mercury in minutes and seconds, ranging from –0;45,0° (for arguments 0° and 180°) to 0;0° (for arguments 90° and 270°). The inclination and slant are functions of the true anomaly, whereas the minutes of proportion and the deviation are functions of the true center. Table 2 displays an excerpt at 10°-intervals from 0° to 180° of Bianchini’s table for the inferior planets. According to Rosińska, this table is found in eight manuscripts in Cracow.28 In the printed edition of 1526, Table 2 is separated into two tables. The first is for Venus (333r–338v) and displays, in this order, columns 1, 2, 3, 4, 5, and 6; the second is for Mercury (339r–344v) and displays, in this order, columns 1, 7, 8, 9, 10, and 11. We note that there is a column for the minutes of proportion (col. 10), not found in Almagest xiii.5, but the three other columns for the minutes of proportion (columns 4, 5, and 9) are the same (but for shifts) as that for the minutes of proportion in the planetary latitude tables of Almagest xiii.5, as was the case for the superior planets. Nevertheless, entries in columns c4 and c9 are shifted +90° with respect to the corresponding entries in the Almagest. Thus, if p(x) is an entry in the table in the Almagest, then p(x) = c4(x + 90) = c5(x) = c9(x + 90). Therefore, column 5 in Table 2 is identical to column 8 in Table 1, and both are the same as the column for the minutes of proportion in Almagest xiii.5. Again, the shifts are introduced to facilitate computation. In fact, this is also the reason for repeating the same column (but for shifts) for each planet. The entries in column 10, the minutes of proportion for the slant of Mercury, decrease from 54;0′ (at 0°) to 0;0′ (at 90°), increase to 66;0′ (at 180°), decrease to 0;0′ (at 270°), and increase back to 54;0′ (at 360°). There is no indication of the algebraic sign, and they may all be taken to be positive. The extremal values for this correction are indeed 1⁄10 more and 1⁄10 less than 60 minutes, which corresponds to the instructions given by Ptolemy in Almagest xiii.6 (but there is no corresponding table in the Almagest). It is readily seen that the entries in col. 10 can be computed from p(x), as follows: c10(x) =

28

9 10

c5(x) for 0° ≤ x ≤ 90°, 270° ≤ x ≤ 360°

Rosińska 1984a, p. 486.

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table 2

Bianchini’s table for the planetary latitudes of the inferior planets

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

0* 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

1; 3 1 1 1 0 0 57 0 52 0 45 0 36 0 25 0 14 0 0 0 16 0 36 0 59 1 25 2 7 3 3 4 12 5 32 7 22

0; 0 0 14 0 27 0 41 0 55 1 8 1 20 1 33 1 45 1 57 2 7 2 17 2 25 2 29 2 29 2 22 2 1 1 15 0 0

0; 0 10 24 20 24 30 0 38 24 45 44 52 0 56 12 58 56 60 0 58 56 56 12 52 0 45 44 38 24 30 0 20 24 10 24 0 0

60; 0 58 56 56 12 52 0 45 40 38 24 30 0 20 24 10 24 0 0 10 24 20 24 30 0 38 24 45 44 52 0 56 12 58 56 59 0

10; 0 9 49 9 22 8 40 7 38 6 24 5 0 3 24 1 44 0 0 1 44 3 24 5 0 6 24 7 38 8 40 9 22 9 50 9 0

1;46 1 44 1 42 1 36 1 26 1 13 0 59 0 42 0 22 0 0 0 25 0 54 1 25 2 0 2 33 3 7 3 38 3 58 4 4

0; 0 0 18 0 36 0 55 1 13 1 29 1 44 1 57 2 9 2 20 2 27 2 29 2 29 2 23 2 8 1 45 1 17 0 40 0 0

0; 0 10 24 20 24 30 0 38 24 45 44 52 0 56 12 58 56 60 0 58 56 56 12 52 0 45 44 38 24 30 0 20 24 10 24 0 0

54; 0 53 2 50 36 46 48 41 10 34 33 27 0 18 22 9 22 0 0 11 26 22 26 33 0 42 15 50 18 57 12 61 50 64 50 66 0

45; 0 44 12 42 9 39 0 34 18 28 48 22 30 15 18 7 48 0 0 7 48 15 18 22 30 28 48 34 18 39 0 42 9 44 13 45 0

* The entries for 0° appear in the table as entries for 360°. c5(180). bj, ms 555, reads 59;0′ instead of 60;0′ (as in ed. 1526). c6(180). bj, ms 555, reads 9;0′ instead of 10;0′ (as in ed. 1526). c5(40). bj, ms 555, reads 45;40′ instead of 45;44′ as in c4(50), c4(130), c5(140), c9(50), and c9(130). This is just one example of miscopying in the table as it is found in this manuscript; we have not indicated the others. The number of errors in these two tables is low despite the fact that they contain more than 7,500 entries (360 × 10 + 360 × 11), most of them with two sexagesimal digits.

and c10(x) =

11 10

c5(x) for 90° ≤ x ≤ 270°.

As for the inclination and slant of the two inferior planets, the entries in columns 2, 3, 7, and 8 follow very closely the corresponding values in Almagest

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xiii.5. Now, in Table 2, columns 6 and 11 give the deviation for Venus and Mercury, respectively: the entries in col. 6 show an extremal value of +0;10,0° and those in col. 11 an extremal value of –0;45,0°.29 It is readily seen that both columns 6 and 11 can be computed from p(x), the fifth column in Almagest xiii.5 or, equivalently, from c5(x) in Table 2, where the following holds: c6(x) = +0;10 c5(x) and c11(x) = −0; 45 c5(x). In sum, all the entries in Bianchini’s Tables 1 and 2 ultimately derive from Almagest xiii.5, but Bianchini presented them differently. In particular, he introduced one column for Venus (Table 2, col. 6) and two columns for Mercury (Table 2, cols. 10 and 11) which are not found in Almagest xiii.5. It is almost impossible to determine the specific source used by Bianchini in compiling his tables for planetary latitudes because, globally, all such tables in the tradition of the Almagest share the same entries (with minor variants), except in some cases for the extremal values and entries near them. Table 3 displays the significant extremal entries in various sets of tables, all of them in the style of the Almagest (e.g. Oxford, Bodleian Library, ms Can. Misc. 27, containing tables by John of Lignères), except for Lisbon, ms Ajuda (mentioned above), and the Oxford Tables of 1348,30 which are in the form of double argument tables. A glance at this table suggests that Bianchini may have depended on a copy of the Toledan Tables. The variation in extremal values from one set of tables to another suggests more variation in the other entries than is the case. For example, in the column for the inclination of Venus in different sets of tables, most of the entries are the same. In Table 4 a sample of such entries is presented (Ptolemy tabulated the inclination of Venus at 6°-intervals from 6° to 90°, and at 3° intervals from 93° to 180°; the Alfonsine Tables [ed. 1483 and ed. 1492] at 6°-intervals from 6° to 180°; and Bianchini at 1°-intervals from 1° to 360°). Both editions of Bianchini’s tables are preceded by canons explaining their use. Chapter 34 includes short comments on planetary latitudes and worked examples for Jupiter, Venus, and Mercury. In the case of the inferior planets,

29 30

For an explanation of these parameters, see Neugebauer 1975, pp. 222–224. For these tables, we have consulted Vienna, Nationalbibliothek, ms Vin. 2440.

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table 3

Extremal values in tables for planetary latitudes

Toledan Can. Ajuda Oxford Alf. t. Almagest Tables Misc. 27 52-xii-35 Tables Bianchini 1483 Sat.

n. s. Jup. n. s. Mars n. s. Ven. Incl. Slant Mer. Incl. Slant

3; 2° 3; 5° 2; 4° 2; 8° 4;21° 7; 7° 6;22° 2;30° 4; 5° 2;30°

3; 4° 3; 5° 2; 5° a 2; 8° 4;21° 7; 7° 6;22° a 2;30° 4; 4° 2;30°

3; 2° 3; 5° 2; 5° 2; 8° 4;21° 7;30° 7;22° 2;30° 4; 5° 2;30°

3; 2° 3; 5° 2; 5° 2; 8° 4;21° 6;30° 7;22° 2;25° 4; 5° 2;44°

3; 2° 3; 4° 2; 5° 2; 8° 4;31° 7; 7° 7;22° 4; 5°

3; 4° 3; 5° 2; 5° 2; 8° 4;21° 7; 7° 7;22° b 2;30° 4; 4° 2;30°

3; 3° c 3; 5° 2; 8° 2; 8° 4;21° 7;30° 7;12° 2;30° 4; 5° 2;30°

a. Other mss read 2;4° for Jupiter and 7;22° for Venus (Pedersen 2002, p. 1326). b. bj, ms 555: 7;22; ed. 1526: 7;12. c. Santritter’s ed. (1492) reads 3;2°.

Bianchini used the terms declinatio, reflexio, and deuiatio for the three components of latitude, but we note that the edition of 1495 renders the first term as declaratio, a mistake that was corrected in the edition of 1526. We present Bianchini’s worked examples for Venus and Mercury, without following them word-for-word. In the example for Venus the true anomaly α = 258° and the true center κ = 121°. For this α, Table 2 gives c2(α) = 0;20° (inclination) and c3(α) = 2;9° (slant). For this κ, Table 2 gives c4(κ) = 51;24 (minutes of proportion for the inclination), c5(κ) = 30;52 (minutes of proportion for the slant), and c6(κ) = 0;5,9° = β3 (deviation). The three components of latitude are β1 = c2(α) · c4(κ) = –0;17,8°, β2 = c3(α) · c5(κ) = 1;6,22°, and β3 = 0;5,9°. The latitude of Venus is thus: β = β1 + β2 + β3 = 0;54,23°, in agreement with the edition of 1495 (although the edition of 1526 gives the result as 0;54,22°).31 In the example for Mercury α = 224° and κ = 189°. Table 2 gives c7(α) = 2;20° (inclination), c8(α) = 2;15° (slant), c9(κ) = 9;24 (minutes of proportion for the

31

If Bianchini had followed the instructions in the Almagest (see below), he would have multiplied the entry in col. 6 by c5(κ) and found β3 = 0;5,9° · 0;30,52 = 0;2,39°; the final result would have been β = 0;50,53°.

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Argument 6 12 18 24 30 36 42 48 … 150 153 156 159 162 165 168 171 174 177 180

Inclination of Venus

Almagest xiii.5 Bianchini Alf. t. 1483 1; 2 1; 1 1; 0 0;59 0;57 0;55 0;51 0;46

1; 2 1; 1 1; 0 0;59 0;57 0;55 0;51 0;46

1; 2 1; 1 1; 0 0;59 0;57 0;55 0;51 0;46

3; 3 3;23 3;44 4; 5 4;26 4;49 5;13 5;36 5;52 6; 7 6;22

3; 3 3;24 3;44 4; 5 4;26 4;49 5;13 5;42 6;12 6;46 7;22 a

3; 3 3;43 4;26 5;24 6;24 7;12

a. bj, ms 555: 7;22; ed. 1526: 7;12.

inclination), c10(κ) = 65;1 (minutes of proportion for the slant), and c11(κ) = –0;44,21° (deviation). The three components of latitude are β1 = c7(α) · c9(κ) = –0;22°, β2 = c8(α) · c10(κ) = –2;26°, and β3 = –0;44°. The latitude of Mercury is thus: β = β1 + β2 + β3 = –3;32°, and this is the value computed by Bianchini.32 Bianchini (as well as Vimond, John of Lignères, and Batecombe before him, Zacut at about the same time, and Santritter afterwards) assumed that the formula for the deviation is

32

Analogously, if Bianchini had followed the instructions in the Almagest (see below), he would have computed β3 = c5(κ) · c6(κ) and found β3 = –0;44°; in this case his final result would have been the same.

88 (1)

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β3 = c5(x) · d

where d = 0;10° for Venus and –0;45° for Mercury. However, Ptolemy’s model requires the formula (2) β3 = c5(x) · c5(x) · d and the instructions in Almagest xiii.6 indicate this (although Ptolemy expressed himself in a rather convoluted way): Then we take these same sixtieths which were found by the second entry with the longitude, calculate the amount which is the same fraction of them as they are of 60, and, for Venus, take 1⁄6 of this and set it out too, always with a northerly direction; but for Mercury we take ¾ of the amount and set it out, always in a southerly direction.33 The critical phrase is “calculate the amount which is the same fraction of them as they are of 60″. Ptolemy seems to indicate taking a fraction of a fraction which is the product of the fractions; in this case, the fractions being the same, the result is the square of the fraction used to compute β2 for Venus, i.e., the square of the sixtieths (c5 · c5). In the medieval Latin translation of the Almagest by Gerard of Cremona (d. 1187), based on an Arabic version (rather than the Greek original), this passage appears as follows: Deinde post illud tendemus ad hec minuta eadem etiam que invenimus mittendo longitudinem secundo, et accipiemus ex eis secundum quantitatem partis qua ipsa sunt ex sexaginta partibus, et eius quod provenerit in Venere accipiemus semper sextam et firmabimus in septentrione; et in Mercurio semper accipiemus medietatem et quartam et firmabimus in meridie.34 After this we shall attend to the same minutes which we found by entering with the longitude for the second time, and we shall take [a part] of them according to the size of the part which they themselves are of 60 parts, and of that which results in [the case of] Venus, we shall always take a

33 34

Toomer 1984, p. 636; cf. Swerdlow and Neugebauer 1984, p. 523. Liechtenstein (ed.) 1515, f. 149v. On Gerard of Cremona’s translation, see Kunitzsch 1974, pp. 83–112.

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sixth and write it down35 as northern; and in [the case of] Mercury we shall always take one half plus one quarter and put it down as southern. Since this translation is reasonably faithful to the original Greek, it cannot be the source for Bianchini’s computation of the deviation according to formula (1). In contrast to the “western” tradition, eastern Islamic astronomers understood Ptolemy’s instructions to require computations of the deviation for Venus and Mercury according to formula (2).36 The “western” tradition for computing β3, accepted by Bianchini, seems to have its origin in the zij of al-Battānī.37 In the canons to the Toledan Tables, which depend on al-Battānī in this respect, the instruction is: After this, if your computation is for Venus, take one sixth of the minutes you wrote in the other place, and this [sixth] is always northern; and if it is for Mercury, take one quarter plus a half [of them], and this is always southern.38 This provides additional evidence in support of the suggestion that Bianchini depended on the Toledan Tables and their canons although he replaced the rule given in words with a table. For those using formula (1), the coefficient c5 was always taken to be positive and, as we noted above, c5(x) ≈ |cos x|. But this function has a singularity at 35 36

37 38

Presumably the underlying Arabic word is a derived form of athbata which means “to write down” although the root means “to be firm”; see Nallino 1903–1907, 2:325, and n. 38, below. We have checked the zij al-Sanjarī (ca. 1120) by al-Khāzinī (London, British Library, ms Or. 6669, ff. 148v, 152r; and Vatican, ms Arab. 761, ff. 179v, 186r), and the zij al-Khāqānī (ca. 1420) by al-Kāshī (Istanbul, Aya Sofia ms 2692, ff. 98v, 99v; and London, India Office, ms 430, ff. 139v, 140v). In both zijes the deviation, called “first latitude”, was computed according to formula (2). Nallino 1903–1907, 1:116. F.S. Pedersen 2002, pp. 268–269: “Post haec accipe ex ipsis minutis quae scripsisti in alio loco, si fuerit numerus tuus Veneris, sextam partem eorum, et est semper septentrionalis; et si fuerit Mercurii, quartam partem [eius] et dimidiam, et est semper meridiana”; cf. p. 515 [§ 05]. Ibn al-Bannāʾ gives the same instructions: “Take a sixth of the minutes of proportion that you wrote down (athbata) in the second place, and it is the third latitude, and it is always to the north [for Venus], and keep it” (Vernet 1952, Arabic text, p. 38); and these instructions are also found, almost verbatim, in the anonymous zij preserved in Hyderabad, Andra Pradesh State Library, ms 298 (see Mestres 1999, Arabic text, p. 91: chapter 18:12).

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90° which has no physical meaning. On the other hand, the function y = cos2 x does not exhibit such a singularity. Figure 4.1 shows the behavior of the functions represented by formulas (1) and (2). Indeed, in De revolutionibus vi.8, Copernicus comments on this very point, arguing that if c5 is the coefficient for the deviation (rather than the square of c5), then for Venus and Mercury the angle between the deferent and the ecliptic would be constant (contrary to Ptolemy’s model for the inferior planets) and “this [component of] latitude (…) would suddenly leap back from the intersection [i.e., the nodal line] into the same latitude that it previously left.”39 Clearly, at the time when this passage was written Copernicus understood the behavior of the deviation properly, but his approach had been different in previous discussions of the latitude for the inferior planets.40

3

Tables for Planetary Latitude in Copernicus

Uppsala, University Library, ms Copernicana 4, contains a set of ten tables for planetary latitudes in the hand of Copernicus: ff. 276v–277v [superior planets]; 278r–279r, 280r [inferior planets]; 279v [all planets]; and 280v–281r [canons for the inferior planets only]. These tables are included in a quire bound together with the 1492 edition of the Alfonsine Tables (edited by Santritter) and Regiomontanus’s Tabulae directionum (Augsburg, 1490), and it is almost certain that they were copied at the time when Copernicus was a student in Cracow.41 Comparison with the tables of Bianchini described above shows that Copernicus copied and rearranged these tables that had been compiled by his Italian predecessor.42 A brief description of the 10 tables follows, indicating the corresponding table of Bianchini: c1. Saturn ( f. 276v) There are two sub-tables with a column for the argument at 1°-intervals, from 1°to 30°, and six columns, one for each of the zodiacal signs. One sub-table displays the entries in Bianchini’s Table 1, col. 3 (northern latitude), and the

39 40 41 42

See Swerdlow and Neugebauer 1984, p. 523; Copernicus 1543, ff. 191v–192r. Cf. Swerdlow and Neugebauer 1984, pp. 508, 536. For a description of this manuscript see Czartoryski 1978, p. 366. Note that Copernicus has arguments at 1°-intervals in agreement with Bianchini, in contrast to the 6°-intervals in the corresponding tables edited by Ratdolt (1483) and by Santritter (1492).

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figure 4.1 The functions y =| cos x| and y = cos2 x in the interval (0°, 180°), where the cosine function represents c5(x) in formulas (1) and (2). The maximum difference between these functions is 0;15 at 60° and 120°

other, those in Table 1, col. 4 (southern latitude). The last few entries for northern latitudes are not exactly the same as those in Bianchini’s table: the maximum value given by Copernicus is 3;3°, and not 3;4°, but this is probably due to miscopying (or a faulty archetype). c2. Jupiter ( f. 277r) There are two sub-tables presented in the same way as the corresponding table for Saturn. One sub-table displays the entries in Bianchini’s Table 1, col. 6 (northern latitude), and the other, those in Table 1, col. 7 (southern latitude). c3. Mars ( f. 277v) There are two sub-tables presented in the same way as the corresponding tables for Saturn and Jupiter. One sub-table displays the entries in Bianchini’s Table 1, col. 9 (northern latitude), and the other, those in Table 1, col. 10 (southern latitude). In the latter, the entries for the minutes corresponding to arguments 151°–180° are all mistakenly shifted downwards 1°, so that, at the end of the column, Copernicus had to write the last number outside the frame of the table. c4. Venus ( f. 278r) This table has a column for the argument at 1°-intervals, from 1° to 30°, and six double columns, one for each of the first six zodiacal signs. In each double

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column we are given the entries in Bianchini’s Table 2, columns 2 (inclination) and 3 (slant). The heading for the left side of the table is Prima pars tabule, and that for the right side, Altera pars. c5. Mercury ( f. 278v) This table is headed Tabella latitudinis Mercurii. It has a column for the argument at 1°-intervals, from 1° to 30°, and six double columns, one for each of the first six zodiacal signs. In each double column we are given the entries in Bianchini’s Table 2, columns 7 (inclination) and 8 (slant). The heading for the left side of the table is Superior pars circuli, and that for the right side, Inferior pars circuli. c6. Venus ( f. 279r) This table is headed northern deviation (deviacio borealis). It has a column for the argument at 1°-intervals, from 0° to 30°, and three columns, one for each of the first three zodiacal signs. The entries are taken from Bianchini’s Table 2, column 6 (deviation). c7. Mercury ( f. 279r) This table is headed southern deviation (deviacio australis). It has a column for the argument at 1°-intervals, from 0° to 30°, and three columns, one for each of the first three zodiacal signs. The entries are taken from Bianchini’s Table 2, column 11 (deviation). c8. Venus and Mercury ( f. 279r) This table is headed Minuta ad declinacionem. It has a column for the argument at 1°-intervals, from 0° to 30°, and three columns, one for each of the first three zodiacal signs. The entries are taken from Bianchini’s Table 2, column 4, which is identical to column 9. c9. The Five Planets ( f. 279v) This table is headed Tabella M(inutorum) proporcionabi(lium) 5 planeta(rum). It displays the minutes of proportion, and it has a column for the argument at 1°-intervals, from 0° to 30°, and three columns, one for each of the first three zodiacal signs. The entries are taken from Bianchini’s Table 1, column 8, which is identical to Table 2, column 5. Part of the column for 0s/6s and 5s/11s has been crossed out, and the first 18 entries have been accurately copied in the left margin. Note that Copernicus correctly gives the entry for 40° (45;44′) rather than 45;40′ (see Table 2).

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c10. Mercury ( f. 280r) This table is headed M(inuta) prop. ad reflectionem, and it has a column for the argument at 1°-intervals, from 0° to 30°, and six columns, one for each of the first six zodiacal signs. The entries are taken from Bianchini’s Table 2, column 10. The heading for the left side of the table is Prima pars ta(bule), and that for the right side, Altera pars. In his set of 10 tables, Copernicus gathered all information contained in Bianchini’s Tables 1 and 2, avoiding all unnecessary repetitions by taking advantage of the numerous symmetries they contain. This is not to say that he presented his tables in a more compact way, for Copernicus split the two tables given by Bianchini into ten, and treated the shifts differently. Moreover, the extremal values in Copernicus’s tables are the same as those of Bianchini in Table 1 (the one exception for Saturn has already been noted above). Copernicus’s canons for Venus and Mercury are not very different from those that we have discussed, beginning with the canons to the Toledan Tables. In particular, he accepts formula (1) for computing β3. Hence, it is not easy to decide whether he depended on Santritter’s canons in the edition of the Alfonsine Tables (1492) with which these tables are bound,43 or on Bianchini’s canons, or on some other set of canons. Copernicus expresses himself in his own way and so linguistic criteria are not sufficient to establish filiation. Santritter’s canons for β3 do not refer to a table; rather, they give instructions for taking fractional parts of p(x) for Venus and for Mercury (with the directions north and south, respectively), as in the canons to the Toledan Tables.44 Copernicus, however, refers to tables of the kind he displays and does not give rules with the fractions, a sixth or three-quarters. This suggests that Copernicus is closer to Bianchini than to Santritter, although it is not impossible that he relied on a different source. 43 44

Santritter (ed.) 1492, f. c2r–v. For Mercury, Santritter’s instructions (again taken, almost verbatim, from John of Lignères) suggest computing with a quarter plus a half of a quarter (i.e., 3⁄8) rather than a half and quarter (3⁄4). In this case, it would have been difficult for Copernicus (or any other reader of Santritter’s canons) to compute the third latitude of Mercury correctly without checking some other relevant text. See Santritter (ed.) 1492, f. c2v: “Et accipe de postea minutis proportionalibus in altero locorum servatis quartam partem et dimidium quarte, que pars cum suo dimidio est latitudo Mercurii tertio examinata, que est semper meridionalis.” Where Santritter has “que pars cum suo dimidio”, John of Lignères has “que 4a pars cum suo dimidio” (Saby 1987, p. 211). Note that John of Lignères’s instruction for using the coefficient 3⁄8 (= 0;22,30) is consistent with the extremal value in his table for the deviation of Mercury: –0;23° (see note 13, above).

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In contrast to these student notes, Copernicus introduced a new theory for planetary latitudes in his Commentariolus (ca. 1512), and modified it in his De revolutionibus (1543). His aim was to transform Ptolemy’s geocentric models into heliocentric models, but his success was far from complete.45 There are no tables in the early work, but among the tables for planetary latitude in De revolutionibus is a column that displays p(x)2 for the deviation.46 Despite the misunderstanding of the deviation for Venus and Mercury by medieval and early modern astronomers, Georg Peurbach (d. 1461) correctly described the models in the Almagest for the latitudes of both superior and inferior planets, including the deviation: see his Theoricae novae planetarum (first printed by Regiomontanus in Nuremberg in 1472). But this work does not contain any tables.47

4

Conclusion

In the West the deviation for Venus and Mercury was generally not included in tables for planetary latitudes and it was probably ignored by many users of those tables as well. Bianchini was one of a very small number of astronomers working in the Latin tradition of the Almagest who called attention to this third component of latitude for Venus and Mercury and produced tables for it. Both the zij of al-Battānī and the canons to the Toledan Tables give instructions for computing the deviation, but no tables accompany them. In the 13th century the Castilian canons to the Alfonsine Tables mention the deviation, but the tables associated with these canons are not extant. In the 14th century we have identified John Vimond as having compiled tables for the deviation. John of Lignères also compiled tables for the deviation but, as we noted, with an idiosyncratic extremal value for Mercury; he also included instructions for computing the deviation in his Priores astrologi but these instructions do not refer to a table. In the 15th century, Peurbach described Ptolemy’s model for the latitudes of Venus and Mercury correctly although he did not produce any tables for this purpose. Moreover, Santritter gave instructions for computing the deviation for Venus and Mercury, but did not display any table for it. We have seen that prior to 1500 Ptolemy’s instructions for computing the

45 46 47

See Swerdlow 1973, pp. 482–489, 494–499, 509–510; and Swerdlow and Neugebauer 1984, pp. 483–537, espec. pp. 535–537. Copernicus 1543, ff. 194v–195r. See Swerdlow and Neugebauer 1984, pp. 530–535. Peurbach 1472, reprinted in Schmeidler (ed.) 1972, pp. 783–787; Aiton 1987, pp. 34–35.

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deviation were usually misunderstood in the West, and computations based, e.g., on Bianchini’s worked examples would yield slightly incorrect results. In his student days Copernicus accepted a table based on formula (1) for the deviation, but in his magnum opus of 1543 he argued that the previous interpretation of Ptolemy’s instructions (which he once shared) “by earlier astronomers” was faulty, and that one needed to invoke formula (2) for the deviation of Venus and Mercury. Given the medieval tradition to which he adhered, Bianchini compiled a set of “user-friendly” tables that simplified the computations required for using the Alfonsine Tables. Many astronomers in the late Middle Ages (e.g., John of Lignères, William Batecombe, John of Gmunden, and Abraham Zacut) also had this as a goal (each interpreting it in his own way), and Bianchini fits nicely in this group. Finally, there can be little doubt that early in his career Copernicus depended on Bianchini’s tables for planetary latitude which, in turn, are based on Ptolemy’s models in the Almagest. Hence, Bianchini’s tables can be considered a source for Copernicus’s knowledge of astronomy.

Acknowledgments We are grateful to Montserrat Lamarca and Neus Verger (Biblioteca de la Universitat de Barcelona, Spain) for providing us with a copy of the Tables of Bianchini (ed. 1526); to the Biblioteka Jagiellońska (Cracow) for a partial copy of ms 555; to Noel M. Swerdlow for a partial copy of Uppsala, University Library, ms Copernicana 4; and to Benno van Dalen for partial copies of the Sanjarī zij by al-Khāzinī and the Khāqānī zij by al-Kāshī. We also thank Fritz S. Pedersen for supplying us with the text and a preliminary translation of the passage in the Latin version of the Almagest (ed. 1515), Julio Samsó for sending us a partial copy of Ibn ʿAzzūz’s zij, and Alan C. Bowen for comments on the instructions for computing the deviation by John of Lignères and J.L. Santritter.

References Aiton, E.J. 1987. “Peurbach’s Theoricae novae planetarum: A Translation with Commentary”, Osiris, 3:5–44. Beuilaqua, S. (ed.) 1495. Tabularum Joannis Blanchini Venice: Augustinus Morauus. Bianchini. See Beuilaqua 1495; Gaurico 1526; and Pruckner 1553. Boffito, G. 1908. “Le Tavole astronomiche di Giovanni Bianchini”, La Bibliofilia, 9:378– 388, 446–460.

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Casulleras, J. and J. Samsó (eds.) 1996. From Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet. Barcelona. Chabás, J. 1996. “Astronomía andalusí en Cataluña: Las Tablas de Barcelona”, in J. Casulleras and J. Samsó (eds.) 1996, pp. 477–525. Chabás, J. and B.R. Goldstein 1994. “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn alKammād”, Archive for History of Exact Sciences, 48:1–41. Chabás, J. and B.R. Goldstein 2000. Astronomy in the Iberian Peninsula: Abraham Zacut and the Transition from Manuscript to Print. Transactions of the American Philosophical Society, 90.2. Philadelphia. Chabás, J. and B.R. Goldstein 2003a. The Alfonsine Table of Toledo. Dordrecht. Chabás, J. and B.R. Goldstein 2003b. “John Vimond and the Alfonsine Trepidation Model”, Journal for the History of Astronomy, 34:163–170. Chabás, J. and B.R. Goldstein 2004. “Early Alfonsine Astronomy in Paris: The Astronomical Tables of John Vimond (1320)”, Suhayl, 4:207–294. [See essay 8, below] Copernicus, N. 1543. De revolutionibus. Nuremberg. Curtze, M. 1875. “Reliquiae Copernicanae”, Zeitschrift für Mathematik und Physik, 20: 221–248. Czartoryski, P. 1978. “The Library of Copernicus”, Studia Copernicana, 16:355–396. Dalen, B. van 1999. “Tables of Planetary Latitude in the ‘Huihui li’: Part ii”, in Yung Sik Kim and Francesca Bray (eds.), Current Perspectives in the History of Science in East Asia. Seoul, pp. 316–329. Federici Vescovini, G. 1968. “Bianchini, Giovanni”, Dizionario Biografico degli Italiani, 10:194– 196. Gaurico, L. (ed.) 1526. Tabule Joa. Blanchini bononiensis … Venice: Lucas Antonius Giunta. Kennedy, E.S. 1956. A Survey of Islamic Astronomical Tables. Transactions of the American Philosophical Society, 46.2. Philadelphia. Kennedy, E.S. 1998. On the Contents and Significance of the Khāqānī Zīj by Jamshīd Ghiyāth al-Dīn al-Kāshī. Frankfurt. Kennedy E.S. and W. Ukashah 1969. “Al-Khwārizmī’s Planetary Latitude Tables”, Centaurus, 14: 86–96. Reprinted in King and Kennedy (eds.) 1983, pp. 125–135. King, D.A. and M.H. Kennedy (eds.) 1983. Studies in the Islamic Exact Sciences by E.S. Kennedy, his colleagues, and former students. Beirut. Kunitzsch, P. 1974. Der Almagest: Die Syntaxis Mathematica des Claudius Ptolemäus in arabisch-lateinischer Überlieferung. Wiesbaden. Liechtenstein, P. (ed.) 1515. Almagestu[m] Cl. Ptolemei Pheludiensis Alexandrini astronomo[rum] principis. Venice. Mestres, A. 1999. Materials Andalusins en el Zīj d’Ibn Isḥāq al-Tūnisī. University of Barcelona (Ph.D. dissertation). Nallino, C.A. 1903–1907. Al-Battānī sive Albatenii Opus Astronomicum, 2 vols. Milan.

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Nauta, L. and A. Vanderjagt (eds.) 1999. Between Demonstration and Imagination: Essays in the History of Science and Philosophy Presented to John D. North. Leiden. Neugebauer, O. 1962. The Astronomical Tables of al-Khwārizmī. Copenhagen. Neugebauer, O. 1975. A History of Ancient Mathematical Astronomy. Berlin. North, J.D. 1977. “The Alfonsine Tables in England”, in Y. Maeyama and W.G. Salzer, (eds.), Prismata: Festschrift für Willy Hartner. Wiesbaden, pp. 269–301. Reprinted in North 1989, pp. 327–359. North, J.D. 1989. Stars, minds and fate: Essays in ancient and medieval cosmology. London and Ronceverte. Pedersen, F.S. 2002. The Toledan Tables: A review of the manuscripts and the textual versions with an edition. Copenhagen. Pedersen, O. 1974. A Survey of the Almagest. Odense. Peurbach, G. 1472. Theoricae novae planetarum. Nuremberg. Reprinted in Schmeidler (ed.) 1972, pp. 755–793. Porres, B. 2003. Les tables astronomiques de Jean de Gmunden: édition et étude comparative. Paris: École pratique des hautes études (Ph.D. dissertation). Prowe, L. 1883–1884. Nicolaus Coppernicus, 2 vols. (v. 1 is in two parts). Berlin. Reprinted Osnabrück 1967. Pruckner, N. (ed.) 1553. Luminarium atque planetarum motuum tabulae octoginta quinque … Basel. Ratdolt, E. (ed.). 1483. Tabulae astronomice illustrissimi Alfontij regis castelle. Venice. Richter-Bernburg, L. 1987. “Ṣāʿid, the Toledan Tables, and Andalusī Science”, Annals of the New York Academy of Sciences, 500:373–401. Riddell, R.C. 1978. “The Latitudes of Venus and Mercury in the Almagest”, Archive for History of Exact Sciences, 19:95–111. Rosińska, G. 1984a. Scientific Writings and Astronomical Tables in Cracow: A Census of Manuscript Sources (xivth–xvith Centuries). Wrocław. Rosińska, G. 1984b. “Identification of Copernican Tables of the Latitudes of the Planets”, Qwartalnik Historii Nauki i Techniki, 29:637–644 [in Polish with an English summary]. Rosińska, G. 1996. “The ‘Fifteenth-Century Roots’ of Modern Mathematics”, Qwartalnik Historii Nauki i Techniki, 41:53–70. Saby, M.-M. 1987. Les canons de Jean de Lignères sur les tables astronomiques de 1321. Unpublished thesis: École Nationale des Chartes, Paris. 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. Samsó, J. 1997. “Andalusian Astronomy in 14th Century Fez: al-Zīj al-Muwāfiq of Ibn ʿAzzūz al-Qusanṭīnī”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, 11:73–110. Samsó, J. 1999. “Horoscopes and History: Ibn ʿAzzūz and His Retrospective Horoscopes

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Related to the Battle of El Salado (1340)”, in L. Nauta and A. Vanderjagt (eds.) 1999, pp. 101–124. Santritter, J.L. (ed.) 1492. Tabule Astronomice Alfonsi Regis. Venice. Schmeidler, F. (ed.) 1972. Joannis Regiomontani Opera Collectanea, facsimiles of nine works by Regiomontanus and one by Peurbach that was printed by Regiomontanus. Osnabrück. Stahlman, W.D. 1959. The Astronomical Tables of Codex Vaticanus Graecus 1291. Providence: Brown University (Ph.D. dissertation). University Microfilms, No. 62–5761. Suter, H. 1914. Die astronomischen Tafeln des Muḥammad ibn Mūsā al-Khwārizmī. Copenhagen. Swerdlow, N.M. 1973. “The Derivation and First Draft of Copernicus’s Planetary Theory: A Translation of the Commentariolus with Commentary”, Proceedings of the American Philosophical Society, 117:423–512. Swerdlow, N.M. 1990, “Regiomontanus on the critical problems of astronomy”, in T.H. Levere and W.R. Shea (eds.), Essays on Galileo and the History of Science in Honour of Stillman Drake. Dordrecht. Swerdlow, N.M. 2005. “Ptolemy’s Theories of the Latitude of the Planets in the Almagest, Handy Tables, and Planetary Hypotheses”, in J. Buchwald and A. Franklin (eds.), Wrong for the Right Reasons. Dordrecht, pp. 41–71. Swerdlow, N.M. and O. Neugebauer 1984. Mathematical Astronomy in Copernicus’s De Revolutionibus. New York. Thorndike, L. 1950. “Giovanni Bianchini in Paris Manuscripts”, Scripta Mathematica 16:5–12, 169–180. Thorndike, L. 1953. “Giovanni Bianchini in Italian Manuscripts”, Scripta Mathematica 19:5–17. Tichenor, M.J. 1967. “Late Medieval Two-Argument Tables for Planetary Longitudes”, Journal of Near Eastern Studies, 26:126–128. Reprinted in King and Kennedy (eds.) 1983, pp. 122–124. Toomer, G.J. 1968. “A Survey of the Toledan Tables”, Osiris, 15:5–174. Toomer, G.J. 1984. Ptolemy’s Almagest. New York. Vernet, J. 1952. Contribución al estudio de la labor astronómica de Ibn al-Bannāʾ. Tetuán. Vernet, J. 1956. “Las Tabulae Probatae”, in Homenaje a Millás-Vallicrosa. Barcelona, pp. 501–522. Reprinted in Vernet 1979, pp. 191–212. Vernet 1979. Estudios sobre historia de la ciencia medieval. Barcelona–Bellaterra. Zinner, E. 1990. Regiomontanus, his life and work. Translated by E. Brown. Amsterdam.

chapter 5

Displaced Tables in Latin: The Tables for the Seven Planets for 1340*

… The displaced tables are typical of a pervasive tendency in Islamic science to provide extensive and elegant numerical tables for the convenience of practitioners. The underlying astronomical theory is neither questioned nor affected.1 edward s. kennedy

∵ Introduction In 2012 we published A Survey of European Astronomical Tables in the Late Middle Ages in which we discussed a wide range of tables, many of which were previously known. In this monograph we mentioned an unusual set of tables, whose significance had not been appreciated hitherto, that depends on a principle of displacement to eliminate subtractions in the course of computing planetary positions. This principle was employed in some zijes in the Islamic world, but until now there was nothing comparable in the Latin West. As we will see, this set of tables is a most ingenious reworking of the Parisian Alfonsine Tables, rather than a translation of an Arabic zij (where the term zij is used in Arabic to refer to a set of astronomical tables with instructions for their use). What makes this set of tables so unusual is that, for example, the mean planetary motions are defined differently from those in the standard Alfonsine tables, and some of the functions for computing true planetary longitudes from their mean motions also differ noticeably from those in the standard

* Archive for History of Exact Sciences 67 (2013), 1–42, communicated by George Saliba. 1 Kennedy 1977, p. 16.

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Alfonsine tables; nevertheless, computation with these displaced tables yield the same results as computation with the standard Alfonsine tables. Moreover, the tables for first station are presented in a completely different way from those we have seen in the Ptolemaic tradition, although the underlying parameters are unchanged. Given the absence of instructions, it was not an easy task to unravel the cleverness of the construction of these tables. We had some helpful guidance from a paper by Kennedy (1977), but it was not sufficient for uncovering various subtleties in these tables for which there are no counterparts in the tables that Kennedy analyzed. Paris, Bibliothèque nationale de France, ms lat. 10262, is a 15th-manuscript containing two sets of astronomical tables: an anonymous set for 1340 (ff. 2r– 46v); and another set, called Tabule frequentine (ff. 47v–71r), composed by Melchion de Friquento of Naples in 1438 (f. 46v: 1437 completus).2 Neither set has previously been examined. In this paper we focus on the first set, entirely composed of tables with no accompanying text. At the bottom of the last table (f. 46v), we read: Expliciunt tabule de septem planetis et de veris locis eius (Here end the tables for the seven planets and their true positions). Therefore, we shall refer to this set as the Tables for the Seven Planets (where the Sun and the Moon are considered planets). The yearly tables begin in March, and we are told that the tables are valid for 1340 completus, or 1340 (complete), meaning that the epoch is noon of February 28, 1341, the last day of the year, counting from March 1, 1340.3 The name of the author of these tables is not given in the manuscript, and no locality is mentioned. However, on f. 9r, at the bottom of a table for the mean motion of the Sun, we are told that the radix for the Sun, anno domini 1340 completo, is 8s 16;41,49° (= 256;41,49°). In contrast to the use of physical signs of 60° in the Parisian Alfonsine Tables, we note the use here of zodiacal signs of 30°. If we compute the mean motion of the Sun using the Parisian Alfonsine Tables for Toledo, we find its mean longitude to be 346;20,53° and its mean argument of center, 256;43,54°.4 Therefore, the given radix refers to the mean

2 For a description of this manuscript, see Thorndike 1957, especially pp. 144–145. On the Tabule frequentine, see Kremer (forthcoming). 3 The last day of February belongs to 1340 according to the convention of this text (in fact, it is the last day of that year), but to 1341 according to our modern convention (it is the 59th day of this year). The same convention applies to all subsequent years in this text. Note also that the common astronomical practice in the Middle Ages was to take noon as the beginning of the day. 4 According to this computation, 89;36,59° is the longitude of the solar apogee, λa, which is the difference between the solar longitude and the solar mean argument of center at a given time.

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argument of center of the Sun. The difference in the mean argument of center, 0;2,5°, is the amount traveled by the Sun in about 0;50h, which corresponds to a difference in geographical longitude of about 12;40° east of Toledo. This figure agrees, although not exactly, with the longitude of the meridian of Paris, taken as 12;0° east of Toledo in many astronomical tables of the 14th and 15th centuries.5 The Tables for the Seven Planets give multiple examples of displaced tables. A table is said to be displaced with respect to another when its entries are the same as those in the standard table after adding a constant to its argument (horizontal displacement), or derive from the entries in the standard table by adding a constant (vertical displacement). This can also be expressed in algebraic terms: if y = f(x) is the function underlying a given table, then the function embedded in the displaced table is y = f(x + kh), for a displacement on the x-axis, or y = f(x) + kv, for a displacement on the y-axis. Of course, both displacements can occur at the same time, leading to an equation such as (1)

y = f(x + kh) + kv.

The purpose of displaced tables is to avoid subtractions, that is, the use of complicated rules for handling negative numbers before they were available to astronomers who computed planetary positions by means of astronomical tables.6 This problem was already felt by astronomers in the 9th century, who called the standard table aṣlī and the displaced table waḍʿī.7 Displaced tables were clever computational devices, with implications for the method of computation of astronomical quantities, but did not challenge either the parameters or the models on which the original tables were based. Kennedy (1977) presented an explanation of displaced planetary tables in the medieval Islamic world, and demonstrated the equivalence of computations using one such set with computations using Ptolemy’s tables. Kennedy focused on the tables of

5 See, e.g., Kremer and Dobrzycki 1998. 6 It is important to distinguish between displaced tables and shifted tables. A shifted table contains the same information as the standard table (the same columns and rows, and the same entries) but the first row is not that for argument 0° or 1° but for some other convenient number, to stress the fact that the entry reaches a specific value such as a maximum or a minimum for that particular value of the argument. Shifts occur, for instance, for the planetary latitudes of certain planets in John of Murs’s Tables of 1321 (Chabás and Goldstein 2009, see esp. pp. 308–309). 7 Salam and Kennedy 1967, p. 497. See also Debarnot 1987, p. 43 (table 9); and Jensen 1971.

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Ibn al-Aclam (d. 985) and called attention to the fundamental relation of three displacements for each planet (see eq. 26, below).8 Byzantine astronomers apparently depended on their Muslim predecessors for displaced tables: see Tihon 1977–1981, espec. 68:76 and 110. As far as we can determine, none of the displaced tables compiled by astronomers in the Islamic World and Byzantium were known in the West. Two Jewish astronomers in Provence, Levi ben Gerson (d. 1344) and Immanuel ben Jacob Bonfils of Tarascon (fl. 1350), composed zijes in Hebrew in which they used the principle of displaced tables for the times and longitudes of syzygies (but not for the motion of the planets); however, there is no evidence to suggest that they depended on Islamic or Byzantine sources.9 In astronomy written in Latin the tables described here are the first to use displaced tables systematically, although we know of an earlier use of this type of table by John Vimond around 1320, limited to his table for trepidation (or “access and recess”).10 As mentioned above, the Tables for the Seven Planets have no accompanying text; therefore, the following comments are based entirely on the information provided by the tables themselves. As will be seen, the terminology in these tables often differs from that commonly used at the time, and many of the tables have a presentation that diverges from other tables with the same parameters that are based on the same model. There follows a table of contents, arranged by section number. 1. 2. 3. 4. 5. 6. 7.

Multiplication table Mean motion of the Sun Solar equation Length of daylight, diurnal seasonal hours, and the equation of time Mean motions of the Moon Lunar latitude Lunar equations

8

Displaced tables in Islamic astronomy are also discussed in Jensen 1971, Saliba 1976, Saliba 1977, Mercier 1989, and Van Brummelen 1998. For displaced tables in the Maghrib, see Samsó and Millás 1998, and Samsó 2003. For Levi ben Gerson, see Goldstein 1974, pp. 136–146, 229–241; for Bonfils, see Solon 1970, pp. 3–4, 11, and Kremer (forthcoming). It is noteworthy that the Tabule frequentine in this very same manuscript contain an adaptation of Bonfils’s tables, where displaced tables occur. On Vimond, see Chabás and Goldstein 2003, pp. 275–277; and Chabás and Goldstein 2004, pp. 265–267.

9

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8. 9. 10. 11.

12. 13. 14. 15. 16.

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Lunar node Precession/trepidation Planetary mean motions Planetary equations and stations 11.1. Equation of center and equatio porcionis 11.2. Equation of anomaly near greatest distance 11.3. Equation of anomaly near least distance Latitudes of the superior planets Planetary visibility Possibility of an eclipse Eclipsed fraction of the solar and lunar disks Latitudes of Venus and Mercury.

Displacements are discussed in §§3, 5, 7, 9, 10, and 11.

1

Multiplication Table

The first table in this manuscript (ff. 2r–7v) is a multiplication table for base 60 arithmetic, with entries for each integer from 1 to 60. This table, presented as a 60 × 60 square matrix, is quite common in manuscripts containing sets of astronomical tables (see, e. g., Chabás and Goldstein 2012, p. 227). The fact that f. 8r–v is blank makes it uncertain if this multiplication table belongs to the set examined here, for it could very well have been inserted by the copyist to facilitate computation.

2

Mean Motion of the Sun

Folio 9r contains five tables for the mean motion of the Sun. The first lists the Radices ad 32 annos post annum 1340, i.e., the values of the mean motion of the Sun at the beginning of each year for a period of 32 consecutive years. The entry for year 1 is 8s 16;26,51°, and it corresponds to the mean argument of center of the Sun at noon of February 28, 1342; the entry for year 32 is 8s 16;36,12°. It should be emphasized that a period of 32 years for the motion of the Sun is very unusual; in any case, we have not found it in the previous literature, where periods of 20, 24, and 28 abound (see Chabás and Goldstein 2012, p. 54). We also note that the usual phrase anni expansi (expanded years, that is, those within a cycle, in contrast to anni collecti for the years at the interval of a cycle) does not appear here or in the headings of the other tables.

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The other four tables for the Sun have a general title, beginning Tabula porcionis solis ad annos radicum …, where porcio has to be understood here as “argument of center”.11 The second table on this folio gives the mean motion of the Sun for accumulated months; its first entry corresponds to March (1s 0;33,18°) and the last one to February (11s 29;45,39°). These entries agree exactly with those given by John of Lignères in his Tabule magne for a year beginning in January (see Chabás and Goldstein 2012, Table 5.1b, pp. 55–56), who also used zodiacal signs of 30°. The third table is headed ad annos perpetuacionis, from 1372 to 1852, at 32-year intervals, and again we see the use of non-standard terminology, for the phrase anni collecti is totally absent from this set of tables. The fourth and fifth tables are, respectively, for the mean motion of the Sun for days from 1 to 31, and for hours and fractions of an hour from 1 to 60. The entry for argument 1d is 0s 0;59,8°. As mentioned above, at the bottom of the folio we are told that the Radix porcionis solis ad anno domini 1340 completo is 8s 16;41,49°. From this value and the entry for year 32 (8s 16;36,12°) we derive a mean motion in the argument of center of 0;59,8,13,32,49°/d. As indicated previously, the entries in these tables represent the mean argument of center of the Sun, κ̄, and this is another peculiar characteristic of the Tables for the Seven Planets, because in other sets the mean solar motion, that is, λ̄ , is usually tabulated. The two quantities are related by means of λa, the longitude of the solar apogee, for λ̄ = κ̄ + λa. A small table, found later in the text (f. 17r), gives the motion of the solar apogee. It is entitled Motus augium equatus and the entries are yearly values, probably corrected for precession. The entry for 32 year is 0;19,42,35°, which implies a daily motion of 0;0,0,6,4°/d. By adding this value to the one derived for the mean motion in the argument of center of the Sun, the result is 0;59,8,19,37°/d, in good agreement both with the value used by Vimond (0;59,8,19,37,4°/d) and the value in the Parisian Alfonsine Tables (0;59,8,19,37,19,13,56°/d) for the mean

11

We know of no prior usage of porcio for the argument of center of the Sun. There are, however, examples where it means “anomaly” in the case of the planets. On porcio (or portio) in the sense of “anomaly”, see Goldstein, Chabás, and Mancha 1994, pp. 63–64 (see also Nallino 1903–1907, 2:328), where a reference is given to John of Lignères’s use of this word in the canons to his tables, whose incipit is Priores astrologi motus corporum celestium (1322). The Arabic for anomaly is ḥiṣṣa, which means “portion” and one of the Hebrew terms for anomaly, manah, also means “portion”. Both the Latin and Hebrew terms apparently come from Arabic. The fact that in our text the argument of center of the Sun is considered the “anomaly” indicates that the author had in mind an epicyclic model for the Sun, which is equivalent to the eccentric model according to Apollonius’s theorem (see Almagest iii.3; Toomer 1984, pp. 141–153).

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solar motion in longitude.12 It would thus seem that the Tables for the Seven Planets belong to this tradition. We will show that the rest of the tables provide additional evidence of this relationship.

3

Solar Equation

The solar equation is given on ff. 9v–10r. The title of the corresponding table (Tabula equacionum solis cum auge eius equata) indicates that the value for the solar apogee has already been taken into consideration. The argument, which we call k̄ , in parallel with the standard usage of κ̄,13 begins at 0°, not at 1° as was the common practice at the time. The entry for 0° is 2s 29;37,9° (= 89;37,9°), and corresponds to the longitude of the solar apogee at epoch, 89;36,59° (see n. 4). The minimum 2s 27;27,9° is reached at 92°–94°, and the maximum 3s 1;47,9° at 267°–269°. Subtracting algebraically either of these two values from the entry for 0°, one gets 2;10,0°, which is the standard Alfonsine parameter for the maximum solar equation.14 Therefore, all entries for the solar equation, c(k̄ ), are displaced upwards by 89;37,9°, compared to the entries in the standard tables in the Alfonsine corpus, c(κ̄), to make all entries positive, thus allowing the user to avoid dealing with a set of complicated rules for adding and subtracting various terms: (2) c(k̄ ) = c(κ̄) + 89;37,9°. This is the first example of a displaced table in this set, where the argument is the mean argument of center, k̄ , and the entries represent c(k̄ ), which is the sum of the Alfonsine solar equation (whether positive or negative), c(κ̄), and the longitude of the solar apogee, λa. In this case the displacement is vertical (see Figure 5.a).

12

13 14

For Vimond, see Chabás and Goldstein 2004, p. 221; for the Parisian Alfonsine Tables, see Ratdolt 1483, f. d5r. For a modern edition of these tables, based on Ratdolt 1483, see Poulle 1984. Throughout this paper capital letters are used to represent variables and functions in the Tables of the Seven Planets. As in the tables of John Vimond (Chabás and Goldstein 2004, p. 223), the solar equation— with the same maximum of 2;10°—is not explicitly given. For the maximum solar equation, 2;10°, in the Parisian Alfonsine Tables, see Ratdolt 1483, f. e3r.

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figure 5.a The solar equation displaced vertically

The canons to this table, which are not extant, should have had an instruction indicating something like “to find the true position of the Sun, enter the table with the mean argument and add what you find to it” or, in algebraic terms, (3) l = k̄ + c(k̄ ). And, indeed, in Ptolemy’s model for the equation of the Sun, λ, the true longitude of the Sun is obtained adding its mean longitude, λ̄ , to the solar equation: λ = λ̄ + c(κ̄). Now the solar equation is also the difference between the true argument of center of the Sun, κ, and its mean argument of center, κ̄: it is negative if 0° ≤ κ̄ ≤ 180°, and positive if 180° ≤ κ̄ ≤ 360°. The Tables of the Seven Planets yield the same result as that deduced from Ptolemy, for λ = λ̄ + c(κ̄) = (κ̄ + λa) + c(κ̄) = κ̄ + (λa + c(κ̄)) = k̄ + c(k̄ ) = l, provided that k̄ = κ̄, which is the case here. The two terms in L are positive and both are explicitly tabulated on ff. 9v–10r.

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Length of Daylight, Diurnal Seasonal Hours, and the Equation of Time

On ff. 10v–11v there is a table with a column for the argument from 1° to 30°, and three other columns for each of the twelve zodiacal signs, beginning with Aries. The headings are Hora equalis, tempus horarum, and equatio dierum. The first entries for Aries 1° are 12;3h, 15;4°, and 8;28 min, respectively. The entries in the first column, under Hora equalis, display the length of daylight (i.e., the time interval from sunrise to sunset) for a given locality; it reaches a maximum, 15;10h, at Gem 28° – Cnc 2°, and a minimum, 8;50h, at Sgr 28° – Cap 2°. If we consider the obliquity of the ecliptic, ε, to be 23;33°, the resulting geographical latitude, φ, for which the table is valid is φ = 42;44°, which agrees fairly well with the parallel through Toulouse (rather than with that of Paris, where the longest daylight is 16;0h). The geographical longitude (see above) and latitude that we derive from the tables do not yield good agreement with any place where astronomy was practiced in the 14th century, but a locality approximately fitting both computed coordinates is Perpignan. The entries in the second column, under tempus horarum, display the length of a diurnal seasonal hour (i.e., a twelfth of the length of daylight); it reaches a maximum, 18;57h, at Gem 27° – Cnc 3°, and a minimum, 11;3h, at Sgr 27° – Cap 3°. These two columns are mutually consistent. The third column displays the equation of time (i.e., the difference between apparent and mean time where apparent time is counted from true noon, that is, the moment that the true Sun crosses the meridian, and mean time is counted from mean noon). The argument is the solar longitude expressed in degrees and the entries are given in minutes and seconds of an hour. The extremal values are 0;21,24h (at Tau 26°), 0;11,16h (at Leo 5°), 0;31,48h (at Sco 8°), and 0;0,0h (at Aqr 20°–23°). When converting these values into time-degrees, that is, multiplying each entry by 360°/24h, we obtain 5;21°, 2;49°, 7;57°, and 0,0°, respectively, which are the characteristic values found in the equation of time ascribed to Peter of Saint Omer and used by John of Lignères, among others (see, e. g., Chabás and Goldstein 2012, pp. 37–40).

5

Mean Motions of the Moon

The mean motions of the Moon are addressed in five tables on ff. 12r–13r. In addition to columns for the various arguments, each table has three columns headed centrum lune (here meaning double elongation), porcio lune (argument

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of lunar anomaly), and medius locus lune (mean longitude of the Moon). Again, for the first two quantities this is not the standard terminology. The first table lists the radices at the beginning of each consecutive year from 1342 for a period of 32 years. The entries for year 1 are 6s 3;36,2° (double elongation), 3s 17;36,22° (anomaly), and 8s 10;16,38° (longitude). The second table gives the mean motion of the Moon for the months; the first entries, for March, are 1s 5;49,36° (double elongation), 1s 15;0,53° (anomaly), and 1s 18;28,6 (longitude). The third and fourth tables are, respectively, from 1372 to 1852, at 32-year intervals, and for hours and parts of an hour, from 1 to 60. The fifth table is for days from 1 to 31, and the first entries for argument 1d for the three variables are 0s 24;22,54° (double elongation), 0s 13;3,54° (anomaly), and 0s 13;10,35° (longitude), confirming that the first column is indeed the double elongation, for 0s 24;22,54° = 2 · (0s 13;10,35° – 0s 0;59,8°). As was the case for the Sun, at the bottom of f. 12v we are given the radices for the three variables, anno Christi 1340 perfecto: 9s 14;21,15,14° (double elongation), 0s 18;53,8,0° (anomaly), and 4s 0;53,34,57° (longitude). We note that the values are given here to thirds and that the last digit for the anomaly is “0”, indicating that the author was sensitive to accuracy. From the values of the radices and those for year 32 in the first table for the mean motions of the Moon, we derive the following values for the mean motions: 24;22,53,23,16°/d (double elongation),15 13;3,53,57,30°/d (anomaly),16 and 13;10,35,1,15°/d (longitude).17 All of them are typical parameters of the Parisian Alfonsine Tables (Ratdolt 1483, ff. d5v–d7r). We have recomputed the radices using the Parisian Alfonsine Tables for noon of February 28, 1341 for the meridian of Paris, 12° east of Toledo, that is, at 0;2d before noon in Toledo. The results are presented in Table 1. The agreement is good for the double elongation and the mean argument of center of the Sun, but much better results are obtained when recomputing the 15

16

17

The value for year 32 given in the text is 4s 15;17,11°, and the difference between this value and that of the radix is 210;55,56°. Thus the amount of the double elongation in 32 years of 365;15 days is 210;55,56° and 791 complete revolutions, leading to a daily mean motion of 24;22,53,23,16 º/d. The value for year 32 given in the text is 12s 22;28,24° (read 2s 22;28,24°), and the difference between this value and that of the radix is 63;35,6°. Thus the amount traveled by the Moon in the argument of anomaly in 32 years of 365;15 days is 63;35,6° and 424 complete revolutions, leading to a daily mean motion of 13;3,53,57,30°/d. The value for year 32 given in the text is 1s 16;35,39°, and the difference between this value and that of the radix is 285;42,4°. Thus the amount traveled by the Moon in longitude in 32 years of 365;15 days is 285;42,4° and 427 complete revolutions, leading to a daily mean motion of 13;10,35,1,15°/d.

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displaced tables in latin table 1

Recomputation for the Moon at epoch: noon, Feb. 28, 1341

Text Mean argument of center of the Sun Double elongation Mean argument of anomaly Lunar mean longitude

table 2

Computation

Text–Comp.

8s 16;41,49° 256;41,46° 0; 0, 3° 9s 14;21,15,14° 284;20,13,41° 0; 1, 1,33° 0s 18;53, 8, 0° 32; 1,34,53° –13; 8,26,53° 4s 0;53,34,57° 128;29, 2, 0° –7;35,27, 3°

More detailed recomputation for the Moon at epoch: noon, Feb. 28, 1341

Text Mean argument of center of the Sun Double elongation Mean argument of anomaly Lunar mean longitude

Computation

Text–Comp.

8s 16;41,49° 256;41,49° 0; 0, 0° 9s 14;21,15,14° 284;21,14,39° +0; 0, 0,35° 0s 18;53, 8, 0° 32; 2, 7,32° –13; 8,59,32° 4s 0;53,34,57° 128;29,34,57° –7;36, 0, 0°

radices for 0;1,57,30d before noon in Toledo, that is, for a locality 11;47,45° east of that Spanish city. The results are presented in Table 2. From Table 2 it follows that the mean argument of anomaly of the Moon tabulated here is diminished by 13;9° from that obtained from the Parisian Alfonsine Tables for a locality near the meridian of Paris, and that the mean lunar longitude is diminished by exactly 7;36°. As will be seen in § 7, the displacements applied to the two quantities correspond, on the one hand, to the maximum value of the lunar equation of center in the Parisian Alfonsine Tables (13;9°) and, on the other hand, to the sum (7;36°) of two parameters, the maximum values for the equation of anomaly (4;56°) and for the increment (2;40°). So, if we call ā the mean argument of anomaly in the Tables of the Seven Planets, and ᾱ that in the Parisian Alfonsine Tables, we have (4) ā = ᾱ – 13;9°. Similarly, if l̄ is the mean lunar longitude in the Tables of the Seven Planets, and λ̄ that in the Parisian Alfonsine Tables, we have (5) l̄ = λ̄ – 7;36°.

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chapter 5 figure 5.b Ptolemy’s second lunar model. o is the observer, d is the center of the deferent circle whose radius is dc, c is the center of the epicycle, s̄ is the direction to the mean Sun, l is the true position of the Moon, v is the direction to the vernal point, and d′ is 180° from d on a circle about o with radius od. The angle η is the elongation of the mean longitude of the Moon from the mean longitude of the Sun, angle ᾱ is the mean argument of anomaly counted from the mean apogee āe , whose position is fixed by the direction d′c, ae is the true lunar apogee, angles c3 and c are the corrections to the mean lunar longitude, and angle λ is the true longitude of the Moon.

6

Lunar Latitude

The table for the lunar latitude (f. 13r) has a maximum of 5;0,0° at 90°; it is the standard table found in many zijes, including those used by Alfonsine astronomers (see Chabás and Goldstein 2012, pp. 103–104).

7

Lunar Equations

The lunar equations are presented in three separate tables. The author, well aware of Ptolemy’s second lunar model (see Figure 5.b), has split the treatment of the lunar equations, c3 and c, according to the two independent variables involved: the double elongation, 2η, and the true argument of anomaly, α. The true longitude of the Moon is the sum of four positive terms, all of which are tabulated (see eq. 14, below), whereas in the standard Alfonsine tables the true longitude of the Moon is the algebraic sum of three terms (see eq. 8, below). Despite the differences in these procedures, we show that the results are the same. The longitude of the Moon, λ, is found by applying an equation, c, to its mean longitude, λ̄ , which is a linear function of time: λ = λ̄ + c. Moreover, the mean argument of anomaly, ᾱ, is also a linear function of time. In Ptolemy’s second lunar model (Almagest v.8), the lunar equation, c, depends on two independent variables: the true argument of anomaly, α (angle aecl), and the double elongation, 2η (twice the angular distance between the mean longitude of the Moon and the mean longitude of the Sun). In this context the lunar equation

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is a combination of functions of one variable represented by the 4th, 5th, and 6th columns of Ptolemy’s table in Almagest v.8. The vast majority of later sets of tables used Ptolemy’s approach, although some of them have slightly modified parameters or rearrangements of the order of the columns. In the Parisian Alfonsine Tables (pat) these three columns correspond respectively to the 6th, 5th, and 4th columns; in the following, we will use the conventions of pat to identify the columns ci: (6) c = c6(α) + c5(α) · c4(2η) where (7) α = ᾱ + c3(2η). In eqs. 6 and 7, c3(2η) is the equation of center, c4(2η) represent the minutes of proportion, c5(α) is called the increment, and c6(α) is the equation of anomaly. Thus, the true longitude of the Moon is given by (8) λ = λ̄ + c6(α) + c5(α) · c4(2η). The first table (ff. 13v–14r) is for the equation of center; it has a column for the argument, from 0° to 29°, and then columns for each sign, from 0s to 11s. For each sign and each degree of the argument, we are given entries for the equation of center and the minutes of proportion (c3 and c4, respectively, in pat’s arrangement, both variables depending on the double elongation). For the tabulated equation of center, the entry for 0s 0° is 13;9°. It reaches a maximum of 26;18° (twice 13;9°) at 114°–115°, and a minimum of 0;0° at 245°–246°. So the equation of center of the Moon is tabulated with a vertical displacement, for its entries are displaced upwards by 13;9° with respect to the standard Alfonsine table (see Chabás and Goldstein 2012, Table 6.2a, p. 71). If we call c3(2η) the equation of center in the corresponding Alfonsine table, the tabulated entries, c3(2η), for the equation of center in the Tables for the Seven Planets are given by (9) c3(2η) = c3(2η) + 13;9°. It is worth noting that eq. 7 and the definition of ā in eq. 4 imply that a = α.18

18

a = ā + c3(2η) = (ᾱ – 13;9°) + c3(2η) = (ᾱ – 13;9°) + (c3(2η) +13;9°) = ᾱ + c3(2η) = α.

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For the minutes of proportion the entry for 0s 0° is 0′. The entries increase monotonically to 60′ at 173°–187°, and decrease to 0′ at 349°–371°. Except for copyist’s errors, they agree with those in the Parisian Alfonsine Tables. Therefore, (10) c4(2η) = c4(2η). Let us now turn to the third table (ff. 14v–15r), which is for the equation of anomaly; it has a column for the argument, from 0° to 29°, and then columns for each zodiacal sign, from 0s to 11s. For each sign and each degree of the argument, we are given entries for the equation of anomaly and the dyversitas dyametri proporcionalis19 (corresponding to c6 and c5, respectively, in pat’s arrangement, both variables depending on the true argument of anomaly). For the equation of anomaly, the entry for 0s 0° is 4;56,0°. It reaches a minimum of 0;0,0° at 91°–99° and a maximum of 9;52,0° (twice 4;56,0°) at 264°, that is, the vertical displacement is 4;56°. This parameter was systematically used by Parisian Alfonsine astronomers, but its origin goes back much earlier (Chabás and Goldstein 2003, pp. 252–253). Thus, if we call c6(α) the equation of anomaly in the corresponding Parisian Alfonsine table, the tabulated entries, c6(a), for the equation of anomaly in the Tables for the Seven Planets are given by (11) c6(a) = c6(α) + 4;56°. The first entry in the column for the dyversitas dyametri proporcionalis, usually called “increment” is 2;40°; it reaches a minimum of 0;0° at 113°–119° and a maximum of 5;20° (twice 2;40°) at 251°–257°. Except for copyist’s errors, the entries agree with those in the Parisian Alfonsine Tables, but for the vertical displacement of 2;40°. Again, if we call c5(α) the increment in the corresponding Parisian Alfonsine table, the tabulated entries, c5(a), for the increment in the Tables for the Seven Planets are given by (12) c5(a) = c5(α) + 2;40°. In the second table (f. 14v), headed dyversitas dyametri centralis, the argument is given at intervals of 1′, from 0′ to 60′, and the entries, d(m), decrease monotoni-

19

The word dyversitas, used in the headings of the lunar and the planetary equations, is often spelled diversitas.

displaced tables in latin table a

Arg. (′) 0 … 10 … 20 … 30 … 40 … 50 … 60

113

dyversitas dyametri centralis (excerpt)

Dyv. dyam. centralis (°) 2;40, 0 2;13,20 1;46,40 1;20, 0 0;53,20 0;26,40 0; 0, 0

cally from 2;40,0° (at 0′) to 0;0,0° (at 60′). Moreover, the entries show a constant decrease of 0;2,40° for each degree of the argument (see Table a); hence, (13) d(m) = 2;40 · (1 – c4(2η)). The tables described above yield the same results as those obtained from the Parisian Alfonsine Tables for the true position of the Moon, and were computed with a similar expression (see eq. 8). If λ and l are the true lunar longitudes in the Parisian Alfonsine Tables and the Tables for the Seven Planets, respectively, then using eqs. 5, 8, 10, 11, and 12, λ can be written as λ = λ̄ + c6(α) + c5(α) · c4(2η) = (l̄ + 7;36°) + (c6(a) – 4;56°) + (c5(a) – 2;40°) · c4(2η) = (l̄ + c6(a) + 2;40°) + c5(a) · c4(2η) – 2;40°· c4(2η) = (l̄ + c6(a) + c5(a) · c4(2η)) + 2;40° – 2;40°· c4(2η) = l. The term, 2;40° · (1 – c4(2η)) = d(m), is found in the table for the dyversitas dyametri centralis, and therefore the expression to compute the true longitude of the Moon is

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(14) l = l̄ + c6(a) + c5(a) · c4(2η) + d(m), where all the terms are tabulated and none of them is to be subtracted. Note that there is no counterpart to the function d(m) in the standard Alfonsine tables. The canons to this set of tables, which are not extant, should have had an instruction more or less in the following terms: “To determine the true position of the Moon, find the mean longitude of the Moon, and keep it; find the double elongation, and enter with it in the table for the equation of center and the minutes of proportion, and keep them. Find the mean argument of anomaly, and add to it the equation of center to obtain the true argument of anomaly, and enter with it in the table for the equation of anomaly and the increment, and keep them. Enter the table for the dyversitas dyametri centralis with the minutes of proportion, and keep what you find there. Then add the mean longitude to the true argument of anomaly, and add the result to what is obtained from multiplying the increment by the minutes of proportion. To the value obtained add what you found in the table for the dyversitas dyametri centralis.” To illustrate eq. 14 consider the Moon at epoch (noon, February 28, 1341), when the double elongation, 2η, was 4,45;9,0° (= 285;9,0°). The equation of center in the standard Alfonsine tables is c3(2η) = –10;26° and, because of eq. 9, c3(2η) = 2;43° (= –10;26° + 13;9°). The corresponding minutes of proportion are c4(2η) = 19 and c4(2η) = 19, because of eq. 10. In this case the term d(m) = 1;49,20°. Now, the argument of anomaly, ᾱ, according to the standard Alfonsine tables, is 32;27,43°, and thus α = 22;1,43° (= 32;27,43° – 10;26°). From eq. 4, ā = 19;18,43°, and thus a = 22;1,43° (= 19;18,43° + 2;43°). The tabulated values for the equation of anomaly, c6(a), and the dyversitas dyametri proporcionalis, c5(a), are 3;13,19° and 1;48°, respectively, whereas the values for the equation of anomaly, c6(α), and the increment, c5(α), found in the standard Alfonsine tables are –1;42,41° and –0;52°, respectively. We note that eqs. 11 and 12 hold. According to eq. 6, using the standard Alfonsine tables the correction to be applied is c = (–1;42,41°) + (–0;52°) · (19/60) = –1;59,9°, whereas using the Tables of The Seven Planets c = 3;13,19° + 1;48° · (19/60) = 3;47,31°. With the standard Alfonsine tables the mean lunar longitude at epoch, λ̄ , is computed to be 128;55,23°; thus, with eq. 8, the true lunar longitude of the Moon at epoch is λ = λ̄ + c6(α) + c5(α) · c4(2η) = 128;55,23° – 1;59,9° = 126;56,14°.

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With the Tables for the Seven Planets we obtain l̄ = 121;19,23° (see eq. 5); thus, with eq. 14, the true longitude of the Moon at epoch is l = l̄ + c6(a) + c5(a) · c4(2η) + d(m) = 121;19,23° + 3;47,31° + 1;49,20° = 126;56,14°, in agreement with λ.

8

Lunar Node

Folio 16r contains five tables for the mean motion of the lunar node. The first lists the values of the mean motion of the node at the beginning of each year, beginning with 1342, for a period of 32 years. The entry for year 1, i.e., 1342, or more precisely, Feb. 28, 1341, is 3s 22;3;17°. The second table gives the mean motion of the Sun for the months; its first entry corresponds to March (0s 1;38,30°) and the last one to February (0s 19;19,42°). The third table lists values from 1372 to 1852, at 32-year intervals. The fourth and fifth tables are, respectively, for the mean motion of the lunar node for days from 1 to 31, and for hours and parts of an hour from 1 to 60. The entry for argument 1d is 0s 0;0,3,11°. As was the case for other mean motions, at the bottom of the folio we are told that the radix at anno Christi 1340 perfecto is 3s 2;43,34,43°. From this value and that for year 32 we derive –0;3,10,38,7°/d as the mean motion of the lunar node,20 in full agreement with the corresponding parameter in the Parisian Alfonsine Tables.

9

Precession/Trepidation

On folios 16v–17r there are various tables for precession/trepidation in the framework of the Parisian Alfonsine Tables. In general, the approach found in the Alfonsine corpus is to treat two terms separately: a linear term, usually called “motion of the apogees and the fixed stars”, based on a mean motion of 0;0,0,4,20,41,17,12°/d, and presented in a single table; and a periodic term,

20

The value for year 32 given in the text is 11s 21;39,20°, and the difference between this value and that of the radix is 258;55,45,17°. Thus the amount traveled by the lunar node in 32 years of 365;15 days is 285;42,4° and one complete revolution, leading to a daily mean motion of –0;3,10,38,7°/d.

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usually called “motion of access and recess of the 8th sphere”, requiring the use of two tables (one for the “mean motion”, based on a value of 0;0,0,30,24,49,0°/d, and another for its “equation”, found in a separate table with a maximum of 9;0,0°): see Chabás and Goldstein 2012, pp. 48–52. The author of the Tables for the Seven Planets follows the same approach, uses the same parameters, but gives a different presentation. Folio 16v has four tables under the general title Tabula motus augium et stellarum fixarum atque 8ª spere, displaying the two mean motions of the components of Alfonsine variable precession. Each table has two columns, in addition to that for the argument: the mean motion of the apogees and the fixed stars on the one hand, and the mean motion of the 8th sphere on the other. The motion of trepidation is to be applied to the positions of the fixed stars as well as to the positions of the planetary apogees. The first table gives entries for both quantities at the beginning of each year for a period of 32 consecutive years. The entries for year 1 are 0s 0;51,12° (apogees) and 2s 8;10;51° (8th sphere), and those for year 32 are 0s 1;4,52° (apogees) and 2s 9;46;31° (8th sphere). The second table gives the mean motion of both quantities for accumulated months. The third table, headed ad annos perpetuacionis, has entries for years 1372 to 1852, at 32-year intervals, of both quantities, whereas the fourth table is for their mean motions for days from 1 to 31. As was the case for other quantities, at the bottom of the folio we are given the radices for both quantities, anno Christi 1340 perfecto: 0s 9;50,45,59,27° (apogees), and 2s 8;7,45,20,10° (8th sphere).21 From these radices and the entries for year 32 we derive the following mean motions: 0;0,26,28 º/y (apogees) and 0;3,5,11°/y (8th sphere), which correspond to about 0;0,0,4,20,51°/d (apogees) and 0;0,0,30,25,12 º/d (8th sphere), that is, 1 revolution in 49,000 years and 7,000 years, respectively. These are the standard values found in Alfonsine astronomy for precession/trepidation. Combining the mean motion of the 8th sphere and its radix, we find that the argument of the 8th sphere was 0° about 1324 years and 9 months before the epoch of these tables (February 28, 1341), that is, in May 16ad. This date is explicitly given by Giovanni Bianchini (15th century) as the epoch of the Alfonsine model; see Chabás and Goldstein 2009, p. 32, and the Tables for the Seven Planets offer a justification of it. It remains to compute the periodic term of Alfonsine trepidation, that is, the equation of the 8th sphere, which is tabulated on f. 17r. Indeed, the entries given in the table are based on the standard Alfonsine equation of the 8th

21

We note that the radix for the apogees has the wrong number of degrees (probably a copyist’s error); to be consistent with the 32 values in the table, it should be 0s 0;50,45,59,27°.

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sphere which reaches a maximum of 9;0,0° at 90°, a minimum of –9;0,0° at 270°, and vanishes at 0° and 180° (see also Ratdolt 1483, f. d3v). The corresponding entries on f. 17r are 9;0,0° at 0°, 18;0,0° (maximum) at 90°, 9;0,0° at 180°, 0;0,0° (minimum) at 270°, and 0;0,0° at 360°; thus, the entries are displaced upwards by 9° from the standard table, and can be represented by the modern expression (where y is the entry and x is the argument): (15) y = 9° + arcsin (sin 9° · sin x). This is another case of a vertically displaced table we find in this manuscript. Interestingly, the only example in Latin of a displaced table previously known to us is that of John Vimond (see n. 10), whose table for the equation of the 8th sphere presents both a vertical displacement (8;17°) and a horizontal displacement (113°). However, Vimond’s purpose was not to avoid “negative” numbers in his table (otherwise he would have used a vertical displacement of at least 9;0°, as here); rather, he wished to set the origin of the equation of the 8th sphere at the value (8;17°) it had at the time he composed his tables (1320). Apparently, the author of the Tables for the Seven Planets did not have the same goal as Vimond, although he used the same principle. As mentioned above, on folio 17r there is also a small table entitled Motus augium equatus ad 32 annos post annum 1340, listing 32 values. The first entry is 0;0,37,11° and the last entry, for 32 years, is 0;19,42,35°. The entries show a steady increase but the line-by-line differences do not reveal a clear pattern. A complement to this table is found on f. 18r among the tables for the mean motions, where we find one headed Motus augium equatus ad annos perpetuacionis, listing entries from 1372 to 1852 at intervals of 32 years. The first entry is 0;19,42,35° but, as in the previous case, the line-by-line differences do not vary in a smooth way. With all these cautions, from the entry for 32 years it is possible to derive a motion of the apogee of 0;0,36,57°/y, or 0;0,0,6,4°/d, which is the amount to be added to the mean motion in solar anomaly to obtain the mean motion in the argument of solar longitude (see §2). The value we deduce from the table is certainly close to Ptolemy’s value for precession of 1° in 100 years, that is 0;0,36° in 1 Egyptian year of 365 days.

10

Planetary Mean Motions

Next come tables for the mean motions, equations, and stations of the five planets, on ff. 17v–22r (Saturn), ff. 22r–26v (Jupiter), ff. 27r–31r (Mars), ff. 31r–35v

118 table 3

chapter 5 Radices for year 1340 (complete)

Centrum Saturn Jupiter Mars Venus Mercury

Porcio

0s 4;56,59° 2s 8;46,40° 0s 1;38,17,48° 4s 29;28,59,11° 1s 25; 6,26, 4° 2s 24;48,42,[..]º 6s 25;41,48,36° 7s 17; 6,11° 3s 19;27,49° 3s 23;27,56°

(Venus), and ff. 35r–40r (Mercury). In this section we review the mean motions of the planets, for each of which we are given five tables. As we will see, all the mean motions are displaced with respect to those in the standard Alfonsine tables. The first of these five tables lists the radices of two variables, centrum (argument of center) and porcio (here meaning the displaced argument of anomaly, as will be seen below), for the beginning of each year after 1340 for a period of 32 consecutive years. The entries for both variables for year 32 are, respectively: 1s 6;7,52° and 1s 14;30,11° (Saturn); 8s 12;53,57° and 8s 18;7,43° (Jupiter); 2s 0;5,0° and 2s 19;44,28° (Mars); 6s 25;36,12° and 7s 22;56,20° (Venus); and 3s 19;22,12° and 2s 4;36,10°(Mercury). The second of these tables gives the mean motion of the planets for accumulated months, beginning in March, for the two variables. The third of these tables is headed ad annos perpetuacionis and displays entries for the two variables from 1372 to 1852, at 32-year intervals. The fourth and fifth of these tables are, respectively, for the mean motion of the planets for days from 1 to 31, and for hours and parts of an hour from 1 to 60. From the values of the radices and those for year 32 in the first table for the mean motions of the planets, we derive the following values for the daily mean motions: 0;2,0,29,13 º/d (Saturn, argument of center); 0;4,59,9,23°/d (Jupiter, argument of center); and 0;31,26,32,35°/d (Mars, argument of center). These three parameters differ by 0;0,0,6,4°/d from the standard Alfonsine mean motions, as was the case for the argument of solar anomaly, indicating that the precession of 1° in 100 years was applied to the planets as well as to the Sun. We have also derived the daily mean motions of anomaly for the inferior planets: 0;36,59,27,24°/d (Venus) and 3;6,24,7,43°/d (Mercury), which agree exactly with the corresponding parameters in the Parisian Alfonsine Tables. As was the case for the Sun and the Moon, we are also given the radices anno christi 1340 completo, which we display in Table 3.

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displaced tables in latin table 4

Recomputation for the planets at epoch: noon, Feb. 28, 1341

Argument of Difference Rounded Argument of Difference Rounded center Text–Comp. value anomaly Text–Comp. value Saturn 18;56,58° Jupiter 19;18,21° Mars 116; 6,24° Venus 256;41,56° Mercury 137;27,46°

–13;59,59° –17;40, 3° * –60;59,58° –51; 0, 7° –27;59,57°

14 18 61 51 28

75;46,38° 155;28,57 96;48,44° 230; 6, 9° 117;27,52°

–6;59,58° –5;59,58° –12; 0, 2° –2;59,58° –3;59,56°

7 6 12 3 4

* The difference is –18;0,3° if we take the radix for the argument of center in Table 3 to be 1;18,17,48°, which is the result one would obtain from a set of standard Alfonsine tables.

We have recomputed the mean positions of the planets using the Parisian Alfonsine Tables for noon of February 28, 1341 for the meridian of Paris, 12° east of Toledo. The results are presented in Table 4. From Table 4 it follows that the arguments of center of the five planets given in these tables are diminished, but for a few seconds, by 14°, 18°, 61°, 51°, and 28° from those obtained from the Parisian Alfonsine Tables for a locality near the meridian of Paris. Table 4 also indicates that the entries for the argument of anomaly are diminished, but for a few seconds, by 7°, 6°, 12°, 3°, and 4° from those derived from the Parisian Alfonsine Tables. As will be shown in § 11, all these integer values correspond to the horizontal and vertical displacements (kh3 and kv3, respectively) used in the tables for the equations of the planets. If we call κ̄ and ᾱ the planetary arguments of center and anomaly in the Parisian Alfonsine Tables, the corresponding quantities in the Tables for the Seven Planets are defined by: (16) k̄ = κ̄ – kh3, and (17) ā = ᾱ – kv3.

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chapter 5 figure 5.c The model for Mars, where o is the observer, d is the center of the deferent circle acp, a is the apogee, p is the perigee, e is the equant point, ov is the direction to the vernal point on the ecliptic, s̄ is the direction to the mean Sun, āe is the mean apogee on the epicycle whose center is c, ae is the true apogee on the epicycle, κ̄ and ᾱ are the mean arguments of center and anomaly, respectively, m̄ is the mean position of the planet, and m is its true position (where, for an outer planet, cm is parallel to os̄).

11

Planetary Equations and Stations

In this section we review the equations and the stations of the five planets, for each of which we are also given five tables, in which the equation of center (one table; see §11.1) and the equation of anomaly (four tables; see §§ 11.2 and 11.3) are treated separately. Figure 5.c displays the model for Mars. As we will see, the true longitude of a planet is found by adding four positive terms, all of which are tabulated (see eqs. 43 and 45, below). These terms are all different from those in the standard Alfonsine tables, where the true longitude is found as the algebraic sum of four terms. The author of these tables has considered two cases: (1) the epicyle is near the apogee of the deferent, and (2) the epicycle is near the perigee of the deferent. In case 1 he has the distance of the center of the epicycle vary from its maximum to its mean value, whereas in case 2 he has the distance of the center of the epicycle vary from its mean to its minimum value. 11.1 Equation of Center and equatio porcionis In the first table the argument ranges from 0° to 29° for each zodiacal sign, and for each degree we are given three entries: equatio centri, equatio porcionis, and minutes of proportion. The first two are given in degrees and minutes, and the third entry is given in minutes (see Table b). Table 5 displays significant values of the equation of center for the five planets. To determine the relation between the equation of center presented here and that in the Parisian Alfonsine Tables, let us consider the case of Mars. The entry for 0° is 2;29°; the minimum (0;36°) is reached at 31°–35°

121

displaced tables in latin table b

(°) 0 1 2 … 60 … 79 80 81 82 … 180 … 254 255 … 264 265 266 267 … 359

equatio centri, equatio porcionis, and minutes of proportion for Saturn (excerpt)

Equatio centri (º)

Equatio Minutes porcionis of proportion (º) (′)

5;30 5;24 5;17

8;30 8;36 8;43

0 0 1

0;48

13;12

37

0;29 0;29 0;30 0;30

13;31 13;31 13;30 13;30

58 59 1 2

8;39

5;21

60

13;31 13;31

0;29 0;29

0 0

13;25 13;23 13;22 13;21

0;35 0;37 0;38 0;39

2 1 59 58

5;36

8;24

0

and the maximum (23;24°) at 203°–207°. The difference between the two extremal values is 22;48°, that is, twice 11;24°, the standard parameter used for Mars in the Parisian Alfonsine Tables. The entries in the table are displaced exactly 12° upwards as compared with the corresponding entries in the Parisian Alfonsine Tables, and 12° is the minimum integer for a vertical displacement ensuring that all entries for the equation of center are positive. But the entries for the equation of center have a second displacement, a horizontal displacement, of 61° with respect to the table in the Parisian Alfonsine

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table 5

Equation of center for all planets

Entry for 0° Saturn

5;30°

Jupiter

4;15°

Mars

2;29°

Venus

1;21°

Mercury 2;47°

Minimum equation of center

Maximum equation of center

0;29° (76°–80°) 0; 3° (72°–78°) 0;36° (31°–35°) 0;50° (37°–47°) 0;58° (65°–69°)

13;31° (252°–256°) 11;57° (246°–252°) 23;24° (203°–207°) 5;10° (211°–221°) 7; 2° (235°–239°)

UnderVertical Horizontal lying displacedisplaceparameter ment (kv3) ment (kh3) 6;31°

7°

14°

5;57°

6°

18°

11;24°

12°

61°

2;10°

3°

51°

3; 2°

4°

28°

Tables. With the notation used above, k̄ = κ̄ – 61°. If we call c3(κ̄) the equation of center in the corresponding Parisian Alfonsine table,22 the tabulated entries, c3(k̄ ), for the equation of center in the Tables for the Seven Planets are given by c3(k̄ ) = c3(k̄ + 61°) + 12°. This is the first example of a horizontal displacement in these tables. For each planet the corresponding displacement applied to κ̄ (see Table 5) accounts for the difference between the entries given in the Tables for the Seven Planets and the Parisian Alfonsine Tables. Thus, the general rule for the equation of center of the planets can be written as (18) c3(k̄ ) = c3(k̄ + kh3) + kv3, where kh3 and kv3 are the horizontal and vertical displacements associated with c3(κ̄), the equation of center in the Parisian Alfonsine Tables. Once c3(k̄ ) is known, the true argument of center, k, is defined as: (19) k = k̄ + c3(k̄ ),

22

In the Parisian Alfonsine Tables the equation of center is displayed in the third column in the tables for planetary equations, hence the notation used here.

displaced tables in latin

123

figure 5.d Mars, equation of center and equatio porcionis

on analogy with the expression for the true argument of center, κ, used in the Alfonsine tables, (20) κ = κ̄ + c3(κ̄), where c3(κ̄) ≤ 0° when 0° ≤ κ̄ ≤ 180°, as shown in Figure 5.d. All the underlying parameters displayed in Table 5 are strictly in the tradition of the Alfonsine corpus. Interestingly, the function for Venus is presented with displacements that differ from that for the Sun (see § 3, above), although they share the same maximum equation, 2;10°. As for the column labeled equatio porcionis, note that for a given argument its entries and those corresponding to them in the equation of center always add up to an integer number of degrees: 14° (Saturn), 12° (Jupiter), 24° (Mars), 6° (Venus), and 8° (Mercury). This is exactly twice the vertical displacement applied to each planet, that is, 2 · kv3. Thus, if we call the equatio porcionis e(k̄ ), it follows that the author of the Tables of the Seven Planets defined it as (21) e(k̄ ) + c3(k̄ ) = 2 · kv3. This definition was intended to eliminate subtractions, as will be justified below: there is no counterpart to the function e(k̄ ) in the standard Alfonsine tables. Figure 5.d shows the equation of center and the equatio porcionis in the case of Mars. The other planets follow the same pattern (see Table 5). When e(k̄ ) is known, the true anomaly, a, is defined as:

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(22) a = ā + e(k̄ ), on analogy with the expression for the true argument of anomaly, α, used in the Alfonsine tables, (23) α = ᾱ – c3(κ̄) where, again, c3(κ̄) ≤ 0° when 0° ≤ κ̄ ≤ 180°. In the third column of this table the entries for the minutes of proportion are also dependent upon those in the Parisian Alfonsine Tables (see Figure 5.e for the case Mars) except for two features: a horizontal displacement, different for each planet, and a vertical displacement of 60′, which is limited to the positions when the planet is near its apogee. Note that in the Parisian Alfonsine Tables the minutes of proportion for these positions are subtractive, whereas they are additive for the positions near perigee. Consequently, in order to avoid subtractions, the author of the Tables for the Seven Planets only applied a vertical displacement of 60′ to half of the column, that is, where the center of the epicycle is near apogee, and left unchanged the additive part (for details see Table 6). With this approach, if an entry, c4(κ), gives the minutes of proportion in the Parisian Alfonsine table,23 the corresponding entry in the Tables for the Seven Planets in the case of Mars is c4(k) = c4(k + 49°) + 60′, near apogee, and c4(k) = c4(k + 49°), near perigee. Thus, the general rule for the equation of center of the planets can be written as (24) c4(k) = c4(k + kh4) + kv4, near apogee, and simply (25) c4(k) = c4(k + kh4), near perigee, where kh4 and kv4 (= 60′) are the horizontal and vertical displacements, respectively, associated with c4(κ), the minutes of proportion. Figure 5.e 23

In the Parisian Alfonsine Tables the minutes of proportion are displayed in the fourth column in the tables for planetary equations; hence the notation used here. Note, however, that in the corresponding table in Ptolemy’s Almagest they are found in the eighth column and depend on the mean argument of center, whereas in the Parisian Alfonsine Tables they are a function of the true argument of center. For this function in the Almagest, see Neugebauer 1975, pp. 185–186.

displaced tables in latin

125

figure 5.e Mars, minutes of proportion

displays the minutes of proportion for Mars; the other planets follow the same pattern (see Table 6). As in previous cases, the vertical displacements are intended to avoid cumbersome rules for addition and subtraction corresponding to the simple rules we now give by means of algebraic signs. The horizontal displacements are intended to counterbalance another displacement in a different column or table. That this is indeed the case can be seen from the fact that the horizontal displacements applied to the argument of center (the column on the right side in Table 5) and to the minutes of proportion (the column on the right side in Table 6) add up algebraically to the vertical displacement applied to the equation of center (see Table 5): (26) kh3 – kh4 = kv3. From eqs. 16, 18, 20, and 26 it follows that (27) k = κ – kh4. Therefore kh4 is the displacement to be applied to the Alfonsine true argument of center to obtain the true argument of center in the Tables of the Seven Planets.24

24

Indeed, k = k̄ + c3(k̄ ) = κ̄ – kh3 + c3(κ̄) + kv3 = κ̄ + c3(κ̄) – kh3+ kv3 = κ – (kh3+ kv3) = κ – kh4.

126 table 6

chapter 5 Displacements for the minutes of proportion

Entry for 0°

Vertical displacement (kv4)

Horizontal displacement (kh4)

Saturn

0

7°

Jupiter

0

60′ (265°–266° to 360° and 0° to 80°–81°) 60′ (259°–260° to 360° and 0° to 77°–78°) 60′ (223°–224° to 360° and 0° to 38°–39°) 60′ (223°–224° to 360° and 0° to 40°–41°) 60′ (271°–272° to 360° and 0° to 41°–42°)

Mars

20

Venus

19

Mercury

10

12° 49° 48° 24°

11.2 Equation of Anomaly near Greatest Distance For each planet the equation of anomaly is displayed in four tables: two, here called (i) and (ii), associated with greatest distance (i.e., between greatest and mean distance) and two, with exactly the same presentation, associated with least distance (i.e., between mean and least distance: see § 11.3). In the Almagest and sets of tables related to it the equation of anomaly is treated separately when the center of the epicycle is near apogee in contrast to the case when the center of the epicycle near perigee. Ptolemy first assumed that the center of the epicycle is at mean distance and computed the equation of anomaly under this condition (Ptolemy’s column 6). He then considered the case where the center of the epicycle lies between maximum distance (apogee), where the equation of anomaly is least, and mean distance. Ptolemy’s column 5 gives the subtractive difference to be applied to the equation of anomaly at mean distance. This is because at maximum distance the equation of anomaly is least. To interpolate between mean distance and maximum distance Ptolemy applied the minutes of proportion in his column 8 to the subtractive difference; thus, in this case, the total equation is (28) c(α) = c6(α) – c5(α) · c8(κ̄) where κ̄ is near apogee. As Neugebauer (1975, pp. 185–186) demonstrated, Ptolemy used different formulas for c8(κ̄) near apogee and near perigee, but put the values in a single column. Between mean distance and perigee Ptolemy com-

displaced tables in latin

127

puted the entries in his column 7 which is the amount to the added to the value for mean distance to get the equation of anomaly at least distance (perigee) where the equation of anomaly is greatest; thus, in this case, the total equation is (29) c(α) = c6(α) + c7(α) · c8(κ̄), where κ̄ is near perigee. So in some cases one has to subtract and in other cases one has to add. The author of the Tables for the Seven Planets wished to avoid subtractions and he took a different approach without changing the model or the parameters. Instead of first tabulating the values for mean distance with a subtractive difference, he tabulated the equation of anomaly at greatest distance, where the equation is least. Hence, all corrections are positive, and the total equation is (30) c(a) = d(a) + c5(a) · c4(k), where d(a) is the equation of anomaly at apogee, c5(a) corresponds to Ptolemy’s c5(α), and c4(k) (defined in eq. 24) corresponds to Ptolemy’s c8(κ̄); all terms are positive. For values of κ near perigee, the author of the Tables for the Seven Planets adhered more closely to Ptolemy’s formula since it did not involve subtraction. Thus, between mean distance and minimum distance, our author used (31) c(a) = c6(a) + c7(a) · c4(k), where c6(a) corresponds to Ptolemy’s c4(α), c7(a) corresponds to Ptolemy’s c7(α), and c4(k) (defined in eq. 25) corresponds to Ptolemy’s c8(κ̄). We will see that the entry for mean distance is the same whether one uses the formula near apogee or the formula near perigee. (i) The argument is the minutes of proportion, as indicated in the heading, displayed here as integers from 0 to 60. There are two other columns, one labeled diversitas dyametri centralis and another for the first station near greatest distance (see Table c). The entries corresponding to 0 refer to greatest distance, and the entries corresponding to 60 refer to mean distance. This table is unprecedented in the astronomical literature and, in the absence of canons explaining its use, it is difficult to interpret the meaning of the expression, diversitas dyametri centralis. Table 7 shows the extremal values for all planets.

128 table c

chapter 5 Diversitas dyametri centralis and first station near greatest distance for Jupiter (excerpt)

Minutes Diversitas of prop. dyametri centralis (′) (s, º)

First station (s, º)

0 1 … 15 … 30 … 45 … 59 60

5 22;37,40 5 22;37,10

4 4; 5 4 4; 7

5 22;30,10

4 4;30

5 22;22,40

4 4;53

5 22;14,40

4 5;17

5 22; 8,10 5 22; 7,40

4 5;38 4 5;39

A close inspection of the entries for the diversitas dyametri centralis indicates that they are strongly dependent on the apogees of the planets at epoch (noon, February 28, 1341, Toledo). A comparison of our computation of the planetary apogees for the city of Toledo at that time and the entries for 0 for each planet is displayed in Table 8. From Table 8 it follows that the entries for 0 tabulated here for the diversitas dyametri centralis are obtained by adding a constant to the longitude of the apogee of each planet at epoch, λ0. This constant is specific to each planet and results in turn from the displacements applied to its variables, kh4 – max (c6 – c5): see Table 10. Therefore, these entries play a crucial role in computing the true planetary longitudes but have no direct astronomical meaning. On the other hand, for each planet the range for the entries is 0;20,42° (= 8s 12;42,18° – 8s 12;21,36°); or 0;21° (if we consider 8s 12;21,18° as the intended entry for 60) for Saturn, 0;30° for Jupiter, 5;38° for Mars, 1;42° for Venus, and 3;12° for Mercury. It turns out that these values agree with the vertical displacement applied to the difference at greatest distance, c5, associated with the argument of anomaly. We have called this quantity kv5 (see Table 11). In the case of Mars, its value is 5;38°, and the diversitas dyametri centralis near greatest distance, d5, which is a function of the minutes of proportion, m, can be written as

129

displaced tables in latin table 7

Diversitas dyametri centralis and first station near greatest distance for all planets

Diversitas dyametri centralis Entry for 0 Entry for 60 Saturn Jupiter Mars Venus Mercury

8s 12;42,18° 5s 22;37,40° 4s 25;38,49° 3s 2;48, 9° 7s 3;50, 9°

8s 12;21,36° * 5s 22; 7,40° 4s 20; 0,49° 3s 1; 6, 9° 7s 0;38, 9°

First station Entry for 0 Entry for 60 3s 22;44° 4s 4; 5° 5s 7;28° 5s 15;51° 4s 27;14°

3s 24; 7° 4s 5;39° 5s 13;11° 5s 17; 9° 4s 25; 8°

* Probably an error for 8s 12;21,18°. For justification, see below. One should also note that the number of seconds in the entries for 0 and 60 in columns 1 and 2 agree for each planet (but for Saturn here), both in this table and in the equivalent, Table 12. table 8

Saturn Jupiter Mars Venus Mercury

Planetary apogees near greatest distance

Computed

Entry for 0 – comp.

Rounded value *

251;35,19° 171;31,37° 133;23,50° 89;37, 0° 208;51,10°

1; 6,59° 1; 6, 3° 12;14,59° 3;11, 9° 4;58,59°

1; 7 (= 7 – 5;53) 1; 6 (= 12 – 10;54) 12;15 (= 49 – 36;45) 3;11 (= 48 – 44;49) 4;59 (= 24 – 19; 1)

* As will be seen below, the rounded value for each planet should result from subtracting the maximum of c6 – c5 (see Table 10) from kv3 (see Table 5). In the case of Jupiter, this gives 1;26° (= 12 – 10;34), whereas we obtain 1;6° when using the entry for 0 in the table, 5s 22;37,40° (see Table 7), and a set of standard Alfonsine tables. If the author had taken 5s 22;57,40° as the entry for 0, the rounded value would have been fully consistent with those for the other planets.

d5(m) = d5(0) – 5;38 · c4(k). Thus, the general rule for the diversitas dyametri centralis near greatest distance is given by d5(m) = d5(0) – kv5· c4(k), that is, (32) d5(m) = λ0 + kh4 – max (c6 – c5) – kv5 · c4(k).

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chapter 5

figure 5.f First station of Mars

The third column in this table displays the first station as a function of the minutes of proportion. The entries for the first station represent the true argument of anomaly of the first stationary point, and those for 0 agree with the entries in the Almagest (see, e.g., Toomer 1984, p. 588) and sets of tables related to it. They correspond to the greatest distance of the center of the epicycle, when the mean argument of center, κ̄, is zero. It should be recalled that the table in Almagest xii.8 gives the positions of the stationary points on the epicycle as a function of κ̄ (see Neugebauer 1975, pp. 202–205), not as a function of the minutes of proportion, as is the case here. The entries increase (except for Mercury) monotonically and vary in a range different for each planet: 1;23° (Saturn), 1;34° (Jupiter), 5;43° (Mars), 1;18° (Venus), and –2;6° (Mercury). Figure 5.f displays the first station of Mars, both near apogee (see Table 7) and near perigee (Table 12). If we call s5(m) the tabulated argument of anomaly of the first station near apogee, the entries are can be computed by means of a linear function of m, the minutes of proportion: (33) s5(m) = s5(0) + [s5(60) – s5(0)] · c4(k), which, in the case of Mars, turns into s5(m) = 157;28° + 5;43° · c4(k). (ii) In the second table the argument ranges from 0° to 29° for each zodiacal sign. There are two columns, one for the equation of anomaly (the heading has argumentum for anomaly) and another for the diversitas dyametri. The

131

displaced tables in latin table d

(º) 0 1 … 90 … 125 126 127 128 … 154 155 156 157 158 … 169

170 …

Equation of anomaly and diversitas dyametri for Mars (excerpt)

Equation of anomaly (º)

Diversitas dyametri (directus or retrogradus) (º)

36;45 37; 7

5;38 5;40

67;40

8; 5

73;29 73;30 73;30 73;29

9;48 9;52 9;55 9;59

66;22 65;42 64;59 64;15

11;16 11;16 11;16 11;15 statio 11;14

63;29 52;11

50;55

9;42 prima retro. 9;26

(º) 180 … 191

Equation of anomaly (º)

Diversitas dyametri (directus or retrogradus) (º)

36;45

5;38

21;19

1;34 statio 1;20

192 … 203

20; 3

204 205 206 … 232 233 234 235 … 270 … 358 359

8;31 7;48 7; 8

9;15

0; 0; 0; 0;

0; 1 secunda directus 0; 0 0; 0 0; 0

1 0 0 0

1;17 1;21 1;24 1;28

5;50

3;11

36; 0 36;23

5;35 5;36

title indicates that the entries are given for greatest distance of the center of the epicycle (ad longitudinem longiorem), that is, near apogee. The heading of the entries for the diversitas dyametri is generally diversitas dyametri directus, and sometimes diversitas dyametri retrogradus, a change which is indicated by the insertion of the words statio, prima, and secunda, among the entries. See Table d. The relevant information for the equation of anomaly is summarized in Table 9, where the “underlying parameter” was derived by taking half the difference between maximum and minimum equation of anomaly.

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table 9

Greatest distance

Equation of anomaly near greatest distance for all planets

Entry for 0°

Minimum equation of anomaly

Maximum equation of anomaly

Underlying parameter

Saturn

5;53° 10;34°

Mars

36;45°

Venus

44;49°

Mercury

19; 1°

11;46° (92°–97°) 21; 8° (97°–102°) 73;30° (126°–127°) 89;38° (135°) 38; 2° (108°)

5;53°

Jupiter

0;0° (261°–266°) 0;0° (258°–263°) 0;0° (233°–234°) 0;0° (225°) 0;0° (250°–252°)

10;34° 36;45° 44;49° 19; 1°

The underlying parameters in this table are the maximum values of the difference c6 – c5, that is, the maximum difference between the equation of anomaly at mean distance and the subtractive difference in the Parisian Alfonsine Tables.25 All these values are shown in Table 10. Figure 5.g displays the tabulated values of the equation of anomaly for Mars near greatest distance, here called d(a) and the difference between c6 and c5 in the Parisian Alfonsine Tables. It is readily seen that d(a) is displaced upwards with respect to c6 – c5. The other planets follow the same pattern. In the case of Mars the expression for the equation of anomaly at greatest distance can be written as d(a) = c6(α) – c5(α) + 36;45°, which, in the case general, corresponds to (34) d(a) = c6(α) – c5(α) + max (c6 – c5).

25

In the Parisian Alfonsine Tables the fifth and sixth columns in the tables for planetary equations display, respectively, the subtractive difference and the equation of anomaly at mean distance; hence the notation used here.

133

displaced tables in latin table 10

Saturn Jupiter Mars Venus Mercury

Maximum values of c6, c5 and c6 – c5 in the Parisian Alfonsine Tables

Maximum of c6

Maximum of c5

Maximum of c6–c5

6;13° (at 94°–99°) 11; 3° (at 99°–102°) 41;10° (at 131°–132°) 45;59° (at 135°–136°) 22; 2° (at 111°–112°)

0;21° (at 100°–106°) 0;30° (at 107°–117°) 5;38° (at 153°–156°) 1;42° (at 161°–162°) 3;12° (at 129°–131°)

5;53° (at 94°–99°) 10;34° (at 97°–102°) 36;45° (at 126°–127°) 44;49° (at 135°) 19; 1° (at 108°)

figure 5.g Equation of anomaly near greatest distance for Mars

The diversitas dyametri tabulated for each planet is the difference near greatest distance. The entries in this column agree with those for the same quantity, c5(α), in the Parisian Alfonsine Tables, but for a vertical displacement, which differs from one planet to another (see Table 11). In the case of Mars this displacement amounts to 5;38°. In general, if c5(α) is the difference near greatest distance in the corresponding Parisian Alfonsine table, that in the Tables for the Seven Planets is given by

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table 11

Difference near greatest distance

Greatest Entry Minimum difference Maximum difference Vertical disdistance for 0° at greatest distance at greatest distance placement (kv5) Saturn

0;21°

Jupiter

0;30°

Mars

5;38°

Venus

1;42°

Mercury

3;12°

0;0° (253°–261°) 0;0° (242°–252°) 0;0° (204°–207°) 0;0° (198°–199°) 0;0° (229°–231°)

0;42° (99°–104°) 1; 0° (108°–118°) 11;16° (153°–156°) 3;24° (161°–162°) 6;24° (129°–131°)

0;21° 0;30° 5;38° 1;42° 3;12°

(35) c5(a) = c5(α) + kv5. We note that a = α, which is a consequence of equations 17, 18, 21, and 22.26 Therefore, the displacements applied by the author of the Tables for the Seven Planets to the standard quantities used in Alfonsine planetary astronomy are such that they keep invariant the argument of anomaly. As in eq. 28, the total correction when the planet is near apogee in Ptolemy’s notation is c(α) = c6(α) – c5(α) · c8(κ̄), which, in standard Alfonsine notation, is equivalent to c(α) = c6(α) – c5(α) · c4(κ̄). In the Tables for the Seven Planets the corresponding expression near apogee is (36) c(a) = d(a) + c5(a) · c4(k),

26

Indeed, a = ā + e(k̄ ) = (ᾱ – kv3) + (2· kv3 – c3(κ̄)) = ᾱ + kv3 – [(c3(κ̄) +kv3)] = ᾱ – c3(κ̄) = α.

135

displaced tables in latin table e

Minutes of prop. (′) 0 1 … 15 … 30 … 45 … 59 60

Diversitas dyametri centralis and first station near least distance for Venus (excerpt)

Diversitas dyametri centralis (s, º)

First station (s, º)

3 3

1;38, 9 1;36,17

5 17; 9 5 17;10

3

1;10, 9

5 17;25

3

0;42, 9

5 17;43

3

0;14, 9

5 18; 4

2 29;48, 1 2 29;46, 9

5 18;20 4 18;21

an expression only involving positive terms and leading to the same true longitude of the planets as the standard Alfonsine procedure. 11.3 Equation of Anomaly near Least Distance The following two tables, for least distance (i.e., between mean and least distance), have the same presentation as those for greatest distance (i.e., between greatest and mean distance), reviewed in §11.2 (i) and (ii). (i) As in Table c the argument is the minutes of proportion from 0′ to 60′, and there are two other columns, one labeled diversitas dyametri centralis and another for the first station near least distance (see Table e). Here the entries for 0′ correspond to mean distance, and the entries for 60′ correspond to least distance. The extremal values of the two columns (diversitas dyametri centralis and first station) for all planets are listed in Table 12. As was the case for greatest distance, the entries for 0′ tabulated here for the diversitas dyametri centralis are obtained by adding a constant to the longitude of the apogee of each planet at epoch, λ0. This constant is specific to each planet and results in turn from the displacements applied to its variables: kh4 – kv6 (see Table 13).

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table 12

Diversitas dyametri centralis and first station near least distance for all planets

Diversitas dyametri centralis Entry for 0 Entry for 60 Saturn Jupiter Mars Venus Mercury

table 13

Saturn Jupiter Mars Venus Mercury

8s 12;22,18° 5s 22; 8,40° 4s 21;13,49° 3s 1;38, 9° 7s 0;49, 9°

8s 11;57,18° 5s 21;35,40° 4s 13;10,49° 2s 29;46, 9° 6s 28;48, 9°

First station Entry for 0 Entry for 60 3s 24; 8° 4s 5;40° 5s 13; 6° 5s 17; 9° 4s 25; 8°

3s 25;27° 4s 7;11° 5s 19;15° 5s 18;21° 4s 24;28°

Planetary apogees at least distance

Computed

Entry for 0 – comp.

Rounded value *

251;35,19° 171;31,37° 133;23,50° 89;37, 0° 208;51,10°

0;46,59° 0;37, 3° 7;49,59° 2; 1, 9° 1;57,59°

0;47 (= 7 – 6;13) 0;37 (= 12 – 11;23) 7;50 (= 49 – 41;10) 2; 1 (= 48 – 45;59) 1;58 (= 24 – 22; 2)

* As will be seen below, the rounded value for each planet should result from subtracting kv6 (see Table 14) from kv3 (see Table 5). In the case of Jupiter, this gives 0;57° (= 12 – 11;3), whereas we obtain 0;37° when using the entry for 0 in the table, 5s 22;8,40° (see Table 12) and a set of standard Alfonsine tables. If the author had taken 5s 22;28,40° as the entry for 0, the rounded value would have been fully consistent with those for the other planets.27

On the other hand, the range for the entries is: 0;25° for Saturn, 0;33° for Jupiter, 8;3° for Mars, 1;52° for Venus, and 2;1° for Mercury. The entries for least distance are not exactly a continuation of those for greatest distance (Table 7), because they were computed by means of different expressions and different ranges

27

This is the same situation we noted in Table 8; thus, it would seem that the author either miscopied an entry for Jupiter on which he built up his two tables for the diversitas dyametri centralis, or that he had at his disposal a table for Jupiter generating a difference of 0;20°.

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were used. In the case of Mars the vertical displacement applied here, kv7, is 8;3° rather than kv5 = 5;38°, as was the case at greatest distance. Then the diversitas dyametri centralis near least distance, d7, can be written as d7(m) = d7(0) – 8;3 · c4(k), where m is a value for the minutes of proportion. Thus, the general rule for the diversitas dyametri centralis near least distance is given by d7(m) = d7(0) – kv7· c4(k) that is, (37) d7(m) = λ0 + kh4 – kv6 – kv7 · c4(k). The third column in this table displays the argument of anomaly of the first station near least distance as a function of the minutes of proportion (Table 12). The entries follow the same pattern as those near greatest distance, but we note that the entries for 60′ at greatest distance (see Table 7) do not coincide with the entries for 0′ near least distance (see Table 12) in three cases: Saturn, Jupiter, and Mars. The entries for 60′ essentially agree with those in the Almagest and sets of tables related to it, corresponding to the least distance of the center of the epicycle, when the mean argument of center, κ̄, is 180°. The entries, graphed in Figure 5.f, increase (except for Mercury) monotonically and vary in a range which is different for each planet: 1;19° (Saturn), 1;31° (Jupiter), 6;9° (Mars), 1;12° (Venus), and –0;40° (Mercury). If we call s7(m) the tabulated argument of anomaly of the first station near perigee, the entries are can be computed by means of a linear function of m, the minutes of proportion, (38) s7(m) = s7(60) + [s7(0) – s7(60)] · (1 – c4(k)), which, in the case of Mars, turns into s7(m) = 159;15° – 6;9° · (1 – c4(k)). (ii) In the second table the argument ranges from 0° to 29° for each zodiacal sign. There are two columns (equation of anomaly and diversitas dyametri). The title indicates that the entries are given for least distance of the center of the epicycle (ad longitudinem propiorem), that is, near perigee (see Table f). As was the case for Table d, the words statio, prima, and secunda are inserted among various entries.

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table f

(º) 0 1 … 90 … 110 111 112 113 … 130 131 … 136 137 … 144 145 146 147

148 …

Equation of anomaly and diversitas dyametri for Mercury (excerpt)

Equation of anomaly (º)

Diversitas dyametri (directus or retrogradus) (º)

22; 2 22;19

2; 1 2; 2

42;35

3;30

44; 44; 44; 44;

3 4 4 3

9;48 9;52 9;55 9;59

42;46 42;37

4; 1 4; 2

41;39 41;26

4; 2 4; 1

39;34

3;59 prima 3;58 3;56 3;54 statio retro. 3;52

39;16 38;57 38;37

38;16

Equation of anomaly (º)

Diversitas dyametri (directus or retrogradus) (º)

180 … 213

22; 2

2; 1

5;27

214 215

5; 7 4;48

216 … 223 224 … 229 230 … 247 248 249 250 … 270 … 358 359

4;30

0; 8 secunda 0; 6 0; 4 statio directus 0; 3

2;38 2;25

0; 1 0; 0

1;27 1;18

0; 0 0; 1

0; 0; 0; 0;

1 0 0 1

0; 9 0;10 0;11 0;12

1;29

0;32

21;29 21;45

1;59 2; 0

(º)

The relevant information for the equation of anomaly is summarized in Table 14, where the “underlying parameter” was derived by taking half the difference between maximum and minimum equation of anomaly. For each planet the underlying parameter agrees with that in the Parisian Alfonsine Tables for mean distance, and so do the entries for the equation of anomaly, but for a vertical displacement. If c6(α) represents the equation of

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displaced tables in latin table 14

Least distance

Equation of anomaly near least distance

Entry for 0°

Saturn

6;13°

Jupiter

11; 3°

Mars

41;10°

Venus

45;59°

Mercury

22; 2°

Minimum equation of anomaly

Maximum equation of anomaly

0;0° (261°–266°) 0;0° (258°–261°) 0;0° (229°) 0;0° (224°–225°) 0;0° (248°–249°)

12;26° (94°–99°) 22; 6° (99°–102°) 82;20° (131°) 91;58° (135°–136°) 44; 4° (111°–112°)

Underlying parameter

Vertical displacement (kv6)

6;13°

6;13°

11; 3°

11; 3°

41;10°

41;10°

45;59°

45;59°

22; 2°

22; 2°

anomaly in the corresponding Parisian Alfonsine table, that in the Tables for the Seven Planets is given by (39) c6(a) = c6(α) + kv6, where kv6 = 45;59° in the case of Venus.28 Figure 5.h displays the tabulated values of the equation of anomaly for Venus near least distance in the Tables for the Seven Planets and at mean distance in the Parisian Alfonsine Tables. The terminology in the Tables of the Seven Planets may be confusing, but the author sought to maintain a parallel structure in his treatment of the equation of anomaly near apogee and near perigee. The entries near apogee range from greatest distance to mean distance, and those near perigee range from mean distance to least distance. Hence, what is here called “the equation of anomaly near least distance” actually refers to the equation of anomaly at mean distance, to be applied to distances between mean and least distance. The diversitas dyametri tabulated for each planet is the difference near least distance to be added to the corresponding equation of anomaly at mean distance (see the extremal values in Table 15). The entries in this column agree

28

This expression is equivalent to c6(a) = c6(a) + kv6, because a = α.

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figure 5.h Venus, equation of anomaly as a function of the true argument of anomaly

table 15

Difference at least distance

Least Entry Minimum difference Maximum difference Vertical disdistance for 0° at least distance at least distance placement (kv7) Saturn Jupiter Mars Venus Mercury

0;25° 0;33° 8; 3° 1;52° 2; 1°

0;0° (249°–258°) 0;0° (240°–253°) 0;0° (201°) 0;0° (197°–199°) 0;0° (224°–229°)

0;50° (102°–111°) 1; 6° (107°–120°) 16; 6° (159°) 3;44° (161°–163°) 4; 2° (131°–136°)

0;25° 0;33° 8; 3° 1;52° 2; 1°

with those for the same quantity, c7(α), in the Parisian Alfonsine Tables, but for a displacement, which differs from one planet to another. If c7(α) is the difference at least distance in the corresponding Parisian Alfonsine table, the difference at least distance in the Tables for the Seven Planets is given by (40) c7(a) = c7(α) + kv7. Note that at least distance of the center of the epicycle, i.e., perigee, the equation of anomaly is greatest.29

29

These tables bear some similarity with the tables of John Vimond, who tabulated c6 –

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141

As indicated previously in eq. 29, in Ptolemy’s notation the total correction when the planet is near perigee is c(α) = c6(α) + c7(α) · c8(κ̄), which, in standard Alfonsine notation, is equivalent to c(α) = c6(α) + c7(α) · c4(κ̄). In the Tables for the Seven Planets, the corresponding expression near perigee is (41) c(a) = c6(a) + c7(a) · c4(k). This is an expression involving only positive terms and leads to the same true longitude of the planets as the standard Alfonsine procedure. To find the true longitude, λ, of the planet at that time according to the Parisian Alfonsine Tables one has to add the correction c(α) to its mean longitude, that is, to the sum of the longitude of the apogee, λ0, and the true argument of center of the planet, κ: λ = λ0 + κ + c(α), where c(α) = c6(α) + c7(α) · c4(κ). Now, the author of the Tables of the Seven Planets introduced an analogous expression to find the true longitude of the planet: (42) l = d7(m) + k + c(a), where c(a) = c6(a) + c7(a) · c4(k): see eq. 41. The term d7(m) is tabulated under diversitas dyametri centralis. The general expression for the longitude of a planet near perigee is therefore given as (43) l = d7(m) + k + c6(a) + c7(a) · c4(k),

c5, but he did not apply any displacements to the planets. Vimond also tabulated c5 + c7, which has no counterpart in these tables (see Chabás and Goldstein 2004, pp. 248–256).

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which can be obtained from (44) λ = λ0 + κ + c6(α) + c7(α) · c4(κ). To be sure, λ = λ0 + κ + c6(α) + c7(α) · c4(κ) = λ0 + (k + kh4) + (c6(Α) – kv6) + (c7(Α) – kv7)· c4(k) = λ0 + kh4 – kv6 – kv7 · c4(k) + k + c6(Α) + c7(Α) · c4(k) = l, provided that d7(m) = λ0 + kh4 – kv6 – kv7 · c4(k), which is the case: see eq. 37. All terms appearing in eq. 43 are positive and are found directly in the tables. Note that d5(m) and d7(m) have no counterparts in the standard Alfonsine tables. To illustrate eq. 43 consider the position of Mars at epoch (noon, February 28, 1341), when the planet is near perigee. The mean argument of center, k̄ , and the mean argument of anomaly, ā, are given as 55;6,26° and 84;48,42°, respectively, rounded to the seconds (see Table 3). From eqs. 16 and 17 it follows that κ̄ and ᾱ in the standard Alfonsine Tables are 116;6,26° and 96;48,42°, respectively (where kh3 = 61° and kv3 = 12°: see Table 5). The corresponding values of the equation of center in both sets of tables are c3(k̄ ) = 1;23° and c3(κ̄) = –10;37°, and eq. 18 holds. Thus, the true arguments of center are k =56;29° and κ = 105;29°, and eq. 27 holds, where kh4 = 49° (see Table 6). The tabulated equatio porcionis for k̄ = 55;6,26° is e(k̄ ) = 22;37°, and thus a = 84;48,42° + 22;37° = 107;26°. The argument of anomaly, α, according to the Parisian Alfonsine Tables, is 107;26° (= 96;48,42° + 10;37°), and we note that a = α. The minutes of proportion corresponding to k = 56;29° and κ = 105;29° are c4(k) = 17/60 and c4(κ) = 17/60, and we note that c4(k) = c4(κ), as indicated in eq. 25, valid near perigee. The tabulated values for the equation of anomaly, c6(a), and the diversitas, c7(a), are 79;11° and 11;48°, respectively, whereas the values for the equation of anomaly, c6(α), and the additive difference, c7(α), found in the Parisian Alfonsine Tables are 38;1° and 3;45°, respectively. We note that eqs. 39 and 40 hold, where kv6 = 41;10° and kv7 = 8;3° (see Tables 14 and 15). From eq. 41, c(a) = 79;11° + 11;48° · (17/60) = 82;32°, whereas the Parisian Alfonsine Tables yield c(α) = 38;1° + 3;45° · (17/60) = 39;5°. Now, the true longitude derived with the standard Alfonsine Tables is λ = 133;23,50° + 105;29° + 39;5° = 277;57,50° (see Table 8). On the other hand, from eq. 37, d7(17) = 141;13,49° – 8;3° · 17/60 = 138;56,58° (see Table 12). Thus, l = 138;56,58° + 56;29° + 82;32° = 277;57,58°, in agreement with the previous result. Similarly, the general expression for the longitude of a planet near apogee is given as

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143

(45) l = d5(m) + k + d(a) + c5(a) · c4(k), and it can be obtained from the standard Alfonsine procedure (46) λ = λ0 + κ + c6(α) – c5(α) · c4(κ). To be sure, λ = λ0 + κ + c6(α) – c5(α) · c4(κ) = λ0 + (k + kh4) + c6(α) – c5(α) + c5(α) + c5(α) · c4(κ) = λ0 + (k + kh4) + d(a) – max (c6 – c5) + c5(α) + c5(α) · c4(κ) = λ0 + kh4 – max (c6 – c5) + k + d(a) + c5(α) · (1 + c4(κ)) = λ0 + kh4 – max (c6 – c5) + k + d(a) + (c5(α) – kv5) · c4(k) = λ0 + kh4 – max (c6 – c5) + k + d(a) – kv5 · c4(k) + c5(a) · c4(k) = λ0 + kh4 – max (c6 – c5) – kv5 · c4(k) + k + d(a) + c5(a) · c4(k) = l, provided that d5(m) = λ0 + kh4 – max (c6 – c5) – kv5 · c4(k), which is the case: see Table 7 and eq. 32. All terms appearing in eq. 44 are positive and are found in the tables. To illustrate eq. 44, consider the position of Mars at noon, October 1, 1340, when κ̄ = 37;31,6°, and thus the planet is near apogee. From eq. 16 it follows that k̄ = 336;31,6° (where kh3 = 61°; see Table 5). In the Parisian Alfonsine Tables, the corresponding equation of center is c3(κ̄) = –6;26,40°, whereas the tabulated equatio centri and equatio porcionis for k̄ = 336;31,6° are c3(k̄ ) = 5;33,20° and e(k̄ ) = 18;26,40°, respectively. We note that eq. 18 holds, where kv3 =12° (see Table 5). We also note that eq. 27 holds, where kh4 = 49° (see Table 6). Then, κ = 31;4,26° because of eq. 18, and k = 342;4,26° because of eq. 19. In the Parisian Alfonsine Tables, the corresponding minutes of proportion is c4(κ) = –51/60, whereas the minutes of proportion tabulated here is c4(k) = 9/60. Therefore eq. 24 holds, where kv4 = 60′. The mean argument of anomaly of Mars for that date is ᾱ = 27;35,24°, and thus the true argument of anomaly is α = 34;2,4°, because of eq. 23, and ā = 15;35,24°, because of eq. 17, where kv3 =12°. Therefore, a = 34;2,4°, because of eq. 22. We note that a = α. In the Parisian Alfonsine Tables, the corresponding entries for the equation of anomaly and the difference at greatest distance are c6(α) = 13;26° and c5(α) = 0;48°, respectively. Thus, the total correction is c(α) = 13;26° + 0;48° (–51/60) = 12;45,12°. On the other hand, the tabulated values for the diversitas dyametri and the equation of anomaly are c5(a) = 6;26° (we note that eq. 35 holds, where kv5 = 5;38°: see Table 11) and d(a) = 49;23°. From eq. 36 it follows that c(a) = 49;23° + 6;26° · 9/60 = 50;20,54°.

144 table 16

chapter 5 Displacements in the Tables for the Seven Planets

kv Sun

Saturn Jupiter Mars Venus Mercury

kv3

kv5

kv6

kv

89;37,9°

Moon

13; 9°

2;40°

4;56°

Eq. 8th sphere

9°

kv3

kh3

kv4

kh4

kv5

kv6

kv7

7° 6° 12° 3° 4°

14° 18° 61° 51° 28°

60′ 60′ 60′ 60′ 60′

7° 12° 49° 48° 24°

0;21° 0;30° 5;38° 1;42° 3;12°

6;13° 11; 3° 41;10° 45;59° 22; 2°

0;25° 0;33° 8; 3° 1;52° 2; 1°

Now, the true longitude derived with the standard Alfonsine Tables is λ = 133;23,50° + 31;4,26° + 12;45,12° = 177;13,28° (see Table 8). On the other hand, from eq. 32, d5(9) = 145;38,49° – 5;38° · 9/60 = 144;48,7° (see Table 7). Thus, l = 144;48,7° + 342;4,26° + 50;20,54° = 177;13,27°, in agreement with the previous result. In Table 16 we present a summary of the values for the displacements, both vertical and horizontal, applied by the anonymous author of the Tables of the Seven Planets to the Sun, the Moon, the 8th sphere, and the planets.

12

Latitudes of the Superior Planets

Folio 40r displays a table for the latitudes of the three superior planets (the inferior planets are addressed in a very different way on ff. 41r–46v: see § 16, below). This table is in the Almagest tradition, and is found in many other sets of tables such as the zij of al-Battānī and the Toledan Tables (see Chabás and Goldstein 2012, Table 9.2b, p. 109). For the zij of al-Battānī, see Nallino 1903–1907; and for the Toledan Tables, see Toomer 1968 and Pedersen 2002.

13

Planetary Visibility

On f. 40v there is a table entitled Tabula visionis et occultationis for the three superior planets and the two inferior planets. It is a table for the visibility of the planets, also called a table of planetary phases, which is already found in

145

displaced tables in latin table g

Lunar latitude at eclipse (excerpt)

Argument of latitude (s, º) 0 0 6 0 12 0 0 1 5 29 11 29 … 0 12 5 18 11 18 0 13 5 17 11 17

Latitude (′) 6 6

0 1

0; 0 5;13

6 12 6 13

62;16 67;23

Almagest xiii.10, as well as in many other sets of astronomical tables, such as the Handy Tables, the zij of al-Battānī, and the Toledan Tables (see Chabás and Goldstein 2012, Table 11.2, p. 125).

14

Possibility of an Eclipse

On f. 40v there is a small table entitled Tabula latitudinis lune in principio medio et fine eclipsis, an excerpt of which we reproduce above (see Table g). The argument, which is the argument of lunar latitude, is presented in four columns and, as indicated in the title, it is restricted to the values for which an eclipse is possible, that is, ±13° from the lunar nodes. The purpose of the table is to show the correspondence between the argument of latitude and the latitude of the Moon, and the entries can be recomputed by means of the modern formula β = arcsin (sin i · sin ω), where ω is the argument of latitude, β is the latitude, and i is the inclination of the lunar orb to the ecliptic, taken here as 5;0° (the same parameter as in the table for lunar latitude on f. 13r).

15

Eclipsed Fraction of the Solar and Lunar Disks

A table for the eclipsed fraction of the solar and lunar disks is also found on f. 40v. The argument ranges from 1 to 12 linear digits (where the diameter of the eclipsed body is 12 digits), and the entries are the corresponding area

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table h

Latitude of Venus (excerpt)

Anomaly 0s6°/11s24° … 3s0°/9s0° … 5s24°/6s6° 6s0°/6s0° Center 1st 2nd 1st 2nd 1st 2nd 1st 2nd 3rd (s, º) (s, º) (s, º) (s, º) (º) (º) (º) (º) (º) (º) (º) (º) (′) 0 0 0 … 1 … 2 … 2 2 3

6 6 6 5 24 11 24 0; 7 0; 1 … 0;0 0;12 … 0;38 0; 5 0;46 0;0 12 6 12 5 18 11 18 0;13 0; 2 … 0;0 0;23 … 1;15 0;10 1;30 0;0 18 6 18 5 12 11 12 0;19 0; 2 … 0;0 0;35 … 1;52 0;14 2;14 0;0

1 2 3

0 7 0 5 0 11 0 0;31 0; 4 … 0;0 0;59 … 3; 6 0;24 3;41 0;0

5

0 8 0 4 0 10 0 0;54 0; 7 … 0;0 1;41 … 5;26 0;42 6;25 0;0

9

18 8 18 3 12 9 12 1; 0 0; 8 … 0;0 1;53 … 6; 1 0;47 7;10 0;0 10 24 8 24 3 6 9 6 1; 1 0; 8 … 0;0 1;55 … 6; 8 0;48 7;16 0;0 10 0 9 0 3 0 9 0 1; 2 0; 8 … 0;0 1;57 … 6;12 0;48 7;22 0;0 10

digits (where the area of the eclipsed body is 12). It is very common in sets of astronomical tables, such as the zij of al-Battānī, the Toledan Tables, and it is already found in Almagest vi.8. However, this was not one of the tables included in the editio princeps of the Alfonsine Tables (see Chabás and Goldstein 2012, Table 15.4, p. 175).

16

Latitudes of Venus and Mercury

The tables for the latitude of Venus and Mercury are presented as six sub-tables for each planet, on ff. 41r–43v and ff. 44r–46v, respectively. Both are double argument tables. The vertical argument (center) is given in four columns at 6°-intervals and altogether there are 30 columns for the horizontal argument (anomaly), also at 6°-intervals. Each of these columns contains entries for the inclination (latitudo prima) and the slant (latitudo secunda). The deviation (latitudo tercia) is presented in another column, which remains invariant in all sub-tables for each planet. Table h displays an excerpt of the table for the latitude of Venus. The maximum values for Venus and Mercury are, respectively, 7;22° and 4;5° for the inclination, 2;30° and 2;30° for the slant, and +10′ and –13′ for the

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147

deviation. This table differs, strongly in presentation and slightly in the basic parameters, from tables by Parisian astronomers who included the third component of latitude, John Vimond and John of Murs.30 For the maximum values of deviation, Vimond had +10′and –45′ (Ptolemy’s values in the Almagest), for Venus and Mercury, respectively, whereas John of Murs had +10′and –23′, in contrast to +10′and –13′ in the tables reviewed here. John of Lignères was aware of the deviation, for he mentions it in the chapters on the latitude of Venus and Mercury in the canons of his Priores astrologi motus corporum celestium,31 but we do not know of any tables by him similar to those presented here. All in all, the tables for planetary latitudes in the Tables for the Seven Planets, although certainly in the same tradition, are not simply related to those by John Vimond and John of Murs, or by any other known tablemaker.

Conclusion Throughout the time from the reception of the Alfonsine Tables in Paris (no later than 1320) to the publication of the editio princeps in 1483 in Venice, an intense effort was made to adapt tables in the Alfonsine framework to the needs of practitioners. These adaptations focused on presentation rather than on the parameters underlying the tables of the Alfonsine corpus of tables. The set of tables which we have called the Tables for the Seven Planets for 1340 is an early example of this kind of work, with zodiacal signs of 30° (rather than physical signs of 60°), vacillation in the beginning of the year, and cyclical radices of 32 years, with special characteristics such as displacements, different parameter for the latitude of Mercury, and different terminology. The use of double argument tables and the extensive use of displacements, both vertical and horizontal, show a deep insight into planetary astronomy and great skill in producing astronomical tables. This set of tables is also the first example that has been discovered in the Latin world of a systematic use of displaced tables, of which only a few examples were previously known in the medieval astronomical literature. Unfortunately, no name is associated with this set, but the author was an astronomer working around 1340, probably in Southern France (judging by the geographical coordinates underlying these tables), who deserves the highest praise for his skill in providing clever and complex solutions to many

30 31

Chabás and Goldstein 2004, pp. 257–258; Chabás and Goldstein 2009, p. 309. An edition of chapters 22 and 23 is found in Saby 1987, pp. 207–211.

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problems, and for constructing a compact and consistent set of tables for the planets, building upon the work done by the Parisian group of astronomers in the early 1320s.

References Chabás, J. and B.R. Goldstein 2003. The Alfonsine Tables of Toledo. Archimedes: New Studies in the History and Philosophy of Science and Technology, 8. Dordrecht and Boston. Chabás, J. and B.R. Goldstein 2004. “Early Alfonsine Astronomy in Paris: The Tables of John Vimond (1320)”, Suhayl 4:207–294. Chabás, J. and B.R. Goldstein 2009. “John of Murs’s Tables of 1321”, Journal for the History of Astronomy 40:297–320. Chabás, J. and B.R. Goldstein 2012. A Survey of European Astronomical Tables in the Late Middle Ages. Boston. Debarnot, M.-T. 1987. “The Zīj of Ḥabash al-Ḥāsib: A Survey of ms Istanbul Yeni Cami 784/2”, in King and Saliba 1987, pp. 35–69. Goldstein, B.R. 1974. The Astronomical Tables of Levi ben Gerson. Transactions of the Connecticut Academy of Arts and Sciences, 45. New Haven. Goldstein, B.R., J. Chabás, and J.L. Mancha 1994. “Planetary and Lunar Velocities in the Castilian Alfonsine Tables”, Proceedings of the American Philosophical Society 138:61–95. Hogendijk, J. and A.I. Sabra 2003. The Enterprise of Science in Islam. Cambridge, ma. Jensen, C. 1971. “The Lunar Theory of al-Baghdādī”, Archive for History of Exact Sciences 8:321–328. Kennedy, E.S. 1977. “The Astronomical Tables of Ibn al-Aclam”, Journal for the History of Arabic Science 1:13–23. King, D.A. and M.H. Kennedy (eds.) 1983. Studies in the Islamic Exact Sciences by E.S. Kennedy, Colleagues and Former Students. Beirut. King, D.A. and G. Saliba (eds.) 1987. From Deferent to Equant: A volume of studies in the of history science in the ancient and medieval Near East in honor of E.S. Kennedy. Annals of the New York Academy of Sciences, 500. Kremer, R.L. (forthcoming). “Melchion de Friquento’s eclipse tables of 1437, a revised Latin version of Immanuel Bonfils’s Six Wings”. Kremer, R.L. and J. Dobrzycki 1998. “Alfonsine meridians: Tradition versus experience in astronomical practice c. 1500”, Journal for the History of Astronomy 29:187–199. Mercier, R. 1989. “The parameters of the Zīj of Ibn al-Aclam”, Archives Internationales d’Histoire des Sciences 39:22–50. Nallino, C.A. 1903–1907. Al-Battānī sive Albatenii Opus Astronomicum, 2 vols. Milan.

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Neugebauer, O. 1975. A History of Ancient Mathematical Astronomy. Berlin. Pedersen, F.S. 2002. The Toledan Tables: A review of the manuscripts and textual versions with an edition. Copenhagen. Poulle, E. 1984. Les tables alphonsines avec les canons de Jean de Saxe. Paris. Ratdolt, E. (ed.) 1483. Tabule astronomice illustrissimi Alfontij regis castelle. Venice. Saby, M.-M. 1987. Les canons de Jean de Lignères sur les tables astronomiques de 1321. Unpublished thesis: École Nationale des Chartes, Paris. 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. Salam, H. and E.S. Kennedy 1967. “Solar and Lunar Tables in Early Islamic Astronomy”, Journal of the American Oriental Society 87:493–497. Reprinted in King and Kennedy 1983, pp. 108–113. Saliba, G. 1976. “The Double-Argument Lunar Tables of Cyriacus”, Journal for the History of Astronomy 7:41–46. Saliba, G. 1977. “Computational Techniques in a Set of Late Medieval Astronomical Tables”, Journal for the History of Arabic Science 1:24–32. Samsó, J. 2003. “On the Lunar Tables in Sanjaq Dār’s Zīj al-Sharīf ”, in Hogendijk and Sabra 2003. Samsó, J. and E. Millás 1998. “The computation of planetary longitudes in the zīj of Ibn al-Bannāʾ”, Arabic Sciences and Philosophy 8: 259–286. Solon, P. 1970. “The Six Wings of Immanuel Bonfils and Michael Chrysokokkes”, Centaurus 15:1–20. Thorndike, L. 1957. “Notes on some Astronomical, Astrological and Mathematical Manuscripts of the Bibliothèque Nationale, Paris”, Journal of the Warburg and Courtauld Institutes 20:112–172. Tihon, A. 1977–1981. “Un traité astronomique chypriote du xive siècle”, Janus 64:281– 308; 66:49–81; 68:65–127. Reprinted in Tihon 1994, Essay vii. Tihon, A. 1994. Études d’astronomie byzantine. Aldershot. Toomer, G.J. 1968. “A survey of the Toledan Tables”, Osiris 15:5–174. Toomer, G.J. 1984. Ptolemy’s Almagest. New York. Van Brummelen, G. 1998. “Mathematical Methods in the Tables of Planetary Motion in Kūshyār ibn Labbān’s Jāmic Zīj”, Historia Mathematica 25:265–280.

chapter 6

Computing Planetary Positions: User-Friendliness and the Alfonsine Corpus* Astronomical tables are ways to turn the treatment of complex problems into elementary arithmetic. Since Antiquity astronomers have addressed many problems by means of tables; among them stands out the treatment of planetary motion as well as that for the motions of the Sun and the Moon. It was customary to assign to the planets constant mean velocities to compute their mean longitudes at any given time in the past or the future, and to add to these mean longitudes corrections, called equations, to determine their true longitudes. In this paper we restrict our attention to the five planets,1 with an emphasis on their equations. Part 1 deals with what we call the standard tradition, beginning with Ptolemy’s Handy Tables, and Part 2 deals with the new presentations that proliferated in Latin Europe in the fourteenth and fifteenth centuries, some of which reflect a high level of competence in mathematical astronomy.2

1

The Standard Tradition

By the middle of the second century ad Ptolemy displayed tables for the equations of the five planets with specific layouts and based on specific models, algorithms, and parameters. We argue that this category of tables, as is the case for many others, provides a clear example of user-friendliness, the driving force that prevailed in the history of table-making. In Almagest xi.11 Ptolemy presented tables for the planetary equations, one for each of the five planets.3 Each table has eight columns, of which the first two

* Journal for the History of Astronomy 44 (2013), 257–276, 479–480. 1 Unless otherwise specified, by planets we mean the five visible planets of Antiquity, although we are well aware that at the time the Sun and the Moon were also considered planets. 2 We do not treat tables in Islamic zijes systematically but, occasionally, we refer to some of them. For a survey of these zijes, see D.A. King and J. Samsó, with a contribution by B.R. Goldstein, “Astronomical Handbooks and Tables from the Islamic World (750–1900): An Interim Report”, Suhayl, ii (2001), 9–105. 3 G.J. Toomer, Ptolemy’s Almagest (New York and Berlin, 1984), 549–553.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_008

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are for the argument (one from 6° to 180° and the other for its complement in 360°). The argument is given at intervals of 6°, from 6° to 90° (and for 270° to 354°), and at intervals of 3°, from 90° to 180° (and for 180° to 270°). According to Toomer, Ptolemy computed the entries at 6°-intervals, even where the function is tabulated at 3°-intervals.4 Columns 3 and 4 are for the equation in longitude and the difference in equation, respectively. Column 3 assumes an eccentric model, which Ptolemy rejected in favor of an equant model. Column 4 displays the difference between the equation for an equant model and the equation for an eccentric model. The sum of corresponding entries in these two columns is the equation of center, which replaced columns 3 and 4 that appear in Almagest xi.11 (see Table a, col. 3).5 Columns 5 and 7 give the subtractive and additive differences to be applied to the equation of anomaly (displayed in col. 6), when the planet is at greatest and least distance, respectively. Column 8 is for the minutes of proportion, to seconds, used for interpolation purposes. We note that, in the case of Venus, the entries for the equation in longitude (col. 3) are exactly the same as those for the solar equation, although Ptolemy does not call attention to this fact.6 We display Ptolemy’s model for Mars to illustrate how a planet’s position can be computed directly from the model: see Figure 6.1. To do this, one must solve plane triangles by means of trigonometric procedures that were already available in Ptolemy’s time. The solution is as follows. Given κ̄, we wish to compute the correction angle, c3, by solving triangle eco. But, before we can do this, we have to find the length of ec, where dc, the radius of the deferent, is 60. So first we must solve triangle edc to find ec, where angle ced is the supplement to angle κ̄ and ed is the eccentricity (a given parameter in the model). With κ̄, ec and eo (twice the eccentricity), we can solve triangle eco, which yields the values for c3 and co. We then have to solve triangle mco to find c(α). In this triangle two sides and an angle are known: angle mco is equal to 180° – (ᾱ – c3), cm is the radius of the epicycle (a given parameter in the model), and co has already been determined. Then λ = λ(a) + κ̄ + c3 + c(α),

4 See Toomer, Almagest (ref. 3), 548, n. 54. 5 O. Neugebauer, A History of Ancient Mathematical Astronomy (Berlin, 1975), 183. 6 Although the equation of center for the Sun and the equation in longitude for Venus are the same in the Almagest (Toomer, Almagest (ref. 3), 167, 552), their apogees differ: the solar apogee is 65;30° and tropically fixed (Toomer, Almagest (ref. 3), 155), whereas the apogee of Venus is 55° in Ptolemy’s time and sidereally fixed, and thus subject to precession (Toomer, Almagest (ref. 3), 470; cf. Neugebauer, Ancient Astronomy (ref. 5), 58, 154, 182).

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chapter 6 figure 6.1 Ptolemy’s model for Mars. The observer is at o; the center of the deferent, ac, is at d; the equant is at e such that ed = do; the direction to the vernal equinox is ov; a is the apogee, c is the center of the epicycle such that dc = 60. The radius of the epicycle is cm; the mean apogee of the epicycle is at āe and the true apogee of the epicycle is at ae ; the mean longitude of Mars is in the direction om̄ , and its true position is at m. The mean argument of center is κ̄ and the mean argument of anomaly is ᾱ ; the equation of center is c3, and the equation of anomaly is c(α). The true longitude of Mars, λ, is angle vom. For an outer planet the direction cm is always parallel to os̄, where s̄ is the direction to the mean Sun.

where λ(a), the longitude of the apogee, is a given parameter in the model. Using the planetary equation tables takes trigonometric functions out of the computational scheme. In the Handy Tables Ptolemy did not modify the models or the parameters for the planetary equations, but he introduced a series of changes to make the tables more suitable for calculation. Firstly, the arguments are now given at intervals of 1°, rather than at intervals of 3° or 6°, as was the case in the Almagest.7 This certainly simplifies interpolation. Secondly, he merged columns 3 and 4 in the Almagest into a single column representing the equation of center, thus reducing the number of operations required for using these tables. This also reduced the number of columns, from 8 to 7. Thirdly, the column for the minutes of proportion was also modified by avoiding unnecessary precision (the entries are given to seconds in the Almagest but only to minutes in the Handy Tables) and by changing the argument (mean argument of center in the Almagest and true argument of center in the Handy Tables).8 This new 7 W.D. Stahlman, The Astronomical Tables of Codex Vaticanus Graecus 1291 (unpublished Ph.D. thesis, Brown University, 1959; University Microfilms no. 62–5761), 295–324. This dissertation includes an edition of Ptolemy’s Handy Tables. A. Tihon and R. Mercier are currently editing the Handy Tables; only two volumes have been published so far, but they do not deal with planetary equations. 8 See Neugebauer, Ancient Astronomy (ref. 5), 183–186 (Almagest) and 1002–1003 (Handy Tables).

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presentation (see Table a for Mars) set the standard for most tables dealing with planetary equations for about 14 centuries. table a

(1)

Equations for Mars in the Handy Tables (excerpt)9

(2)

Argument (°) (°) 1 2 3 … 86 87 88 89 90 … 92 93 … 96 97 … 130 131 132 … 152 153 … 156 157 158 159

359 358 357

(3) Equation of center (º)

(4) Min. prop. (′)

(5) Subtractive difference (º)

(6) Equation of anom. (º)

(7) Additive difference (º)

0;11 0;22 0;33

60 60 60

0; 2 0; 3 0; 4

0;24 0;48 1;12

0; 2 0; 3 0; 4

2;28

33;22

2;49

274 273 272 271 270

2 1 1 2

268 267

11;24 11;25

264 263

11;25 11;24

230 229 228

41; 9 41;10 41; 9

208 207

5;37 5;38

204 203 202 201

5;38 5;37

9 Stahlman, Tables (ref. 7), 307–312.

8; 2 8; 3

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table a

(1)

Equations for Mars in the Handy Tables (excerpt) (cont.)

(2)

Argument (°) (°) 160 … 178 179 180

(3) Equation of center (º)

(4) Min. prop. (′)

(5) Subtractive difference (º)

(6) Equation of anom. (º)

200 182 181 180

(7) Additive difference (º) 8; 2

0;27 0;14 0; 0

60 60 60

0;51 0;26 0; 0

3;52 1;57 0; 0

1;35 0;48 0; 0

In Table a, the mean argument of center, κ̄, serves as argument (columns 1 and 2) for the equation of center (col. 3), c3; the true argument of center, κ, can then be computed, for κ = κ̄ + c3(κ̄), where c3(κ̄) ≤ 0° when 0° ≤ κ̄ ≤ 180°. The mean argument of center serves also as argument for the minutes of proportion (col. 4), which are necessary to compute the true position of the planet when not found at maximum or minimum distance of the epicycle from the observer. Now the true argument of anomaly, α, serves as argument (columns 1 and 2) for the equation of anomaly (col. 6), and is obtained from the mean argument of anomaly: α = ᾱ – c3(κ̄). The tabulated equation of anomaly, c6(α), was originally computed assuming that the center of the epicycle is at mean distance. When the epicycle lies between maximum distance (apogee) and mean distance, a subtractive difference (col. 5) must be applied. Similarly, when the epicycle is between minimum distance (perigee) and mean distance, an additive difference (col. 7) must be applied. The true argument of anomaly serves as argument for both subtractive and additive differences. Then, the total equation of anomaly is c(α, κ̄) = c6(α) – c5(α) · c4(κ̄), when κ̄ ranges from 270° to 90°, that is, when the planet is near apogee, and

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c(α, κ̄) = c6(α) + c7(α) · c4(κ̄), when κ̄ ranges from 90° to 270°, that is, when the planet is near perigee. The combined effect of the equation of center and the total equation of anomaly is thus c3(κ̄) + c(α, κ̄), and the true position of the planet, λ, at a given time is: λ = λ̄ + κ̄ + c3(κ̄) + c(α, κ̄), where λ̄ , the mean longitude of that planet at a given time t since epoch, is defined as: λ̄ = λ0 + Δλ · t, λ0 being the planet’s mean longitude at epoch, and Δλ the planet’s mean motion in longitude. In the early Islamic world, the Zīj al-Sindhind of al-Khwārizmī (fl. 830) followed the Indo-Iranian tradition, which was not based on Ptolemaic models and parameters, and made no use of equants.10 This tradition was represented by the Zīj al-Shāh, a work composed in Sasanian Persia and translated into Arabic c. 790, where the maximum value for the equation of Venus is set equal to that of the solar equation (2;13° or 2;14°); the identity of these parameters is also found in the Almagest.11 Accordingly, the tables for the planetary equations are quite different, both with respect to the entries and the presentation, from those in the Almagest or the Handy Tables. The Greek tradition was represented in the eastern Islamic world by the Zīj al-Ṣābiʾ of al-Battānī (d. 929) which is strongly Ptolemaic; indeed, the tables in it for the planetary equations followed exactly those in the Handy Tables, but for the equation of center of Venus.12 Both the Almagest and the Handy Tables have 2;24° as the maximum value for Venus’s equation of center, whereas it is 1;59° in the zij of al-Battānī. This change in the equation of center of Venus was not due to new observations of Venus; rather, it was the result of a new value found from observations for the eccentricity of the solar model that 10 11

12

O. Neugebauer, The Astronomical Tables of al-Khwārizmī (Copenhagen, 1962). In this tradition the apogees of Venus and the Sun are the same and both are sidereally fixed: see B.R. Goldstein and F.W. Sawyer, “Remarks on Ptolemy’s Equant Model in Islamic Astronomy”, in Prismata: Festschrift für Willy Hartner, ed. by Y. Maeyama and W.G. Salzer (Wiesbaden, 1977), 165–181, p. 168. C.A. Nallino, Al-Battānī sive Albatenii Opus astronomicum (2 vols, Milan, 1903–1907), ii, 108–137.

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implied a maximum solar equation of 1;59,10°. This new solar parameter was simply applied to the equation of center for Venus, where it was rounded to 1;59°. In this modification al-Battānī followed other Islamic zijes, such as that of Ḥabash al-Ḥāsib (fl. 850).13 Toomer pointed out that modifying the entries for the equation of Venus was inconsistent with leaving unchanged the entries for the subtractive and additive differences (at greatest and least distances, respectively), because they also depend on eccentricity.14 In any case, in the zij of al-Battānī only the entries for the equation of center of Venus differ from those in the Handy Tables whereas all the rest remain unchanged. The Toledan Tables were compiled in the second half of the eleventh century, but the original Arabic version is not extant. In the Latin versions of the Toledan Tables the presentation and the numerical entries agree with those in the zij of al-Battānī, but for (in most cases) an added column for the first station of each of the planets.15 In Almagest xii.8 Ptolemy displayed the first and second stations of the five planets in a single table, using the mean center as argument, with entries at intervals of 6°.16 In the Handy Tables, Ptolemy gave more entries, at 3°-intervals, and presented a table for the two stations for each planet. He also introduced a change in the argument (true argument of center, instead of mean argument of center), thus making the entries slightly different from those in the Almagest.17 In his zij al-Battani reproduced in a separate table the entries for the first and second stations in the Handy Tables, and only displayed them at 6°-intervals. The compilers of the Toledan Tables probably realized that it was unnecessary to give entries for both the first and the second stations (because corresponding entries add up to 360°) and just included a specific column for the first station of each of the planets. Thus, in the tables for the planetary equations, ultimately derived from the Handy Tables, the number of entries

13

14 15 16 17

See Goldstein and Sawyer, “Equant” (ref. 11), 168. Ḥabash identified both the eccentricities and the apogees of Venus and the Sun, despite the lack of justification based on observations or based on anything said by Ptolemy in the Almagest (or elsewhere). In modern terms, this would mean that the solar orb serves as the deferent for Venus; but this claim was not made by any medieval scholar. Nevertheless, the medieval tradition was to keep the apogee and eccentricity of Venus equal to those of the Sun, such that whenever the parameters for the Sun were changed, the same changes were applied to Venus. See G.J. Toomer, “A survey of the Toledan Tables”, Osiris, xv (1968) 5–174, p. 67. Toomer, “Toledan Tables” (ref. 14), 60–68; and F.S. Pedersen, The Toledan Tables: A review of the manuscripts and the textual versions with an edition (Copenhagen, 2002), 1265–1306. Toomer, Almagest (ref. 3), 588. See Neugebauer, Ancient Astronomy (ref. 5), 1005–1006; and J. Chabás and B.R. Goldstein, A Survey of European Astronomical Tables in the Late Middle Ages (Leiden, 2012), 118.

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increased, for they are given here at intervals of one degree, and gained one column which was eliminated as a separate table.18 The Toledan Tables were by far the most popular tables in Latin Europe, and the presentation in them for tables of planetary equations can be considered standard. The Maghribī astronomers Ibn Isḥāq al-Tūnisī (c. 1193–1222), Ibn al-Bannāʾ of Marrakesh (1265–1321), and Ibn al-Raqqām (Tunis and Granada, d. 1315) used new parameters for the equations of center of Saturn, Jupiter, and Venus. In contrast, the values given to the equations of anomaly agreed precisely with those in the standard tradition, namely that of the Handy Tables, the zij of al-Battānī, and the Toledan Tables, but for the fact that in this tradition the columns displayed are combinations of cols. 5, 6, and 7.19 We further note that the tables for the planetary equations of Ibn Isḥāq and his followers depart from the standard tradition not only in the three basic parameters already mentioned, but also in presentation. Indeed, for each of the planets there are two tables of equations: one for quantities that depend on the argument of center and one for those that depend on the argument of anomaly.20 The Castilian Alfonsine Tables were produced in Castile by two astronomers working under the patronage of Alfonso x (reigned: 1252–1284), Judah ben Moses ha-Cohen and Isaac ben Sid. We do not know how the tables for the planetary equations were presented in these tables, because the tables themselves are not extant. However, the canons have been preserved in Castilian, and chapter 18 (De la equaçion de los v planetas) describes the way to compute planetary longitudes by means of tables. Although no numerical values are given, the description agrees perfectly with the layout of tables in the standard tradition of the Handy Tables, the zij of al-Battānī, and the Toledan Tables.21 This tradition was transmitted from the Iberian Peninsula to the rest of Europe. The earliest astronomer to depend on this Iberian tradition was John Vimond, who was active in Paris c. 1320. He compiled a set of tables which

18

19

20 21

The only known example of this kind of table where the entries are given at intervals of half a degree is preserved in a double folio now in the General Archive of Navarre: see J. Chabás, “The Toledan Tables in Castilian: Excerpts of the planetary equations”, Suhayl, xi (2012), 179–188. See J. Samsó and E. Millás, “The computation of planetary longtiudes in the zīj of Ibn al-Bannāʾ”, Arabic Sciences and Philosophy, viii (1998), 259–286; reprinted in J. Samsó, Astronomy and Astrology in al-Andalus and the Maghrib (Aldershot, 2007), Essay viii. A. Mestres, Materials Andalusins en el Zīj d’Ibn Isḥāq al-Tūnisī (unpublished Ph.D. thesis, Universitat de Barcelona, 1999), 50–51 and 234–235. J. Chabás and B.R. Goldstein, The Alfonsine Tables of Toledo (Dordrecht, 2003), 38–39 and 157–160.

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appear to be at the interface of the astronomy rooted in al-Andalus and the Maghrib and developed in Castile in the late thirteenth century on the one hand, and the activity of the astronomers working in Paris in the 1320s and the 1330s that resulted in the Parisian Alfonsine Tables on the other.22 In many ways Vimond’s tables follow a tradition unattested in Latin prior to 1320; for example, his tables for the planetary equations are also split into two tables for each planet, much as the Maghribī-Andalusian astronomers did. In addition to changes in structure, which will be examined later, the main modification in Vimond’s tables is found in the entries for the equations of center of Jupiter and Venus, with maximum values of 5;57° and 2;10°, respectively. Not much can be said about the value 5;57° other than it does not appear in any text or table prior to Vimond’s tables, and no medieval discussion of its origin has been found. However, the value 2;10°, also used by Vimond as the maximum solar equation, appears in previous texts: implicitly in a table for the daily solar positions for 1278 contained in the Libro del astrolabio llano composed by the astronomers in the service of King Alfonso x of Castile,23 and explicitly in an account in John of Murs’s Expositio of two observations of autumnal equinox, one by Ptolemy in 132 and the other attributed to Alfonso in 1252, where John explains that he has seen this observational report in what he calls the “Tables of Alfonso”.24 We thus think it likely that these two new values for Jupiter and Venus/Sun were taken from an earlier work, and the most reasonable candidate is the Alfonsine Tables in the original Castilian version.25 The Parisian Alfonsine Tables, produced in the 1320s by a group of astronomers working in Paris, were built on material coming from the Iberian Peninsula. They are best known today from the editio princeps that appeared in Venice in 1483. While each part of this printed edition has a complicated history, the planetary equation tables in it are faithful to the Parisian Alfonsine Tables as they were presented in the 1320s. The layout of the tables for the planetary equations conforms to the standard tradition, although they have no additional column for the first station. We will refer to the presentation and parameters of this version of the Alfonsine Tables as “standard”. The entries

22 23 24 25

J. Chabás and B.R. Goldstein, “Early Alfonsine Astronomy in Paris: The Tables of John Vimond (1320)”, Suhayl, iv (2004), 207–294, pp. 236–256. J. Chabás, “Were the Alfonsine Tables of Toledo first used by their authors?”, Centaurus, 45 (2003), 142–150. E. Poulle, “Jean de Murs et les tables alphonsines”, Archives d’histoire doctrinale et littéraire du moyen âge, xlvii (1980), 241–271, p. 253. Chabás and Goldstein, Toledo (ref. 21), 251–254.

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Maximum values of the equations of center and anomaly in various sets of tables (new values are shown in boldface type)

Saturn Jupiter Mars Venus Mercury Eq. of Eq. of Eq. of Eq. of Eq. of Eq. of Eq. of Eq. of Eq. of Eq. of center anom. center anom. center anom. center anom. center anom. Almagest * Handy Tables al-Khwārizmī al-Battānī Toledan Tab. Maghribī astr. Vimond ** Parisian Alf.

6;31° 6;31° 8;36° 6;31° 6;31° 5;48° 6;31° 6;31°

6;13° 6;13° 5;44° 6;13° 6;13° 6;13° 6;13° 6;13°

5;15° 5;15° 5; 6° 5;15° 5;15° 5;41° 5;57° 5;57°

11; 3° 11; 3° 10;52° 11; 3° 11; 3° 11; 3° 11; 3° 11; 3°

11;25° 11;25° 11;13° 11;25° 11;24° 11;25° 11;24° 11;24°

41; 9° 41;10° 40;31° 41; 9° 41; 9° 41; 9° 41; 9° 41;10°

2;24° 2;24° 2;14° 1;59° 1;59° 1;51° 2;10° 2;10°

45;59° 45;59° 47;11° 45;59° 45;59° 45;59° 45;59° 45;59°

3; 3; 4; 3; 3; 3; 3; 3;

2° 2° 2° 2° 2° 2° 2° 2°

22; 2° 22; 2° 21;30° 22; 2° 22; 2° 22; 2° 22; 2° 22; 2°

* The values for the equation of center shown here are found by adding algebraically the equation in longitude and the difference in equation in Alm. xi.11 (cols. 3 and 4). ** The values for the equation of center shown here result from subtracting the motus completus (col. 2) from the mean argument of center (col. 1). The values for the equation of anomaly shown here result from adding the motus completus (col. 2) to the correction for maximum distance (col. 5 in the standard tradition); see below.

are given at intervals of one degree, as was already established in the Handy Tables.26 Moreover, out of ten basic parameters for the five planets, only two differ from those defined by Ptolemy, namely, the equations of center of Jupiter and Venus, and both of these parameters are already found in John Vimond’s tables. It is difficult to find other examples of such great stability in the transmission of astronomical tables for more than 13 centuries. Table b provides a summary of the main parameters for the equations of center and anomaly used by different authors.

26

Characteristic of the Parisian Alfonsine Tables is the consistent use of sexagesimal days and angles. Angles are given in physical signs of 60° (in contrast to zodiacal signs of 30°): an angle a,b means a · 60 + b degrees (where a and b are integers such that 0 ≤ a ≤ 5 and 0 ≤ b ≤ 59; and 6,0° = 360°). In our notation 10s 25° means 10 · 30° + 25° = 325°, that is, “s” signifies a zodiacal sign of 30°. Sexagesimal fractions of a degree are used in the same way with both physical signs and zodiacal signs.

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A Proliferation of New Presentations

Prior to the first edition of the Parisian Alfonsine Tables in 1483, a variety of original approaches for presenting tables for the planetary equations were undertaken within the Alfonsine corpus. They coexisted with the standard tradition, which is preserved in a number of manuscripts dating from the fourteenth and fifteenth centuries.27 The goal of these new approaches that depart from the standard tradition was, once again, to facilitate computation. Let us return to about 1320, the date of John Vimond’s tables, in which the two equations for each planet are displayed in different tables. In those where all the tabulated functions depend on the mean argument of center (see Table c), the entries are given at 6°-intervals. Vimond displayed the true argument of center (col. 2: motus completus) and added columns for the increment of the true argument per degree of the argument (col. 3: motus gradus), planetary velocity (col. 4: motus diei), minutes of proportion (col. 5: diametri), and first station (col. 6). Moreover, the equation of center incorporates a displacement which is the difference between the apogee of each of the planets and that of the Sun (no displacement is therefore needed in the case of Venus, for its apogee is assumed to be the same as that for the Sun). Analysis of Vimond’s tables shows that the motion of the solar apogee was included in the motions of the planetary apogees, thus following a theory for which there was no previous evidence outside al-Andalus and the Maghrib.28 With this particular arrangement Vimond intended to present a more user-friendly table than the standard table for the equation of center. Now, in the tables where all the tabulated functions depend on the argument of anomaly (given at 6°-intervals for Saturn, Jupiter, and Mercury, and at 3°-intervals—and at 2°-intervals in the vicinity of 180°—for the other two planets), Vimond also added columns for planetary velocities and other corrections, such as col. 5 (see Table d), which results from adding the correction for maximum distance to the correction for minimum distance (cols. 5 and 7, respectively, in the Handy Tables, the zij of al-Battānī, and the Toledan Tables). As was the case for the equation of center, the entries for the equation of anomaly are not explicitly displayed. Rather, we are given entries for the motus completus (col. 2), which is the difference between the equation of anomaly and the

27 28

For example, for a list of manuscripts extant in Spain or of Spanish origin, see Chabás and Goldstein, Toledo (ref. 21), 292–303. Samsó and Millás, “Ibn al-Bannāʾ” (ref. 19), 268–270.

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computing planetary positions table c

(1)

(2) Motus completus s (°)

(3) Motus gradus min

(4) Motus diei min

Diametri min

6 12

0 12;31 0 17;36

50;50 50; 0

26 26

6 4

5 5

8;41 8;21

12 18

1 12;22 1 17;16

49; 0 49;10

26 25

0 0

5 5

7;29 7;31

18

4

6;36

60;30

31

32

5 13;46

12 18

7 11;33 7 18;54

73;30 73;10

38 38

60 59

5 19;14 5 19;13

12

10 23;24

58;50

30

31

5 13;36

24 0

12 12

51;50 51;10

27 26

10 8

5 5

Argument s (°) 0 0 … 1 1 … 4 … 7 7 … 10 … 11 12

John Vimond’s equation of center and first station of Mars (excerpt)

2;13 7;24

(5)

(6) First station s (°)

9;31 9; 6

correction for maximum distance. When we compute the differences between cols. 6 and 5 in the standard tradition, we find agreement with Vimond’s motus completus, indicating that he kept all the basic parameters for the equation of anomaly in this tradition. It is noteworthy that, as indicated by North, this implies that Ptolemy’s eccentricities underlie these tables even though, in the case of Venus and Jupiter, the eccentricities were modified for computing the equation of center.29 The only text of which we are aware that treats the equation of anomaly in this way is of Maghribī origin: the Minhāj of Ibn al-Bannāʾ, dependent on the zij of Ibn Isḥāq. In the Minhāj the tables for the equations of anomaly of Saturn and Jupiter give entries for al-mufrad (c6 – c5 in the standard terminology for columns) and al-bucd (c5 + c7).30 These are

29 30

J.D. North, Richard of Wallingford (3 vols, Oxford, 1976), iii, 196. Samsó and Millás, “Ibn al-Bannāʾ” (ref. 19), 278–285.

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table d

John Vimond’s equation of anomaly for Mars (excerpt)

(1)

s 0 0 … 4 4 4 … 5 5 5 … 5 6

Argument (°) s (°)

(2) Motus compl. (°)

(3) Motus gradus min

(4) Motus diei min

(5) Diametri min

(6) Motus grad. sec

(7) Motus diei sec

3 6

11 11

27 24

1; 8 2;16

23 23

11 11

0; 8 0;17

3 3

1 1

3 6 9

7 7 7

27 24 21

36;40 36;44 36;43

1 0 3

1 0 1

8;53 9;19 9;46

9 9 9

4 4 4

6 9 12

6 6 6

24 21 18

28;15 25;56 23;17

46 53 62

21 25 29

13;30 13;37 13;19

0 6 13

0 2 6

28 0

6 6

2 0

3; 1 0; 0

90 90

42 42

2;29 0; 0

74 74

35 35

precisely two of the columns found in Vimond’s tables (cols. 2 and 5). This particular choice of columns was intended to facilitate the computation of the planetary equations of anomaly. In addition to the Expositio, already mentioned, John of Murs, a key figure in the Parisian milieu for the transmission of Alfonsine astronomy, was responsible for a set of tables, called the Tables of 1321, devoted exclusively to the planets and the two luminaries. With these tables the computation of true planetary positions is entirely different from that described in any other text of which we are aware.31 The most significant feature of the Tables of 1321 is a new organizational principle, which does not require the equations of the planets to be displayed explicitly. To be sure, the mean motions of the planets are here presented in tables for the mean conjunctions of each planet with the Sun (tabula principalis), and the corrections to be applied for times between consecutive conjunctions are given in double argument tables

31

J. Chabás and B.R. Goldstein, “John of Murs’s Tables of 1321”, Journal for the History of Astronomy, xl (2009), 297–320.

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163

(contratabula). This particular approach meant that astronomers could avoid the typically cumbersome computations for determining true planetary positions, compared with using tables previously available in Latin. One unusual feature of these double argument tables is that the horizontal argument is the mean argument of center of the planet (at intervals of 12°) and the vertical argument is the “age of the planet”, that is, the time after a mean conjunction with the Sun, expressed as a number of days. It is also noteworthy that for each planet, besides the tabula and contratabula, we are given a table for its equation of center and first station. The values of the maximum equation of center agree in all cases with those used by Vimond, and so do the rest of the entries (given at 6°-intervals in both sets of tables, but presented differently). The tables of Vimond and those of John of Murs for 1321 certainly made the computation of the true positions of the planets much easier, but their approaches do not seem to have been very popular. The main improvement in that direction came from double argument tables, which greatly simplified computations and only required linear interpolation.32 John of Lignères (also active in Paris) was probably the first astronomer in Latin Europe to draw up a double argument table combining the equations of center and anomaly

32

There were a few double argument tables in Islamic zijes prior to 1320, e.g., Ibn Yūnus (c. 990) had such a table for the lunar equations as did al-Baghdādī (c. 1285): see D.A. King, “A Double-Argument Table for the Lunar Equation Attributed to Ibn Yūnus”, Centaurus, xviii (1974), 129–146 and C. Jensen, “The Lunar Theories of al-Baghdādī”, Archive for History of Exact Sciences, viii (1972), 321–328. Double argument tables for planetary latitudes attributed to Ibn al-Bayṭār, who is otherwise unknown, are preserved in Hyderabad, Andra Pradesh State Library, ms 298, Tables 66–77, where the horizontal headings are the true arguments of center and the vertical headings are the true arguments of anomaly (both at intervals of 6°): see A. Mestres, “Maghribī Astronomy in the 13th Century: a Description of Manuscript Hyderabad Andra Pradesh State Library 298”, in From Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet, ed. by J. Casulleras and J. Samsó (Barcelona, 1996), 383–443, p. 428. These tables are also cited in B. van Dalen, “Tables of Planetary Latitude in the Huihui li (ii)”, in Current Perspectives in the History of Science in East Asia, ed. by Y.S. Kim and F. Bray (Seoul, 1999), 316–329, p. 327. This manuscript contains the zij of Ibn Isḥāq (early thirteenth century), and Ibn al-Bayṭār is mentioned in chapter 18 of the canons to this zij. This implies that Ibn al-Bayṭār was active no later than the time of Ibn Isḥāq. A summary of chapter 18 appears in Mestres 1996, pp. 396–397. We are most grateful to van Dalen for sharing with us his translation of the Arabic text of chapter 18, the Arabic text of chapter 18 and of Tables 66–77, as well as his notes on this material.

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in a single table for each planet in his Tabule magne (c. 1325).33 The vertical argument is the mean argument of anomaly (at 6°-intervals in all planets, and also at 3°-intervals from 150° to 180° in the case of Mars and Venus),34 and the horizontal argument is the mean argument of center (at 6°-intervals). In Table e we reproduce an excerpt from the table for the combined equation of Venus in John of Lignères’s Tabule magne, as found in Lisbon, ms Ajuda 52-xii-35, ff. 83r–87v, with the title Tabula equationum ultimarum veneris. We note the use of physical signs of 60°, and the inclusion of columns for the differences, to minutes, of successive entries for a fixed value of the argument of anomaly (not displayed here).35 For each planet there is a total of at least 1860 entries (2160 in the case of Mars and Venus) presented as 31 × 60 or 36 × 60 matrices, not taking into account the columns and rows that display the successive differences. None of the entries explicitly corresponds to the maximum values of the equations of center or anomaly, which could lead to the identification of the tradition to which it

33

34

35

E. Poulle, “John of Lignères”, Dictionary of Scientific Biography (16 vols, New York, 1970– 1980), vii (1973), 122–128, pp. 123–124; J.D. North “The Alfonsine Tables in England”, in Prismata: Festschrift für Willy Hartner, ed. by Y. Maeyama and W.G. Salzer (Wiesbaden, 1977), 269–301, pp. 273–274, 278; reprinted in J.D. North, The universal frame: historical essays in astronomy, natural philosophy, and scientific method (London, 1989), 327–359. The reason is that in the range 150°–210° the entries for these two planets vary quite rapidly, and thus the accuracy of interpolation is increased by doubling the number of entries. Vimond was already aware of this rapid variation and, in his tables for the equation of anomaly for Mars and Venus given at 3°-intervals, he displayed entries in the range 168°–192° at 2°-intervals. Only three manuscripts containing theses tables are known. The other two use zodiacal signs of 30° (Erfurt, Biblioteca Amploniana, ms ca 2° 388, and London, British Library, ms add. 24070). All three have a column at the far right for the mean argument of anomaly from 180° to 360°, but the numbers in this column, and only in this one, are inverted in different ways in two manuscripts; for example, 234° is written as 54 3 (meaning 3,54°) in ms Lisbon; 24° 7s in ms Erfurt; and 7s 24° in ms London. It is possible that this reflects an archetype in Arabic or Hebrew, but see C. Burnett, “Why we read Arabic numerals backwards”, in Ancient and Medieval Traditions in the Exact Sciences, ed. by P. Suppes et al. (Stanford, 2000), reprinted in C. Burnett, Numerals and arithmetic in the Middle Ages (Aldershot, 2010), Essay vii, for some examples of this inversion in medieval Latin texts. Moreover, the three manuscripts differ in another respect: ms Lisbon has a column for successive differences, ms Erfurt has no such column, and ms London has columns and rows for successive differences. This is definitely a good example of the intervention of copyists when transmitting the very same table, without altering its presentation or any of its essential features.

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computing planetary positions table e

ᾱ (º)

0, 0 0, 6 0,12 … 2, 6 2,12 2,18 2,24 … 2,54 2,57 3, 0

John of Lignères’s double argument table for the combined equation of Venus (excerpt)

3,54° (º)

κ̄ 4,0° (º)

4,6° (º)

…

a 1; 0 **

a 1; 4

a 1; 8

… …

3;35 6; 6

3;39 6;10

43;51 44;20 40;18 41;44

… … … …

47;23 48;10 48;33 48;13

12;26 5;38 m 1;28

… … …

0,6° (º)

0,12° (º)

m* 0; 8 a 2;22 4;50

m 0;16 a 2;14 4;43

44; 2 44;32 44;29 43;41 13; 9 6;23 m 0;44

…

…

5,54° (º)

0,0° (º)

…

a 0; 8

a 0; 0

3;42 6;13

… …

2;38 5; 7

2;30 4;58

47;24 48;10 48;33 48;16

47;22 48; 8 48;30 48;14

… … … …

44;26 44;57 44;59 44;13

44;14 44;44 44;43 43;57

22;21 27;14

16;21 19;14

12;21 15;14

… …

14;32 7;44

13;52 7; 7

31; 6

24; 7

19; 7

…

0;39

0; 0

* m stands for minue (to be subtracted) and a for adde (to be added). ** ms Erfurt: 1;1.

belongs (see Table b), but a few entries are easy to track. Let us consider the case when κ̄ = 0° or 180°. Then c3(κ̄) = 0° and ᾱ = α, and the entries for ᾱ = 90° reduce to c6(90) – c5(90) and c6(90) + c7(90), respectively, in the usual terminology for columns. We find agreement in all cases, except for the equation for Mercury at greatest distance (the entry reads 21;32°, whereas computation with the tables in the standard tradition give 22;2°).36 In all other cases there is good agreement, but it is not always perfect because columns 5, 6, and 7, which depend exclusively on the argument of anomaly sometimes vary in the

36

In the case of Mars, for instance, c5(90) = 2;28°, c6(90) = 33;22°, and c7(90) = 2;49° in the standard tradition beginning in the Handy Tables (see Table a). Thus, c6(90) – c5(90) = 30;54° and c6(90) + c7(90) = 36;11°. The entry in John of Lignères’s table of Mars for κ̄ = 0° and ᾱ = 90° is 30;54° and that for κ̄ = 180° and ᾱ = 90° is 36;11°.

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minutes. To show that the entries in John of Lignères’s table are specifically based on the values used by John Vimond and, in particular, on those maximum values for the equation of center found for the first time in Vimond’s tables, we have recomputed a few critical entries.37 The maximum entries in John of Lignères’s tables could not have been computed with values as low as those in the tradition represented by the Toledan Tables, and we conclude that they were calculated with Vimond’s tables, or that both astronomers had a common source. It should be noted that in his tables of 1322 John of Lignères had used the parameters 1;59° (Venus) and 5;15° (Jupiter) that are found in the Toledan Tables for the maximum equations of center, replacing them with 2;10° (Venus) and 5;57° (Jupiter) in his double argument tables for the planetary equations in 1325. As a matter of fact, John of Lignères’s tables for planetary equations for 1322, as presented in Bibliothèque nationale de France, ms 7286c (ff. 33r–47v),38 share the same entries and layout, including a column for first station, with the Toledan Tables. This change was much the same as John of Murs had done a few years previously, given that in his earliest astronomical work of 1317, beginning Auctores calendarii …, he had praised the Tables of Toulouse and seemed unaware of Alfonsine material.39

37

38

39

For Venus, the maximum entry in John of Lignères’s table of 1325 is 48;33° (at κ̄ = 3,54° and 4,0°; and ᾱ = 2,18°) as displayed in Table e. If κ̄ = 4,0° = 240°, then c3(240) = 1;55°, using an equation of center with a maximum of 2;10° (Vimond’s value), and c4(240) = 31. Thus, α = 138 – 1;55 = 136;5°. Therefore, c5(136;5) = 1;11°, c6(136;5) = 45;59°, and c7(136;5) = 1;16°. Finally, the combined equation is 1;55° + 45;59° + (1;16 · 31/60) = 48;32°, in agreement with the entry. However, when performing the same calculation using an equation of center with a maximum of 1;59° (as in the Toledan Tables), one finds a combined equation of 48;19°. As we shall see in the computation that follows, the results are also unambiguous in the case of Jupiter. The maximum entry in John of Lignères’s table is 17;1° (at κ̄ = 4,24° and ᾱ = 1,48°). If κ̄ = 4,24° = 264°, then c3(264) = 5;57°, which is the maximum equation of center in Vimond’s tables, and c4(264) = 7. Thus, α = 108° – 5;57° = 102;3°. Therefore, c5(102;3) = 0;29°, c6(102;3) = 11;3°, and c7(102;3) = 0;32°. Finally, the combined equation is 5;57° + 11;3° + (0;32 · 7/60) = 17;4°, very close to the entry in John of Lignères’s Tabule magne. However, if we use a table with a maximum equation of center of 5;15° (as in the Toledan Tables), one finds a combined equation of 16;21°. For a description of this manuscript, see M.-M. Saby, Les canons de Jean de Lignères sur les tables astronomiques de 1321, (Unpublished thesis: École Nationale des Chartes, Paris, 1987), 516–520. A summary appeared as “Les canons de Jean de Lignères sur les table astronomiques de 1321”, École Nationale des Chartes: Positions des thèses (1987), 183–190. Chabás and Goldstein, Toledo (ref. 21), 278. The Tables of Toulouse are an adaptation of the Toledan Tables for the Christian calendar (instead of the Muslim calendar): see Poulle,

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Double argument tables undoubtedly facilitated computation, because they displayed in a compact and clever way intermediate calculations needed to obtain a final numerical result.40 This kind of presentation was not an invention of the Parisian astronomers, for it is already found in Arabic sources, e.g., it was used by Ibn al-Kammād (Córdoba, c. 1100) in his tables for the time from mean to true syzygy as a function of the difference between the hourly velocities of the Moon and the Sun and the elongation.41 Double argument tables proliferated in fourteenth-century Europe and were not restricted to the planetary equations: they were also used to display true planetary positions (the Tabule anglicane, also called the Oxford Tables of 1348, associated with William Batecombe); planetary conjunctions (John of Murs’s Tables of 1321); planetary latitudes (John of Murs’s Tables of 1321, and the Oxford Tables); syzygies (John of Murs and Firmin of Beauval in their Tabulae permanentes, Immanuel ben Jacob Bonfils of Tarascon, Levi ben Gerson, Juan Gil of Burgos, Joseph Ibn Waqār of Seville, and the Tables of Barcelona); lunar motion (Levi ben Gerson); and lunar and planetary velocities (Judah ben Asher ii of Burgos).42

40

41

42

“Un témoin de l’ astronomie latine du xiiie siècle: les Tables de Toulouse”, in Comprendre et maîtriser la nature au moyen âge: Mélanges d’histoire des sciences offerts à Guy Beaujouan (Geneva and Paris, 1994), 55–81. For interpolation in double argument tables, see M. Husson, “Ways to read a table: reading and interpolation techniques of early fourteenth-century double-argument tables”, Journal for the History of Astronomy, xliii (2012), 299–319. These tables are only extant in Latin and Hebrew versions; the Latin version was composed by John of Dumpno in 1260 in Palermo and survives uniquely in Madrid, Biblioteca Nacional de España, ms 10023. See J. Chabás and B.R. Goldstein, “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for History of Exact Sciences, xlviii (1994), 1–41; and B.R. Goldstein, “Solomon Franco on the Zero Point for Trepidation”, Suhayl, x (2011), 77–83. For the Tabule anglicane see J.D. North, “England” (ref. 33); for the Tables of 1321, see J. Chabás and B.R. Goldstein, “John of Murs” (ref. 31); for the Tabulae permanentes, see B. Porres and J. Chabás, “John of Murs’s Tabulae permanentes for finding true syzygies”, Journal for the History of Astronomy, xxxii (2001), 63–72; for Immanuel ben Jacob Bonfils of Tarascon, see P. Solon, The ‘Hexapterygon’ of Michael Chrysokokkes (Brown University, Ph.D. thesis (unpublished), 1968; Proquest, umi, aat 6910019), and P. Solon, “The Six Wings of Immanuel Bonfils and Michael Chrysokokkes”, Centaurus, xv (1970), 1–20; for Levi ben Gerson, see B.R. Goldstein, The Astronomical Tables of Levi ben Gerson (Hamden, ct, 1974); for Juan Gil of Burgos and Joseph Ibn Waqār of Seville, see J. Chabás and B.R. Goldstein, “Computational Astronomy: Five Centuries of Finding True Syzygy”, Journal for the History of Astronomy, xxviii (1997), 93–105, pp. 94–96; for the Tables of Barcelona, see J.M. Millás, Las Tablas Astronómicas del Rey Don Pedro el Ceremonioso (Madrid and Barcelona, 1962), and J. Chabás, “Astronomia andalusí en Cataluña: Las Tablas de Barcelona”, in From

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Another set of tables in the Alfonsine corpus that adheres strictly to its parameters and models is the set we call the Tables for the Seven Planets for 1340; they are a most ingenious reworking of the Parisian Alfonsine Tables and include several displaced tables. The purpose of displaced tables is to eliminate all subtractions in the derivation of planetary positions, thus facilitating computations.43 This anonymous set of tables, most likely of French origin, is uniquely preserved in Paris, Bibliothèque nationale de France, ms 10262 (ff. 2r– 46v). The two planetary equations are given in separate tables for each planet, and are not explicitly displayed. Rather, for the equation of center we are given entries which are displaced both vertically and horizontally with respect to those in the standard Parisian Alfonsine Tables, whereas the entries for the equation of anomaly are only displaced vertically. In modern algebraic terms, the vertical and horizontal displacements of a function underlying a displaced table are such that y = f(x + kh) + kv, where y = f(x) is the original function to which the displaced table is compared, kh is the displacement on the x-axis, and kv is the displacement on the y-axis. Tables f and g display excerpts of the equation of center of Jupiter in the Tables for the Seven Planets and in the Parisian Alfonsine Tables, respectively (where kh = 18° and kv = 6°). Figure 6.2 illustrates the situation for Jupiter. The graph labeled ms 10262 displays the entries in the Tables for the Seven Planets for 1340, and that labeled pat corresponds to those in the Parisian Alfonsine Tables. In general the vertical displacements are intended to avoid complicated rules for addition and subtraction corresponding to the simple rules we now give by means of algebraic signs. The horizontal displacements are intended to counterbalance other displacements, such as those applied to the minutes of proportion. It is easy to recognize that the vertical displacements of the entries for the equation of anomaly agree with the parameters found in the Parisian Alfonsine Tables: see Table b. The Tables for the Seven Planets use a total of 40 different displacements for the planets (including the Sun and the Moon): see Table h.

43

Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet, ed. by J. Casulleras and J. Samsó (Barcelona, 1996), 477–525; for Judah ben Asher ii of Burgos, see B.R. Goldstein, “Abraham Zacut and the Medieval Hebrew Astronomical Tradition”, Journal for the History of Astronomy, xxix (1998), 177–186, pp. 179–181. For details, see J. Chabás and B.R. Goldstein, “Displaced Tables in Latin: The Tables for the Seven Planets for 1340”, Archive for History of Exact Sciences, lxvii (2013), 1–42. Note that the term “displaced” applied to tables was coined by E.S. Kennedy in 1977 as a translation of the Arabic waḍʿī (see his “The Astronomical Tables of Ibn al-Aclam”, Journal for the History of Arabic Science, i (1977), 13–23, espec. p. 16).

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computing planetary positions table f

(º) 0 1 2 … 71 72 … 77 78 79 … 168 … 245 246 … 252 253 … 259 260 … 359

44

45

Equation of center of Jupiter in the Tables for the Seven Planets44

Equatio Minutes centri of proportion (º) … (′) 4;15 4; 9 4; 3

0 1 1

0; 4 0; 3

53 54

0; 3 0; 3 0; 4

59 2 3

6;39

60

11;56 11;57

14 13

11;57 11;56

7 6

11;52 11;51

1 58

4;20

0

table g

(º) 1 2 3 … 88 89 90 … 96 97 … 180 … 263 264 … 270 271 272 273 … 358 359

Equation of center of Jupiter in the Parisian Alfonsine Tables45

Equation Minutes of center of proportion (º) (′) … 0; 6 0;12 0;12

60 60 60

5;56 5;56 5;57

1 1 2

5;57 5;56

7 8

0; 0

60

5;56 5;57

8 7

5;57 5;56 5;56 5;55

2 1 1 2

0;12 0; 6

60 60

Note that the equatio centri is always positive; it reaches a minimum of 0;3° at 72°–78°, and a maximum of 11;57° at 246°–252°. For the minutes of proportion there are two discontinuities (from 60′ to 0′ between 77° and 78°, and from 0′ to 60′ between 259° and 260°), to keep them positive in all cases. Note that the equation of center is negative between 0° and 180°, and positive between 180° and 360°; it reaches a minimum of –5;57° at 90°–96°, and a maximum of 5;57° at 264°–270°. The minutes of proportion are positive between 89° and 271°, and negative for the rest of the arguments.

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figure 6.2 The equation of center of Jupiter. The graph labeled ms 10262 is displaced vertically by 6° and horizontally by 18° with respect to the graph labeled pat. The maximum of the upper graph is 11;57° and takes place at arguments 246°–252°; the maximum of the lower graph is 5;57° and takes place at arguments 264°–270°.

In any case, computation with this compact and consistent set of tables gives the same results as those obtained with the Parisian Alfonsine Tables, while avoiding subtractions at any stage in the computation. In the fifteenth century the Paduan astronomer, Prosdocimo de’ Beldomandi (d. 1428), compiled a new set of tables that belong to the Alfonsine corpus.46 His tables for the planetary equations follow the Parisian Alfonsine Tables, including the 2;10° and 5;57° used by Vimond for Venus and Jupiter, in agreement with those that were printed in 1483 in the editio princeps. Giovanni Bianchini (d. after 1469) spent most of his life in Ferrara where he served as administrator for the estate of the prominent d’Este family. About 1442 he compiled an extensive set of astronomical tables which depend on the Alfonsine Tables, but have a completely different presentation.47 Bianchini’s tables offer a whole new approach for computing the true positions of the

46 47

J. Chabás, “From Toledo to Venice: The Alfonsine Tables of Prosdocimo de’ Beldomandi”, Journal for the History of Astronomy, xxxviii (2007), 269–281. J. Chabás and B.R. Goldstein, The Astronomical Tables of Giovanni Bianchini (Leiden, 2009).

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Displacements of the planetary equations in the Tables for the Seven Planets for 1340

Eq. of center Eq. of anomaly Vert. displac. Horiz. displac. Vert. displac. Saturn Jupiter Mars Venus Mercury

7° 6° 12° 3° 4°

14° 18° 61° 51° 28°

6;13° 11; 3° 41;10° 45;59° 22; 2°

planets. Although tables for the planetary equations are not explicitly given, the true positions of the planets are computed by means of double argument tables where the vertical argument is the mean anomaly, represented here by the time within an anomalistic period for each planet, and the horizontal argument is the mean center. These tables were first published in 1495 in Venice under the title Tabulae astronomiae, and again in 1526 and 1553.48 The Tabulae resolutae were compiled in central Europe, and circulated widely in manuscripts during the fifteenth century and in print during the sixteenth century.49 One of their characteristics is that the mean motions are arranged according a system of cyclical radices at intervals of 20 year. The Tabulae resolutae are also strictly based in the Parisian Alfonsine Tables; in fact, they are a particular form of presenting them. The tables for the planetary equations display the same parameters as the Parisian Alfonsine Tables but, unlike them, add a column for first station, thus following the layout of most versions of the Toledan Tables. In Vienna John of Gmunden (c. 1380–1442) collected a great variety of tables within the framework of the Parisian Alfonsine Tables. He displayed them in

48

49

For a list of manuscripts that contain Bianchini’s tables, see Chabás and Goldstein, Bianchini (ref. 47), 14. The owners of manuscript copies of these tables include Johannes Regiomontanus (Nuremberg, Stadtbibliothek, Cent v 57) and Johannes Virdung (Vatican, Biblioteca Apostolica, ms Pal. lat. 1375). J. Dobrzycki, “The Tabulae Resolutae”, in De Astronomia Alphonsi Regis, ed. by M. Comes, R. Puig, and J. Samsó (Barcelona, 1987), 71–77; J. Chabás, “Astronomy in Salamanca in the Mid-Fifteenth Century: The Tabulae Resolutae”, Journal for the History of Astronomy, xxix (1998),167–175; and J. Chabás, “The Diffusion of the Alfonsine Tables: The Case of the Tabulae resolutae”, Perspectives on Science, x (2002), 168–178.

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various sets, called “First Version”, Tabulae maiores, and Tabulae breviores.50 He presented his tables for the equations of the planets in three different ways: at 1°-intervals following the standard tradition; at 3°-intervals in an abridged form of the latter; and as double argument tables, reproducing those by John of Lignères. Therefore, with respect to the planetary equations, John of Gmunden was not an innovator; rather, he offered table users several possibilities that were already known in Latin Europe. In the early sixteenth century Johannes Angelus, a follower of Peurbach and Regiomontanus, claimed that these two authors had compiled a new table of planetary equations giving better results than the standard Alfonsine Tables, but this “new” table has not been found in any manuscript or printed edition.51 As already mentioned, the Alfonsine Tables were first printed in 1483 by Erhard Ratdolt in Venice. A few years later (1492) and in the same town, a second edition appeared, edited by Johannes Lucilius Santritter. The entries for the planetary equations are the same in both sets of tables but, in the second edition, the planets were inexplicably presented in the order Venus, Mercury, Mars, Jupiter, and Saturn (rather than in the usual order where Mercury precedes Venus). Also in the 1492 edition, the second column for the argument, displaying the complement in 360°, was eliminated; this left enough space on the page to include five extra columns for the differences between successive entries in the remaining columns. In 1503 Petrus Liechtenstein printed another set of tables in Venice, the Tabule Astronomice Elisabeth Regine. It was much less popular than the standard version of the Parisian Alfonsine Tables, but it is historically significant because in his Commentariolus Copernicus cited its author, Alfonso de Córdoba, who was in the service of Pope Alexander vi in Rome.52 The tables for the planetary

50 51

52

B. Porres, Les tables astronomiques de Jean de Gmunden: édition et étude comparative (unpublished Ph.D. thesis, École pratique des hautes études, Section iv, Paris, 2003). J. Dobrzycki and R.L. Kremer, “Peurbach and Marāgha Astronomy? The Ephemerides of Johannes Angelus and Their Implications”, Journal for the History of Astronomy, xxvii (1996), 187–237, pp. 187–188. Alfonso de Córdoba, known as “Hispalensis”, came from Seville (Latin: Hispalis) and his origin is well attested in various printed texts of the early sixteenth century. His place of origin is explicitly given as “patria hispalensis” and he is cited as “Alfonso hispalensi de Corduba”. Hence there is no reason to emend the text to “hispaniensis”, that is, from Spain, as has been suggested: see N.M. Swerdlow, “The Derivation and First Draft of the Copernicus’s Planetary Theory: A Translation of the Commentariolus with Commentary”, Proceedings of the American Philosophical Society, cxvii (1973), 423–512, pp. 451–452. Copernicus’s Commentariolus is undated, but it is usually taken to be from about 1514. Alfonso dedicated his

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Alfonso de Córdoba’s equation of center of Mars (excerpt)

Mars Longitude Leo 15 Leo 20 Leo 25 … Sco 15 Sco 20 Sco 25 … Aqr 5 Aqr 10 Aqr 15

Leo 15 Leo 10 Leo 5 … Tau 15 Tau 10 Tau 5 … Aqr 25 Aqr 20 Aqr 15

(°)

min

0; 0 0;55 1;49

60 60 59

11;23 11;24 11;21

3 3 8

2;13 1; 7 0; 0

58 59 60

equations, as well as all the others in this set, depend on the Parisian Alfonsine Tables, both for models and parameters. However, this is not true for the presentation. First, for each planet the equation of center is given in a different table from that for the equation of anomaly. As we have seen, John Vimond had also used this two-fold presentation, which was most uncommon in Latin astronomy,53 but the two sets differ in several important aspects (see Table i). Second, the argument in the tables for the equation of center is given at 5°-intervals (as is the case in the tables for the equation of anomaly), in contrast to the tables in the standard tradition (1°-intervals). But most important of all is the fact that the argument in the table for the equation of center represents

53

work to Queen Isabella of Castile and Aragon (1451–1504), whose name in Latin was Elisabeth. On this set of tables, see J. Chabás, “Astronomy for the Court in the Early Sixteenth Century: Alfonso de Córdoba and his Tabule Astronomice Elisabeth Regine”, Archive for History of Exact Sciences, lviii (2004), 183–217. This separation was intended to distinguish clearly between the columns that depend on one variable from those that depend on the other. We know of another example of this two-fold presentation in Erfurt, Biblioteca Amploniana, ms q 362 (ff. 28r–36r), also in the Alfonsine corpus, whose layout seems unrelated to those by either John Vimond or Alfonso de Córdoba.

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the mean longitude of the planet, λ̄ , that is, the mean argument of center plus the longitude of the planet’s apogee. Thus, the argument is shifted by a quantity that, in each case, corresponds to the longitude of the apogee (Leo 15° in the case of Mars). Again, the purpose is to facilitate calculation. In turn, the tables for the equation of anomaly display the usual columns of the tables in the standard tradition (cols. 1, 2, 5, 6, and 7).54 The tabular innovations developed to facilitate computation of the true longitude of the planets paved the way to a substantial increase in the number of almanacs in the fourteenth and fifteenth centuries. Although there are some earlier examples of this genre, the various new presentations of the tables for planetary equations (such as double argument tables) made almanacs much easier to compile. In turn, since almanacs and ephemerides display directly the true positions of the planets at successive times, the user did not have the difficult task of computing planetary equations; hence, they were very popular, for they could be used even by those who had not mastered all the subtleties of astronomy.55 Perhaps the most elaborate and influential almanac in the late Middle Ages was the Almanach perpetuum.56 Its tables, together with a short explanatory text, were first printed in two editions (one in Latin and the other in Castilian) in Leiria, Portugal, in 1496. The tables were derived from a set of astronomical tables in Hebrew called ha-Ḥibbur ha-gadol (The Great Composition) compiled by Abraham Zacut of Salamanca (1452–1515).57 As regards the positions of the planets, Zacut’s work was compiled in the framework of the Parisian Alfonsine Tables with 1473 as epoch. For each planet it gives the true longitude, the true argument of center, and the true argument of anomaly for several days in each

54

55

56

57

The fact that the argument in the table for the equation of center was chosen to be the mean longitude of the planet makes the table less useful in the long term, for it does not take into account the slow motion of the apogee due to precession. An almanac lists successive true positions of each planet at intervals of one day or a few days, that is, each planet is listed separately; hence, the information for a given day is scattered among various tables. An ephemeris displays the true positions of all planets in a single row at intervals of one day or a few days for a certain number of years. That is, the difference between an almanac and an ephemeris is only one of presentation. J. Chabás and B.R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the Transition from Manuscript to Print (Transactions of the American Philosophical Society, 90.2, Philadelphia, 2000). On the various editions of the Almanach perpetuum and its impact on the Jewish community as well as on Christian and Muslim scholars, see Chabás and Goldstein, Zacut (ref. 56), 161–171.

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month (sometimes daily) for periods as long as 125 years in the case of the longitude of Mercury. For these three quantities there is a total of more than 42,100 entries and, in each case, the sign, the degrees, and the minutes are specified. We have certainly come a long way from the tables for the planetary equations in the Handy Tables!

Conclusion As regards planetary equations, the standard tradition, going back to Ptolemy’s Handy Tables, survived at least until the first printed editions of the Parisian Alfonsine Tables. Ptolemy’s underlying models and most of the parameters involved were rarely challenged from about the middle of the second century to the end of the fifteenth century. Only two parameters appearing in the tables were changed in that period, and John Vimond seems to have been the first astronomer to have used them in Latin Europe.58 Vimond depended on material from the Iberian Peninsula, most likely of Arabic origin. Astronomers in Latin Europe in the fourteenth and fifteenth centuries were actively engaged with this well-defined tradition, but they did not simply reproduce the tables and texts of their predecessors, and many of them developed innovative approaches to facilitate computational tasks, such as double argument tables, displaced tables, separated tables, or shifted variables. Userfriendliness, rather than improvement of the models or enhancement of precision, was the driving force for most of the efforts developed by table-makers in the computation of planetary positions. Nevertheless, the editio princeps of the Parisian Alfonsine Tables did not incorporate any of the various new presentations. These innovations in presentation have only been recognized in recent years and, taken together, they indicate that astronomers in Latin Europe reached a high level of mathematical competence in the late Middle Ages. 58

To be sure, the apogee of Venus was also changed, but this does not modify the table for the equations for this planet.

part 3 Sets of Tables

∵

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Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād*1 Introduction In 1956 E.S. Kennedy published his A Survey of Islamic Astronomical Tables in which he described, briefly in most cases, over 100 sets of tables, called zijes (after the Arabic: zīj), from the 8th to the 15th centuries and from all parts of the medieval Islamic world. At that time only two of them had been published, and it was clear that our understanding of scientific activity in the Middle Ages would be greatly enhanced by detailed treatment of the others. Indeed, such has proven to be the case, as we learn from the many studies that have followed this pioneering essay. Astronomers in Islamic Spain, al-Andalus, composed zijes and, beginning in the 12th century, they were adapted and translated into Hebrew, Latin, Castilian, and Catalan, the most famous examples being the Toledan Tables (see Toomer (1968)) and the Alfonsine Tables. In Spain, as elsewhere in the Islamic world, these zijes were largely based on the work of predecessors going back to Ptolemy on the one hand, and Hindu astronomers on the other. More often than not, a table comes with instructions for using it, rather than the method used to construct it. For this reason, much scholarly energy has been devoted to describing the methods underlying these tables, as well as their lines of descent. By such analysis, guided by textual material, one can now distinguish tables that are based on entirely new models, tables that are merely copies of tables in previous zijes (at the two extremes), from tables based on previous models but with new parameters and tables composed by modified or new mathematical methods. The work of Ibn al-Kammād, an Andalusian astronomer of the 12th century, illustrates most of these characteristics. He composed 3 zijes, none of which survives in the original Arabic, but a Latin manuscript contains a translation of what appears to be one complete zij with references to the others. Ibn

* Archive for History of Exact Sciences 48 (1994), 1–41, communicated by J. North. 1 The authors thank Professor J. Samsó (Barcelona) for his valuable comments.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_009

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al-Kammād depended on sources that ultimately go back to Ptolemy and to Hindu astronomers as they were known in zijes prior to his time, and his influence was felt by later astronomers writing in Arabic, Latin, and Hebrew. As is often the case, this work preserves parts of otherwise lost texts: in particular, we gain valuable information on the solar theory of the Andalusian astronomer Azarquiel (or Ibn al-Zarqālluh), who lived in the 11th century (see Toomer (1969)). In this article our primary intention is to describe the astronomical work of Ibn al-Kammād as it is preserved in Latin in ms Madrid 10023. Ibn al-Kammād is cited or criticized by a number of his successors: Ibn al-Hāʾim writing in Arabic ca. 1205 (see Samsó (1992), pp. 321 ff.); Abū l-Ḥasan cAlī al-Marrākushī (ca. 1262) (see Millás (1950), p. 347); Juan Gil (ca. 1350) whose astronomical tables are preserved in a Hebrew version (see Goldstein (1985), p. 237); Ibn al-Ḥadib, a 14th-century astronomer from Spain who went to Sicily where he wrote astronomical works in Hebrew (see Goldstein (1985), p. 239); Joseph Ibn Waqār, a Spanish astronomer of the mid-14th century who wrote in Arabic and Hebrew (see Goldstein (1985), p. 237); and most importantly, as we shall see, Ibn al-Kammād had a profound influence on the Tables of Barcelona (see Millás (1962)). In the discussions that follow new information of particular interest for the history of astronomy in Spain is presented: see, e.g., the solar equation table (Section ii); the preservation of material that goes back to the zij al-Mumtaḥan of Yaḥyā ibn Abī Manṣūr who lived in the 9th century (Section iii, c, f and j); the table for trepidation (Section iv, b); and the tables for planetary latitude (Section iv, e). In sum, Ibn al-Kammād was a major player in medieval Spanish astronomy; his achievements and the extent of his legacy have not yet been sufficiently appreciated.

i

Ibn al-Kammād: Life and Works

This is the name by which the astronomer Abū Jacfar Aḥmad ben Yūsuf Ibn al-Kammād is known, although a variety of names has been associated with him (Vernet (1949), pp. 72–73). He was probably from Sevilla, but active in Córdoba in the 12th century. In a remarkable study, Millás (1942), pp. 231–247, called attention to a 14th-century Latin manuscript at the Biblioteca Nacional de Madrid, ms 10023, containing one of Ibn al-Kammād’s works; he described it and gathered all the information then available on this astronomer. Not much more is known now. Ibn al-Kammād is the author of 3 zijes:

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1. al-Kawr ʿalā al-dawr (the periodic rotations [?]) (in 60 chapters). This treatise has been partially preserved in Arabic in ms Escorial Ar. 939,4 (not seen by us). There is also a short text in Castilian in Segovia, Biblioteca de la Catedral, ms 115, ff. 218vb–220vb, attributed to Yuçaf Benacomed, and entitled “Libro sobre çircunferencia de moto”. 2. al-Amad ʿalā al-abad (the eternally valid [tables]) 3. al-Muqtabis (the compilation [of the two previous works]). ms Madrid 10023 contains the Latin translation of al-Muqtabis. In the explicit it is clearly stated that the translation was done by John of Dumpno in 1260 in Palermo. In Arabic only chapter 28 has survived: see ms Alger 1454,2, ff. 62–63. The dates for Ibn al-Kammād are uncertain. Millás (1950), p. 346, considered the period “towards the end of the 12th century”, and suggested the year 1195 as that of his death, probably following Ahlwardt (1893), p. 219, where this date is given with no specification of his source. More recently, Ibn al-Kammād has been taken to be a “direct disciple of Azarquiel”. This claim is based on a note in the margin of f. 30r of ms lat. 7281, a 15th-century manuscript at the Bibliothèque nationale de France, and it has been argued that this claim is supported by the date ah480 (1087–1088ad) that appears in ms Madrid 10023, f. 65v: see Section v, m, below. The marginal note, already transcribed in Millás (1950), p. 14, reads: “Post uenit Alcamet discipulus …”, referring to Azarquiel. According to Millás, the same hand, or a similar one, has added: “Similiter discipulus Messalle”. It does not seem at all warranted to deduce from this expression that Ibn al-Kammād was a “direct pupil” of Azarquiel. Instead, we understand this to mean only that Ibn al-Kammād was a follower of Azarquiel’s methods. On the other hand, the date in ms Madrid 10023, f. 65v, is not the only one mentioned in the last section of this manuscript (e.g., on f. 66r there is a table for ah550: see Section v, r, below); as we shall see, the last section of this ms contains a variety of tabular material not directly related to al-Muqtabis. Samsó (1992), p. 322, noted that Ibn al­ Hāʾim (fl. 1205) criticized Ibn al-Kammād. Since the available evidence suggests that Ibn al-Kammād lived after Azarquiel and before Ibn al-Hāʾim, we conclude that Ibn al-Kammād lived in the 12th century, without offering any greater precision. Texts in ms Madrid 10023 a al-Muqtabis Text in Latin, presented in two columns. – Introduction (f. 1ra–va). Transcribed in Millás (1942), pp. 231–232. – Index (ff. 1va–2rb). Transcribed in Millás (1942), pp. 234–235.

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– Canons, 1 to 30, each of them called “porta” (ff. 2rb–18vb). Three chapters have been published so far: canon 1 is transcribed in Millás (1942), pp. 235– 236, canon 28 in Vernet (1949), pp. 74–78, and canon 30 in Millás (1942), pp. 237–238. Note that canon 30 explicitly mentions the other two works of Ibn al-Kammād: al-Kawr ʿalā al-dawr and al-Amad ʿalā al-abad. – Explicit (f. 18vb). Transcribed in Millás (1942), p. 238. b Other Texts From f. 18vb to f. 24rb there is a set of chapters, in some disorder, that are distinct from those of al-Muqtabis. Some of them are associated with al-Kawr ʿalā al-dawr, and were also translated by John of Dumpno in 1262 in Palermo; their incipits and explicits were published in Millás (1942), pp. 238–242. Millás also transcribed some of the texts therein, and Toomer (1969), pp. 323–324, transcribed and translated a text concerning the variation of solar eccentricity. Tables in ms Madrid 10023 Two sets of tables can easily be distinguished in the manuscript: a al-Zīj al-Muqtabis (ff. 27r–54v) The tables are mentioned, or their use is explained, in the text in 30 canons called al-Muqtabis. The tables are calculated for the meridian of Córdoba. Folio 54v contains the last table of al-Muqtabis (a geographical table), as we learn from canon 10 (f. 6va): “… tabulam longitudinum terrarum positam in ultimo huius canonis”. We will discuss all the tables in al-Muqtabis as follows: the solar equation in Section ii, eclipse theory in Section iii, and the remaining tables in Section iv. b Other Tables (ff. 55r–66r) These tables do not form a homogeneous set. They are not mentioned in the 30 canons of al-Muqtabis, and are presumably distinct from that work. Some of the tables are related to al-Kawr ʿalā al-dawr, some are attributed to astronomers other than Ibn al-Kammād, and still others are calculated for places other than Córdoba. These tables will be discussed in Section v.

ii

The Solar Equation in Al-Muqtabis Tabula directionis centri solis et augis eius ad primordium annorum seductionis (f. 35r)

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Above the heading, the longitude of the solar apogee, presumably for the Hijra, is given: “Aux” 2s 16;45,21°. This table displays the solar equation in degrees, minutes, and seconds as a function of mean solar anomaly. The maximum solar equation, which occurs at 92°, is 1;52,44°, thus differing from the more common values: 1;59° (Ḥabash al-Ḥāsib, Yaḥyā ibn Abī Manṣūr), 1;59,10° (Toledan Tables, al-Battānī), 2;14° (al-Khwārizmī), 2;23° (Ptolemy). The use of this table is explained in canon 13 (f. 7vb). In his Tractatus super totam astrologiam, Bernardus de Virduno (ca. 1300) attributes an eccentricity of 1;58 to Azarquiel, which yields a maximum equation of 1;52,42° (see Toomer (1987), pp. 515–517 and Samsó (1992), p. 213). However, this is not the only parameter used by Azarquiel for, in the Alfonsine translation of his treatise on the construction of the equatorium, 1;52,30° is explicitly called a rounded parameter for the maximum solar equation (see Samsó (1987), p. 468). It is therefore likely that Ibn al-Kammād accepted a parameter from Azarquiel. The columns in Table 1, The Solar Equation in al-Muqtabis, are arranged as follows: (1) Mean solar anomaly, κ̄, for each integer degree from 1° to 180°. (2) Entries in the text (f. 35r): solar equation, c(κ̄), in degrees, minutes, and seconds. (3) Line-by-line differences in (2): c(κ̄ + 1) – c(κ̄). This column allows us to recognize quite a number of errors in the entries for the solar equation in the text, for these should increase monotonically from anomaly 0° to reach a maximum at anomaly 92°, and then decrease monotonically to anomaly 180°. Consequently, the set of line-by-line differences should follow a smooth decreasing pattern. In a great number of cases, pairs of successive line-by-line differences in (3), whether erroneous or not, are identical, thus strongly suggesting that the table for the solar equation was originally computed for every other degree, and that interpolation was used to derive the rest of the entries. (4) Reconstructed line-by-line differences. Not all the line-by-line differences which do not fit smoothly in (3) have been reconstructed; we have only made suggestions in those cases where the tabulated values for the solar anomaly can be easily derived from the reconstructed values. A question mark indicates that, although there is a peculiar value in (3), we do not offer any alternative value. (5) Reconstructed values for the solar equation. These values result from an easy explanation of the way the copying, or the calculational error was made. There follows a list of various kinds of such errors:

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– misreading of a digit: c(22) = 0;40,19 instead of 0;40,59 (19 and 59 are easily confused in Arabic); c(28) = 0;50,27 instead of 0;51,27; – inversion of the order of the entries: in the text, the seconds for c(175) and c(176) are apparently inverted; – incorrect interpolation between computed entries: c(30) and c(32) were computed correctly, but c(31) seems to result from an incorrect interpolation between them; – displacement of columns; from c(145) to c(150) the entries for the minutes have been shifted one line downwards. We have left unchanged those “erroneous” entries of the solar equation for which we do not have an easy explanation; some of them may be due to incorrect computation by the author of the table rather than to copyist errors. (6) Recomputed values: for the recomputation of the entries, we used a simple eccentric model, and the following expression: [1]

tan(c) = e · sin(κ̄)/(60 + e · cos(κ̄))

where the eccentricity used is e = 1;58,2 = 60 · sin (1;52,44). (7) Differences in seconds: T(ext)–C(omp.), i.e., col. (2) – col. (6); in those cases where we have confidence in our reconstructed values, this column displays the differences between the reconstructed values in col. (5), and computation in col. (6). table 1

(1)

The Solar equation in al-Muqtabis

(2)

(3)

1 2 3 4 5 6 7 8 9 10

0; 1,53 3,47 5,40 7,34 9,25 11,25 13,18 15,11 17, 4 0;18,57

0; 1,53 1,54 1,53 1,54 1,51 2, 0 1,53 1,53 1,53 0; 1,53

11

20,50

1,53

(4)

1,56 1,55

(5)

(6)

(7)

0; 1,54 3,49 5,43 7,37 9,30 9,31 11,24 13,18 15,12 17, 5 0;18,58

–1 –2 –3 –3 –1 +1 0 –1 –1 –1

20,50

0

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(1)

(2)

(3)

12 13 14 15 16 17 18 19 20

22,42 24,39 26, 9 28,13 30,17 32, 1 33,45 35,35 0;37,25

1,52 1,57 1,30 2, 4 2, 4 1,44 1,44 1,50 0; 1,50

21 22 23 24 25 26 27 28 29 30

39,12 40,19 42,45 44,30 46,– 48,16 49,44 50,27 53, 7 0;54,48

1,47 1, 7 2,26 1,45 – – 1,28 0,44 2,40 0; 1,41

31 32 33 34 35 36 37 38 39 40

56,58 58, 7 0;59,45 1; 1,23 2,38 4,33 6, 7 7,37 9, 9 1;10,41

2,10 1, 9 1,38 1,38 1,15 1,55 1,34 1,30 1,32 0; 1,32

41 42 43 44 45

12,10 13,39 15, 7 16,35 17,56

1,29 1,29 1,28 1,28 1,21

(4)

? ? ? ? ? ?

1,47 1,46 1,46 1,44 1,44 1,44 1,40

1,40 1,39

1,35 1,35

(5)

(6)

(7)

22,42 0 24,34 +5 26,26 –15 28,16 –3 30, 7 30, 7 0 31,57 –4 33,46 –1 35,35 0 0;37,24 +1 39,11 40,58 42,45 44,30 46,16 46,15 48, 0 47,59 49,43 51,27 51,25 53, 7 0;54,48

+1 +1 0 0 +1 –1 +1 +2 0 0

56,28

56,27 58, 7 0;59,44 1; 1,21 2,58 2,57 4,32 6, 6 7,38 9,10 1;10,40

+1 0 +1 +2 +1 +1 +1 –1 –1 +1

12, 9 13,37 15, 4 16,29 17,53

+1 +2 +3 +6 +3

40,59

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table 1

(1)

The Solar equation in al-Muqtabis (cont.)

(2)

(3)

46 47 48 49 50

19,17 20,33 21,58 23,45 1;24,32

1,21 1,16 1,25 1,47 0; 0,47

51 52 53 54 55 56 57 58 59 60

25,47 27, 2 28,19 29,27 30,37 31,46 32,50 33,54 34, 7 1;36, 0

1,15 1,15 1,17 1, 6 1,10 1, 9 1, 4 1, 4 0,13 0;1,53

61 62 63 64 65 66 67 68 69 70

36,58 37,59 38,53 39,49 40,48 41,37 42,25 43,13 43,59 1;44,43

0,58 1, 1 0,54 0,56 0,59 0,49 0,48 0,48 0,47 0; 0,44

71 72 73 74 75 76 77 78

45,25 46, 6 46,44 47,21 47,57 48,33 49, 1 49,29

0,42 0,41 0,38 0,37 0,36 0,36 0,28 0,28

(4)

1,21 1,20 1,17 0; 1,17

1,13 1,12

1, 3 0; 1, 3 ? ?

(5)

(6)

(7)

19,15 20,38 20,37 21,57 23,15 23,15 1;24,33

+2 +1 +1 0 –1

25,48 27, 3 28,15 28,15 29,26 30,36 31,44 32.51 33,56 34,57 34,59 1;36, 1

–1 –1 0 +1 +1 +2 –1 –2 –2 –1

37, 1 37,59 38,56 39,51 40,44 41,35 42,25 43,12 43,58 1;44,43

–3 0 –3 –2 +4 +2 0 +1 +1 0

45,25 46, 5 46,44 47,21 47,55 48,28 48,59 49,28

0 +1 0 0 +2 +5 +2 +1

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(1)

(2)

(3)

79 80

49,54 1;50,19

0,25 0; 0,25

81 82 83 84 85 86 87 88 89 90

50,51 51, 3 51,22 51,41 51,58 52,15 52,14 52,19 52,29 1;52,40

0,32 0,12 0,19 0,19 0,17 0,17 –0, 1 0, 5 0,10 0; 0,11

91 92 93 94 95 96 97 98 99 100

52,42 0, 2 52,44 0, 2 52,42 –0, 2 52,39 –0, 3 52,33 –0, 6 52,27 –0, 6 52,16 –0,11 52, 6 –0,10 51,50 –0,16 1;51,34 –0; 0,16

101 102 103 104 105 106 107 108 109 110

51,22 –0,12 50,59 –0,23 50,41 –0,18 50,22 –0,19 49,48 –0,34 49,18 –0,30 48,50 –0,28 48,16 –0,34 47,31 –0,45 1;47, 5 –0; 0,26

111

46,55

–0,10

(4)

(5)

(6)

(7)

49,55 1;50,20

–1 –1

50,43 51, 4 51,23 51,40 51,56 52, 9 52,20 52,29 52,35 1;52,40

–2 –1 –1 +1 +2 +1 –1 0 0 0

52,43 52,44 52,43 52,39 52,34 52,26 52,17 52, 6 51,52 1;51,36

–1 0 –1 0 –1 +1 –1 0 –2 –2

–0,35 –0,36

51,18 50,58 50,36 50,12 50,12 49,46 49,18 48,48 48,16 47,41 47,41 1;47, 5

+4 +1 +5 0 +2 0 +2 0 0 0

–0,40

46,25

–2

0,22 0,22

50,41

0,12 0, 9 0,10 0, 6 0; 0, 5

52,10 52,19 52,29 52,35

? ?

–0,26 ?

46,27

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table 1

(1)

The Solar equation in al-Muqtabis (cont.)

(2)

(3)

(4)

(5)

(6)

(7)

112 113 114 115 116 117 118 119 120

46,45 –0,10 –0,40 45,45 45,41 45,33 –1,12 –0,42 45, 3 45, 4 44,20 –1,13 –0,43 44,20 43,12 –1, 8 –0,48 43,32 43,33 42,22 –0,50 42,42 42,45 42,22 0, 0 –0,50 41,52 41,55 41, 1 –1,21 –0,51 3 41, 3 40,56 –0, 5 –0,55 40, 6 40, 9 1;39,52 –0; 1, 4 –0; 0,54 1;39,12 1;39,13

+4 –1 0 –1 –3 –3 –2 –3 –1

121 122 123 124 125 126 127 128 129 130

38,52 –1, 0 37,53 –0,59 36,51 –1, 2 35,51 –1, 0 33,25 –1,26 32,58 –0,27 31,50 –1, 8 30,39 –1,11 29,25 –1,14 1;28,11 –0; 1,14

38,15 37,15 36,13 35,10 34, 5 32,57 31,48 30,38 29,25 1;28,11

–3 –2 –2 +1 0 +1 +2 +1 0 0

131 132 133 134 135 136 137 138 139 140

27,25 26,40 24,50 23, 0 21,54 20,11 18,44 17,18 15,49 1;14,20

–0,46 –0,45 –1,50 –1,50 –1, 6 –1,43 –1,27 –1,26 –1,29 –1,29

26,55 25,37 24,20 22,57 21,34 21,35 20,10 18,45 17,17 15,49 1;14,18

0 +3 0 +3 –1 +1 –1 +1 0 +2

141 142 143 144

12,47 11,13 9,39 8, 6

–1,33 –1,34 –1,34 –1,33

12,47 11,13 9,39 8, 3

0 0 0 +3

–1, 6 –1, 7

–1,15 –1,15 –1,20 –1,20 –1,26 –1,23

38,12 37,13 36,11 35,11 34, 5

26,55 25,40 24,20

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andalusian astronomy

(1)

(2)

(3)

(4)

145 146 147 148 149 150

7,26 6,49 5, 8 3,27 1; 1,10 0;59,19

–0,40 –0,37 –1,41 –1,41 –2,17 –1,51

–1,40 –1,37

151 152 153 154 155 156 157 158 159 160

57,21 54,31 52,28 50,37 48,47 47,17 44,55 43,32 41,39 0;39,46

–1,58 –2,50 –2, 3 –1,51 –1,50 –1,30 –2,22 –1,23 –1,53 –1,53

–1,38 –1,50 ? ? ? ? –1,52 –1,53

161 162 163 164 165 166 167 168 169 170

37,51 35,57 34, 2 32, 6 30,14 28,19 26,16 24,13 22,14 0;20,15

–1,55 –1,54 –1,55 –1,56 –1,52 –1,55 –2, 3 –2, 3 –1,59 –1,59

171 172 173 174 175 176 177 178

18,14 16,43 14,12 12,11 10, 8 8,10 6, 6 4, 4

–2, 1 –1,31 –2,31 –2, 1 –2, 3 –1,58 –2, 4 –2, 2

(5)

(6)

6,26 6,26 4,49 4,47 3, 8 3, 7 1; 1,27 1; 1,26 –1,37 0;59,50 0;59,43 0;57,59 0;58, 0

(7) 0 +2 +1 +1 +7 –1

56,21

56,15 +6 54,29 +2 52,42 –14 50,54 –17 49, 6 –19 47,15 +2 45,25 45,24 +1 43,33 +1 41,40 –1 0;39,46 0

? ? ?

–2, 1 –2, 1

16,13

–2, 1 –2, 2 –2, 2

10,10 8, 8

37,52 35,57 34, 1 32, 5 30, 8 28,10 26,11 24,13 22,13 0;20,14

–1 0 +1 +1 +6 +9 +5 0 +1 +1

18,13 16,13 14,12 12,11 10,10 8, 8 6, 6 4, 4

+1 0 0 0 0 0 0 0

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table 1

The Solar equation in al-Muqtabis (cont.)

(1)

(2)

179 180

2, 2 0; 0, 0

iii

(3)

(4)

(5)

–2, 2 –2, 2

(6) 2, 2 0; 0, 0

(7) 0 0

Eclipse Theory in Al-Muqtabis

All the tables related to eclipse theory are mentioned in canons 25 (ff. 13rb–14rb) and 27 (ff. 15ra–16va). Some specific terminology in Latin is used in canon 25: “longitudo” for elongation, “preuentio” for opposition, “peruenencia solis” for the fraction of the elongation that “belongs to” the Sun, “precessio” for the hourly relative velocity of the luminaries. a

Solar and Lunar Velocities Tabula diuersi motus solis in una hora quod est respectus (f. 51v) Tabula diuersi motus lune in una hora quod est respectus (f. 51v)

The use of these two tables is explained in canon 25 (f. 13va). Both tables coincide, except for copying errors, with those in al-Khwārizmī (Suter (1914), pp. 175–180, tables 61–66). The extremal values are: vs(1°) = 0;2,22° /h, and vs(180°) = 0;2,24° /h (read: 0;2,34° /h). vm(1°) = 0;30,12° /h, and vm(180°) = 0;35,40° /h. The same two tables are also found in the tables of Juan Gil (London, Jews College, ms Heb. 135, f. 91r), and the Tables of Barcelona (Millás (1962), table 43), as well as in a manuscript containing the Tables of Toulouse (Paris, BnF, ms Lat. 16658, ff. 90v–93r). The Toledan Tables (cf. Toomer (1968), p. 82) and the Almanac of Azarquiel (Millás (1950), p. 174) have tables for solar and lunar velocities that agree with those in al-Battānī but differ from the present ones.

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For the solar velocity table, the formula used in recomputing the entries is: [2] v = v̄ + v̄ · Δ. where v̄ = 0;59,8/24 and Δ = c(κ̄ + 1) – c(κ̄); c is the solar equation as a function of the mean anomaly, κ̄, taken from al-Khwārizmī’s tables 21–26, col. 2. The recomputed values (which are not displayed here) are in good agreement with the entries in the table. Equation [2] has been used by analogy with equation [3], used for computing the lunar velocity table (cf. Goldstein (1992); Goldstein et al.): [3] v(α) = 0;32,56 + 0;32,40 · Δ, where Δ = c(α + 1) – c(α); c is the lunar equation as a function of the anomaly, α, taken from al-Khwārizmī’s tables 21–26, col. 3. The results are shown in Table 2, together with a comparison between the entries in the tables of Ibn al-Kammād and al-Khwārizmī. The original table seems to be better preserved by al-Khwārizmī than by Ibn al-Kammād. lt should be noticed that previously this table had not been recomputed successfully (cf. as-Saleh (1970), p. 162, and Neugebauer (1962), p. 106). The columns in table 2, Lunar Velocity, are arranged as follows: (1) Lunar anomaly, α, for each integer degree from 1° to 180°. (2) Entries in the text (f. 51v): lunar velocity in minutes and seconds of are per hour. (3) Variant readings in al-Khwārizmī’s lunar velocity table (Suter (1914), tables 61–66, col. 3); only the seconds are displayed here, except for 72°. (4) Recomputed values at multiples of 5° of anomaly; only the seconds are displayed here, except for 135°. table 2

(1)

(2)

1 2 3 4 5

30;12 30;12 30;12 30;13 30;13

Lunar velocity

(3)

(4)

(1)

(2)

12

91 92 93 94 95

32;58 33; 0 33; 2 33; 5 33; 7

13

(3)

(4)

15

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table 2

(1)

(2)

6 7 8 9 10

30;13 30;14 30;14 30;14 30;15

11 12 13 14 15 16 17 18 19 20

30;15 30;15 30;16 30;17 30;18 30,19 30;20 30;21 30;22 30;23

21 22 23 24 25 26 27 28 29 30

30;24 30;25 30;26 30;27 30;28 30;29 30;30 30;32 30;33 30;34

31 32 33 34 35 36 37 38

30;36 30;37 30;38 30;40 30;41 30;43 30;44 30;46

Lunar velocity (cont.)

(3)

(4)

(1)

(2)

14

96 97 98 99 100

33;10 33;13 33;16 33;20 33;23

101 102 103 104 105 106 107 108 109 110

33;26 33;29 33;32 33;36 33;39 33;42 33;45 33;48 33;51 33;54

111 112 113 114 115 116 117 118 119 120

33;57 34; 0 34; 3 34; 5 34; 9 34;12 34;15 34;17 34;20 34;22

121 122 123 124 125 126 127 128

34;26 34;27 34;30 34;32 34;36 34;37 34;40 34;45

13

14

17 18 19 20 21 22

17

22

23 24 25 26

34

39

(3)

(4)

26

44

54

6 12

24 25

34

42

36

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andalusian astronomy

(1)

(2)

39 40

30;47 30;49

41 42 43 44 45 46 47 48 49 50

30;51 30;52 30;54 30;56 30;58 31; 0 31; 2 31; 4 31; 6 31; 8

51 52 53 54 55 56 57 58 59 60

31;10 31;12 31;14 31;16 31;21 31;23 31;25 31;27 31;29 31;29

61 62 63 64 65 66 67 68 69 70

31;32 31;35 31;37 31;40 31;42 31;45 31;47 31;49 31;51 31;54

71

31;57

(3)

(4)

(1)

47

129 34;45 130 34;48

56

7

18 21 23 25 27

17

29

42

52 55

(2)

131 132 133 134 135 136 137 138 139 140

34;50 34;53 34;55 34;58 35; 0 35; 1 35; 3 35; 4 35; 5 35; 6

141 142 143 144 145 146 147 148 149 150

35; 8 35; 9 35;10 35;12 35;13 35;14 35;16 35;17 35;18 35;20

151 152 153 154 155 156 157 158 159 160

35;21 35;22 35;24 35;25 35;26 35;27 35;28 35;29 35;30 35;31

161 35;32

(3)

(4)

49

34,57

7

17

22

49

31

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table 2

(1)

Lunar velocity (cont.)

(2)

(3)

72 73 74 75 76 77 78 79 80

31; 1 32; 1 32; 5 32; 8 32;12 32;17 16 32;23 20 32;27 24 32;27 32;31

81 82 83 84 85 86 87 88 89 90

32;34 32;37 32;39 32;42 32;46 32;47 32;49 32;52 32;54 32;56

b

45

(4)

12

32

55

57

(1)

(2)

162 163 164 165 166 167 168 169 170

35;33 35;33 35;34 35;35 35;35 35;36 35;37 35;37 35;38

171 172 173 174 175 176 177 178 179 180

35;38 35;38 35;39 35;39 35;39 35;39 35;39 35;40 35;40 35;40

(3)

(4)

36

38

39

40

Time from Mean to True Syzygy Tabula horarum longitudinis ad dirigendum tempus coniunctionis et preuentionis (f. 52r)

The use of this table is explained in canon 25 (f. 13va). It is a double argument table: the vertical argument is the elongation (e), given in degrees and minutes, from 0;30° to 12;0° at intervals of 0;30°. The horizontal argument is the velocity of the Moon relative to that of the Sun (vm – vs), in minutes and seconds of arc per hour, from 0;27,30° /h to 0;33,30° /h, at intervals of 0;0,30°/h. Each entry (t) can be computed from the following equation: [4] t = e/(vm – vs),

andalusian astronomy

195

where t is the time, given in hours and minutes, that the Moon takes to travel the longitudinal arc between the Sun and the Moon at mean syzygy, i.e., the time interval from mean to true syzygy. For a discussion of tables for finding the time from mean to true syzygy, see Chabás and Goldstein. The entries in the column for 0;32,0° /h are also copied, erroneously, in the column for 0;32,30°/h. A similar, but not quite identical, table is found in the Tables of Barcelona (Millás (1962), table 42). c

Solar Eclipses Tabula rectitudinum ad eclipses solares (f. 52v)

This table in 7 columns (see table 3) is mentioned in canon 27 (f. 15rb), and gives the declination of midheaven, i.e., the intersection of the ecliptic and the local meridian, in degrees and minutes, as a function of the longitude of the ascendant. The manuscript uses “septentrionalis” for north, and “meridionalis” for south, which we have transcribed by assigning positive or negative signs, respectively, to the tabulated entries. The declination reaches its maximum of 23;50° at Libra 1° and its minimum of –23;51° at Aries 0°. Note that 23;51° is the value used by Ptolemy in the Handy Tables for the obliquity of the ecliptic. This table is closely related to a table found in an Arabic manuscript (ms Escorial Ar. 927, ff. 9v, 12r, 12v: see Kennedy (1986)), where it is called “the Table of samt for determining solar eclipses” ( jadwal al-samt li-cilm kusūf al-shams). Kennedy and Faris (pp. 21–24) have described—and presented a graph—of this table, based on the copy in the Escorial where it appears among tables attributed to Yaḥyā ibn Abī Manṣūr (9th century). The entries in this table in ms Madrid 10023, f. 52v, are displayed in table 3 and, for comparison, those in the Arabic manuscript in the Escorial are displayed in a separate table (see table 4), where the underlining of entries indicates that they differ from those in table 3. Both tables are clearly variants of the same archetype despite many discrepancies that can be ascribed to copyist errors. The Arabic copy displays 180 additional entries because it fails to recognize the symmetries (see Kennedy and Faris, p. 24). In addition to the value adopted for the obliquity of the ecliptic, the table depends on the latitude of the place for which it is intended. In this case, the latitude used seems to be 35;55,48° (cf. ms Escorial Ar. 927, f. 8v), even though the use of seconds for geographical latitude was not meaningful at the time; this latitude corresponds to Yaḥyā’s native Ṭabaristan (cf. Kennedy and Faris, p. 24). However, it is quite likely that Ibn al-Kammād, who is associated with Córdoba (latitude = 38;30° as it appears in the geographical table on f. 54v), did not know

196 table 3

chapter 7 Table of samt for determining solar eclipses (ms Madrid 10023, f. 52v)

Degree of the ascendant 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

Lib

Sco

Sgr

Cap

Aqr

Psc

+23;50 23;50 23;48 23;47 23;45 23;42 23;39 23;36 23;32 23;23 23;16 23;12 23; 5 22;59 22;52 22;44 22;37 22;23 22;18 22;11 21;51 21;40 21;38 21;18 21; 7 20;55 20;42 20;29 20;16 +20; 3

+19;25 19; 5 18;50 18;35 18;33 18; 2 17;46 17;29 17;12 16;55 16;36 16;18 15;41 15; 3 14;24 14; 5 13;24 13; 4 12;22 12;39 11;39 11;33 10;36 10;12 9;52 9;44 8;42 8;19 7;23 +7;10

+6;47 6;23 6; 0 5;53 5;50 4; 1 3;37 3;33 2;50 2; 1 1;36 0;48 +0;24 0; 0 –0;24 0;48 1;36 1; 1 2;20 3;53 3;37 4; 1 4;25 5;13 5;36 6; 0 6;24 7;10 7;18 –7;16

–8;35 8;32 9;27 9;50 10;12 10;32 10;53 11;18 11;59 12; 1 12;22 12;48 13; 4 13;23 13;43 14; 5 14;24 14;44 15; 3 15;23 15;41 16; 0 16;18 16;36 16;44 16;56 17;12 17;27 17;46 –18; 2

–18;34 18;50 19; 5 19;19 19;20 19;35 19;50 20; 3 20;16 20;33 20;35 20;39 20;47 20;52 20;55 20;58 21; 0 21; 4 21; 7 21;18 21;29 21;40 21;41 21;51 21; 1 22;10 22;10 22;19 22;28 –22;35

–22;47 22;49 22;52 22;57 22;59 23; 0 23; 5 23;12 23;17 23;18 23;23 23;24 23;26 23;27 23;32 23;36 23;37 23;39 23;39 23;42 23;42 23;45 23;45 23;47 23;48 23;48 23;50 23;50 23;51 –23;51

Vir

Leo

Cnc

Gem

Tau

Ari

197

andalusian astronomy table 4

Table of samt for determining solar eclipses (ms Escorial Ar. 927, ff. 9v, 12r, and 12v)*

Degree of the ascendant 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

Lib

Sco

Sgr

Cap

Aqr

Psc

+23;50 – 23;48 23;46 23;45 23;42 23;39 23;36 23;32 23;23 23;16 23;12 23; 5 22;59 22;52 22;44 22;37 22;27 22;18 22;11 21;51 21;40 21;30 21;18 21; 7 20;55 20;42 20;29 20;16 +20; 3

+19;25 19; 5 18;50 18;35 18;33 18; 2 17;56 17;29 17;12 16;55 16;36 16;18 15;41 15; 3 14;24 14; 5 13;24 13; 4 12;22 12;30 11;39 11;33 10;36 10;12 9;50 9; 5 8;42 8;19 7;13 +7;10

+6;47 6;23 6; 0 5;53 5; 7 5; 1 4;37 4;13 3;50 3; 1 2;36 1;48 +0;24 0; 0 –0;24 0;48 1;36 2; 1 2;25 3;53 3;36 4; 1 4;24 5;13 5;36 6; 0 6;24 7;10 7;13 –7;16

–8;39 8;42 9;27 9;50 10;12 10;32 10;56 11;18 11;39 12; 1 12;22 12;48 13; 5 13;23 13;43 14; 5 14;24 14;44 15; 3 15;23 15;41 16; 0 16;18 16;36 16;44 16;55 17;12 17;26 17;46 –18;10

–18;34 18;50 19; 5 19;15 19;25 19;35 19;50 20; 3 20;18 20;33 20;35 20;39 20;42 20;47 20;55 20;58 21; 0 21; 5 21; 7 21;18 21;29 21;40 21;41 21;51 21; 1 22;10 22;14 22;19 22;28 –22;32

–22;39 22;45 22;51 22;56 23; 0 23; 4 23; 8 23;11 23;14 23;17 23;20 23;23 23;27 23;28 23;30 23;32 23;34 23;36 23;38 23;40 23;41 23;42 23;43 23;44 23;45 23;46 23;47 23;48 23;49 –23;50

Vir

Leo

Cnc

Gem

Tau

Ari

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* The columns for Libra, Scorpio, and Sagittarius are transcribed from f. 9v; the columns for Capricorn and Aquarius from f. 12v; and the column for Pisces from f. 12r, for only one entry appears on f. 12v while the rest of the column on that page is blank. In the following list we display all entries on f. 12r, that differ from those of f. 12v. Variant readings from f. 12r: Cap Cap Aqu Aqu Aqu Aqu Aqu

2 = 9; 2 14 = 13;25 3 = 19; 1 8 = 20; 4 14 = 20;47 19 = 21;19 24 = 21;54

Cap Cap Aqu Aqu Aqu Aqu Aqu

6 = 10;34 15 = 13;45 4 = 19;15 9 = 20;18 15 = 20;52 20 = 21;26 26 = 22; 7

Cap Cap Aqu Aqu Aqu Aqu Aqu

8 = 11;17 18 = 14;43 5 = 19;26 11 = 20;36 17 = 21; 5 21 = 21;33 27 = 22;18

Cap Aqu Aqu Aqu Aqu Aqu Aqu

12 = 12;43 2 = 18;47 6 = 19;38 13 = 20;42 18 = 21;12 23 = 21;47 29 = 22;25

the geographical latitude for which the table was computed. It is worth noting that the table for solar declination (f. 35v) is based on a different value for the obliquity of the ecliptic: 23;33°. Clearly, in the Arabic copy f. 12v has the better readings; the copyist apparently realized that f. 12r had many errors, and so he tried again on f. 12v. The table of the samt discussed here is also found in the Tables of Barcelona (Millás (1962), table 47), and suffers from the same errors as the table of al-Muqtabis in ms Madrid 10023. A method for recomputing the entries of the table has been suggested by Neugebauer (cf. Kennedy and Faris, p. 24), and it consists of 3 steps: (i)

for each integer value of the argument (the longitude of the ascendant) find its oblique ascension by means of a table for the appropriate latitude (Neugebauer suggested using a table for 36° since no such table is known for 35;55,48°); (ii) in a table for normed right ascensions (see ff. 48v–49r; cf. al-Khwārizmī’s table in Suter (1914), pp. 171–173), find the longitude for which its normed right ascension equals the value obtained previously; (iii) the declination corresponding to that longitude in a table of declinations is the entry sought.

199

andalusian astronomy table 5

Table for lunar eclipses

Col. 2 (d) f. 52v f. 57v 12 11 10 9 8 7 6 5 4 3 2 1

d

0 1 3 4 6 8 10 12 12 12 12 12

55 35 50 16 47 19 29 0 0 0 0 0

0 1 3 4 6 8 10 11 12 12 12 12

54 35 50 56 47 39 29 0 0 0 0 0

Col. 3 (h, min, s) f. 52v f. 57v 0 0 0 1 1 1 2 2 2 2 2 2

6 14 50 6 43 57 0 0 0 20 25 30

0 0 40 50 9 50 9 50 0 0 0 0

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

6 54 50 6 2 36 0 0 0 20 40 50

Col. 4 (h, min, s) f. 52v f. 57v 0 5 40 50 9 50 20 50 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 6 1 8 1 10

0 0 0 0 0 0 0 48 40 48 48 12

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 6 1 8 1 10

0 0 0 0 0 0 0 40 40 48 43 42

Lunar Eclipses Tabula eclipsium lunarium (f. 52v)

The use of this table (see table 5) is described in canon 27 (f. 15vb). Columns 2–4 display the lunar eclipse magnitude (digits), the duration of the eclipse (hours) and the duration of totality (hours), as functions of the argument of latitude of the Moon (degrees). The headings are “prima porta”, “secunda porta”, “tercia porta”, respectively, while that for the argument is “longitudo a capite et cauda”. Among the material appearing at the end of this manuscript after the tables associated with al-Muqtabis, folio 57v has the same table under the title: “Hec tabula est quam extraxit et composuit Alkemed, in eclipsibus lunaris in canone suo que est extracta a canone Ebi Iusufi cognoscitur Byn Tarach, que est ualde uerax”. This author is probably to be identified with the late 8th-century astronomer Yacqūb ibn Ṭāriq, a collaborator of al-Fazārī at Baghdad, and whose zij was called the Sindhind (cf. Pingree (1968b) and (1970)). Note that Ebi Iusufi, or Abū Yūsuf, means the father of Joseph, and, in Arabic nomenclature, this “nickname” can be substituted for Jacob, who was the father of the Biblical Joseph. Previously, Millás (1942), p. 245, had suggested that the name here was a corrupt form of the name of the 11th-century astronomer Muḥammad ben Yūsuf ben Aḥmad ibn Mucādh, from Jaén.

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The two versions of this table are displayed in table 5. The entries in both versions of this table seem to be quite corrupt, and do not allow us to derive the parameters underlying them. Another version of this table is found in the Tables of Barcelona (Millás (1962), table 50), and the entries in it are also corrupt. Note that there is a single table for lunar eclipses, which is quite uncommon, for almost all zijes have two such tables (one for minimum, and one for maximum, lunar distance). The astronomical work of Jacob ben David Bonjorn, an astronomer of the 14th century from Perpignan, has only one table for lunar eclipses, but the entries in it are unrelated to those here (cf. Chabás (1991), p. 309). e

Color of Eclipses Colores (f. 52v)

This table (see table 6) is arranged in 6 columns: column 1 displays the argument of lunar latitude; col. 2 gives the color of solar eclipses as a function of the argument of lunar latitude, in degrees; col. 3 displays lunar latitude in minutes; col. 4 gives the color of lunar eclipses in terms of lunar latitude; cols. 5 and 6 (not shown here) display the magnitudes of solar eclipses in area digits as a function of the magnitude of the eclipse in linear digits. Canon 27 (f. 16va) refers to the first four columns, whereas the magnitudes of eclipses are treated on f. 15vb. All six columns are also found in the Tables of Barcelona (Millás (1962): cols. 1–4 appear in table 51, and cols. 5–6 in table 48). Kennedy (1956a), p. 159, has noted that Treatise viii of al-Bīrūnī’s Qānūn al-Masc ūdī has a chapter on the colors of solar and lunar eclipses. On the colors of lunar eclipses, see Goldstein (1967), pp. 234–235. For Ibn al-Muthannā (10th century), the color changes during the eclipse, whereas for Ibn Ezra (Millás (1947), p. 167) color is a function of latitude, as is the case here. Chapter 151 of Kitāb alcAmal biʾl-Asturlāb, by the Persian al-Ṣūfī (903–986), contains a similar list for the colors of lunar eclipses, but with different entries from those presented here (see Kennedy and Destombes, p. 413). Chapter 35 of the Libro de las Taulas Alfonsies, “De qué color sera ell eclipsy”, concerns lunar eclipses, and gives two different rules for their colors. These rules show similarities with those in the above-mentioned tables, but do not fully agree with them.

201

andalusian astronomy table 6

Table for the color of eclipses

(1) (2) Arg. Solar Lat. eclipse

(3) (4) Lat. Lunar eclipse

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

10

valde niger niger clarus turbatus rubeus turbatus croceus turbatus clarus turbatus cinereus cinereus cinereus cinereus cinereus croceus rubeus albus

50

niger valde in nigredine niger cum rubedine niger cum rubedine niger cum croceo turbatus

60

cinereus

20 30 40

The columns concerning the areas of eclipses (cols. 5–6) appear in a number of earlier tables: the Toledan Tables (Toomer (1968), p. 113, table 76), the zij of al-Battānī (Nallino, ii, p. 89), the Almanac of Azarquiel (Millás (1950), p. 233), and in al-Khwārizmī’s zij (Suter (1914), p. 190, table 76, columns 6–7). In fact, this table is already found in Ptolemy’s Almagest (vi, 8) and Handy Tables (Stahlman (1959), p. 258). The entries in all these tables agree, except in the Toledan Tables, where the entries for 3 and 5 (linear) digits are, respectively, 1;50 d and 3;20 d instead of 1;45 d and 3;40 d. f

Parallax in Latitude Tabula latitudinis solis iudicate que est diuersitas respectus lune in latitudine specialiter (f. 53r)

The use of this table (see table 7, below) is described in canon 27 (f. 15rb). The table displays the adjusted parallax in latitude, pβ, in minutes and seconds of arc, where “adjusted parallax” means the difference between the lunar and the solar parallax (see Kennedy (1956b), p. 35). Its maximum value is 0;48,32° at 90°. The same table appears in al-Khwārizmī’s zij (Suter (1914), pp. 191–192, tables 77 and 77a, col. “Diversitas respectus in latitudine”), but the values tabulated there

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table 7

Table for the adjusted parallax in latitude

(1)

(2)

(3)

1 2 3 4 5 6 7 8 9 10

0;50 1;41 2;32 3;23 4;13 5; 3 5;55 6;45 7;37 8;23

11 12 13 14 15 16 17 18 19 20

9;16 10; 5 10;54 11;44 12;33 13,23 14;12 15; 1 15;51 16;41

21 22 23 24 25 26 27 28 29 30

17;32 18;22 18;12 18;58 19;44 20;31 21;17 22; 3 22;49 23;35

17;38 18;26 19;14 19;58 20;44

31 32 33

23;35 24;22 25;46

24;22 25; 4

(4)

(1)

(2)

4;14

46 47 48 49 50

34;30 35;56 35; 0 35;52 36; 8

51 52 53 54 55 56 57 58 59 60

36;44 37;21 38; 7 38;59 39;28 39;22 39;36 39;50 40;56 40;44

61 62 63 64 65 66 67 68 69 70

40;55 41;43 41;54 42;54 42;54 43;57 43;59 44; 0 44;22 44;43

71 72 73 74 75 76 77 78

45; 5 45;27 45;48 46; 2 46;16 46;29 46;42 46;55

2;33

(3)

(4)

34;56

37;11

5; 4

8;27

8;26

12;34

16;36

18;11 18;58 19;44 20;31 21;17 22; 2 22;47 23;32 24;16 25; 0 25;43 26;26

38;39 38;48 39;21

39;45

40;16 42; 2

41;53 42;34 43;17 43;39

43;37 43;59 44;20

45;36

46;39

46;39 46;53 47; 6

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(1)

(2)

34 35 36 37 38 39 40

26;28 27;10 28;32 29;34 29;56 30;37 31;38

41 42 43 44 45

31;22 32;48 33;52 33;58 34; 4

(3)

(4)

(1)

(2)

27; 8 27;50 28;32

79 80

47; 8 47;21

81 82 83 84 85 86 87 88 89 90

47;34 47;40 47; 0 47; 5 47; 9 47;14 47;18 47;22 48;27 48;32

28;44

31;22

31;12

31;42 33;12 34;19

(3)

(4)

47;48

48; 0 48; 5 48; 9 48;14 48;18 48;22 48;34

48;32

differ in all cases, e.g., the entry for 90° is 0;48,45°. This specific table was discussed by Neugebauer (1962), pp. 121–123. Among the eclipse tables attributed to Yaḥyā ibn Abī Manṣūr, and analysed by Kennedy and Faris (pp. 20–38), there is a table entitled “table for the solar latitude” ( jadwal ʿard al-shams) which coincides with this one (ms Escorial Ar. 927, ff. 10v and 71v). Note the absurdity of the title for this table which, in fact, deals with the latitudinal component of the adjusted parallax. Kennedy and Faris (p. 25) give the function that underlies the entries of the table: [5] pβ = 0;48,32 · sin(θ), where θ is the solar zenith distance. They also state that the author of the calculations necessary for this table did “an extraordinarily bad job”, as the results almost never agree with the recomputed ones. These irregularities, which also occur in this table by Ibn al-Kammād, allow us to relate it with confidence to that of Yaḥyā. The same table is also found in the Tables of Barcelona (Millás (1962), table 46), and it exhibits the same inconsistencies as our text. Comparison between the entries in al-Muqtabis and the recomputed values seem to indicate that some shifts of entries occurred in the copying process (entries for 23°–30° have been shifted one place downwards, entries for 33°–35° have also been

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shifted one place, and entries around 67° may have been shifted two places downwards). In any case, we are far from the smoothness of the analogous table in al-Khwārizmī where the maximum value is 0;48,45°. The columns in table 7 for the adjusted parallax in latitude are the following: (1) Solar zenith distance for each integer degree from 1° to 90°. (2) Entries in the text: adjusted parallax in latitude (pβ), in minutes and seconds of arc. (3) Variant readings in the Tables of Barcelona (ms Ripoll 21). (4) Recomputed values by means of equation [5]. g

Lunar Latitude Tabula latitudinis lune iudicate (f. 53r)

This table is mentioned in canon 27 (f. 15rb), and it gives the lunar latitude (β) as a function of the argument of lunar latitude (ω). It is also found, with minor variant readings, in al-Khwārizmī’s zíj (Suter (1914), pp. 132–134, tables 21–26; cf. Neugebauer (1962), pp. 95–98), where the maximum latitude is 4;30°. Kennedy and Ukashah, pp. 95–96, have shown that the entries in this table were computed according to the “method of sines” given by the formula: [6] β = 4;30 · sin (ω). Kennedy (1956a), p. 146, mentions that a similar table, with the same maximum value, appears in the zij of Yaḥyā ibn Abī Manṣūr. The maximum value in the table on f. 53r is 4;29° instead of 4;30°. This does not suggest a different parameter; rather, it should be interpreted as a variant reading of 4;30°. The same table is also found in the Tables of Barcelona (Millás (1962), table 44), and exhibits the same characteristics. The table for the lunar latitude on f. 35v of this ms displays a different maximum, 5;0°, which is the value used by Ptolemy, al-Battānī, Azarquiel, among many others.

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h

205

Elongation Tabula longitudinis et dimidii sexti eius (f. 53v)

Canon 27 (f. 13rb) explains the use of this table. Two sets of entries are tabulated, the lunar longitude (lm) and the solar longitude (ls); both sets of entries are functions of the elongation (e) between the Moon and the Sun, given in degrees and minutes, from 0;30° to 12;0°, at intervals of 0;30°. The entries lm and ls are such that e= lm – ls, where lm = 13e/12 and ls = e/12. A similar, but more extensive table, is found in the Tables of Barcelona (Millás (1962), table 41): the elongation is given at intervals of 0;6° instead of 0;30°, and it ranges from 0° to 13;12° instead of from 0° to 12;0°. i

Parallax in Longitude Tabula diuersitatis respectus lune in longitudine (f. 53v)

This table, mentioned in canon 25 (f. 14ra), gives the longitudinal component of adjusted parallax (pλ), in hours and minutes, as a function of the argument, given in time from 0;15 h to 9 h, at intervals of 0;15 h. The entries reach a maximum of 1;36 h, and can be easily derived from the column with the heading “Horae diversorum/diversitatis respectuum lunae [in longitudine]” in al-Khwārizmī’s zij (Suter (1914), pp. 191–192, tables 77 and 77a; cf. Neugebauer (1962), pp. 121–126). However, in al-Khwārizmī’s zij, (i) the argument is not given in time, but in degrees, and (ii) the parallax in longitude is given to seconds. Kennedy (1956b), pp. 49–50, has shown that al-Khwārizmī’s table can be computed by means of the following formula: [7] pλ = 1;36 · sin (θ(t)) where [8] t = θ – (ε · sin (θ)), such that θ, the argument (in degrees), meets the condition that 0° ≤ θ ≤ 150°. The coefficient 1;36 = 0;4 · 24 = 24/15 contains the standard Hindu value for the obliquity of the ecliptic, 24°. In the case of Ibn al-Kammād’s table, the entries can be recomputed by means of equation [7], where

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[9] t = θ – (24 · sin (θ))/15, for all 0° ≤ θ ≤ 135°; now, 135° = 9 h · 15°/h, and 9 h is the maximum value of the argument, expressed in time. Only two such adjusted longitudinal parallax tables of this kind are known: that of al-Khwārizmī and the one discovered by Kennedy (1956b, p. 48) in the zij of Ibn al-Shāṭir (ca. 1350), explicitly paraphrasing an early Islamic source that has not been identified (Kennedy and Faris, pp. 33–38). We can now add to that short list the parallax table of Ibn al-Kammād and that in the Tables of Barcelona (Millás (1962), table 45). j “Tabula eclipsium solarium” ( f. 54r) Column 2 of this table gives the magnitude of the eclipse, in (linear) digits and minutes, as a function of the adjusted latitude (at conjunction) of the Moon displayed in column 1, in minutes and seconds of arc, from 0;34,13° to 0°. The explanation of this table in canon 27 (f. 15va) confirms the above value for the eclipse limit: “si fuerit minus 34 minutis et 13 secundis erit eclipsis”. Columns 3–9 form a double argument table. The vertical argument is the eclipse magnitude in digits (col. 2); the horizontal argument is the relative velocity of the Moon with respect to the Sun (vm – vs) in minutes and seconds of arc per hour, from 0;27,30° /h to 0;33,30°/h, at intervals of 0;1° /h. This table, as some previous ones, seems to derive from Yaḥyā ibn Abī Mansūr (ms Escorial Ar. 927, f. 13r). Kennedy and Faris (pp. 27–30), once again, have transcribed and explained this table. The adjusted latitude of the Moon is a linear function of the magnitude of the eclipse, so that the graph of the function relating columns 1 and 2 should be a straight line. However, this is not the case: there are “jumps” at 4;20 and 9;20 digits; and one should consider the first three entries in col. 2 and 0 d, 20 d, and 40 d, instead of 0;15 d, 0;30 d, and 0;45 d (these errors appear in this table in the Arabic ms as well as in our Latin text). The second part of the table displays the half-duration of the eclipse, in hours and minutes, as a function of its magnitude and the relative velocity of the Moon with respect to the Sun. The recomputations made by Kennedy and Faris (p. 29) give good results, but fail to reproduce the entries of the table precisely.

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207

Other Tables in Al-Muqtabis

a Calendaric Tables ( ff. 27r –v) The purpose of these tables is to convert dates from the Arabic calendar to “Roman” (i.e., Julian) and Egyptian calendars. The epoch of the radix given is the Hijra: noon of July 14 ad 622. Canon 9 (f. 6ra) states that the epoch of the radix is in “the day of Mercury” (Wednesday). The tables display two correspondences: ah0 = 932 Julian years 9 months 17;0 days from the beginning of the Seleucid era = 9 Egyptian years 11 months 9 days from the beginning of the Yazdijird era. These tables entirely or partially reproduce calendaric tables in the zij of al-Khwārizmī/Maslama (Suter (1914), p. 110, table 2; p. 111, table 2a; and p. 113, table 3), and/or in pseudo-Battānī (Maslama) (Nallino, ii, pp. 301, 304–305). b

Trepidation ( ff. 28v, 35v) Tabula aduenctionis puncti capitis arietis (f. 28v)

The same radix and equivalent entries for the mean motion of the vernal point are found in Azarquiel’s Treatise on the motion of the fixed stars (Paris, BnF, ms Heb. 1036; see Millás (1950), pp. 266, 324); the Liber de motu octave sphere, attributed to Thābit ibn Qurra displays a similar table (Millás (1950), p. 507); cf. Morelon (1987), p. xix. On the theory of trepidation see Goldstein (1964), Dobrzycki (1965), North (1967), North (1976), vol. 3, pp. 155–158, Mercier (1976–1977) and Samsó (1992). Tabula directionis aduenctionis capitis arietis (f. 35v) The entries (see table 8) display very nearly a sine function whose maximum is 9;59° at 90°; we have not succeeded in explaining the deviations from the sine function, e.g., the entry for 30° is not half the entry for 90°. Toomer (1968), 118, gives two tables, which are also sine functions, associated with the Toledan Tables, and in fact they already appear in the Liber de motu octave sphere (Millás (1950), pp. 507–508). The radix and the mean motion of the first point in Aries (see f. 28v and the comments on it, above) are taken from Azarquiel’s Treatise on the motion of the fixed stars. The use of this table is briefly outlined in canon 12 (f. 7v), where another work by Ibn al-Kammād is explicitly mentioned: al-Amad ʿalā al-abad. This table is also found in the Tables of Barcelona (Millás (1962), table 20), a treatise by al-Marrākushī (Sédillot (1834), p. 131), and in the tables of Juan Gil (London, Jews College, ms Heb. 135, f. 78b).

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We have tried to recompute this table in many ways, and by far the best fit comes from formula [10] which is intended to represent Azarquiel’s somewhat vaguely defined second model (Millás (1950), pp. 287–289, 317–318; cf. Samsó (1992), p. 230): [10] sin (e) = r · sin (i)/60 where e is the entry in the table, r = 10;24, and i is the argument. The maximum entry in the table, 9;59°, is indeed a rounded value for arcsin (10;24/60) = 9;58,54°. table 8

Table for the equation of the motion of the first point in Aries

Degrees 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8

0/6

1/7

2/8

0;10 0;20 0;31 0;41 0;53 1; 3 1;54 1;25 1;35 1;45 1;56 2; 8 2;19 2;30 2;41 2;50 2;18 3; 6 3;15 3;25 3;36 3;47

5;16 5;25 5;34 5;43 5;52 6; 4 6;16 6;29 6;41 7;53 7;57 7; 2 7; 6 7;10 7;14 7;21 7;28 7;35 7;42 7;48 7;54 8; 0

8;52 8;56 9; 1 9; 5 9;10 9;14 9;17 9;21 9;24 9;28 9;31 9;35 9;38 9;41 9;45 9;47 9;49 9;51 9;52 9;55 9;56 9;56

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Degrees 23 24 25 26 27 28 29 30

7 6 5 4 3 2 1 0

0/6

1/7

2/8

3;57 4; 8 4;19 4;29 4;39 4;49 4;38 5; 7 5/11

8; 7 8;13 8;20 8;25 8;31 8;37 8;41 8;47 4/10

9;57 9;58 9;59 9;59 9;59 9;59 9;59 9;59 3/9

Variant readings in the Tables of Barcelona (ms Ripoll 21, f. 137r): e (7) = 1;14 e(16) = 2;51 e(17) = 2;58 e(29) = 4;58 e(40) = 6;43 e(41) = 6;57 e(50) = 7;54 e(51) = 7;58 e(53) = 8; 6 e(55) = 8;19

figure 7.1 The geometrical model underlying Ibn al-Kammād’s table for trepidation, as reconstructed

Geometrically, we can understand the underlying model by referring to Figure 7.1: a small circle or epicycle, bce, whose center is a, lies in the plane of the ecliptic, circle ag, and the center of the sphere is o. Note that the small circle bce is partly inside, and partly outside, the sphere. The angle i is equal to arc bc; radius ab = 10;24, and oa = 60. Through c, draw a line parallel to ba, reaching the large circle ag at f. Draw of, and let aof be the angle e, which we seek. To compute angle e, we drop a perpendicular fh from f to ao; then fh = cd,

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table 9

Trepidation according to Ibn al-Kammād

(1)

(2)

(3)

(4)

10 20 30 35 40 45 50 60 70 80 90

1;45 3;25 5; 7 5;52 6;43 * 7;14 7;48 8;47 9;28 9;55 9;59

1;43 3;24 4;58 5;42 6;24 7; 2 7;38 8;38 9;22 9;50 9;59

2 1 9 10 19 12 10 9 6 5 0

* There is a textual problem with this entry (see Table 8); hence we display surrounding values.

and cd = r · sin (i). In right triangle fho, fo = 60, and sin (e) = fh/fo; equation [10] follows. We cannot account for the remaining differences between text and computation. The columns in table 9 for trepidation according to Ibn al-Kammād are as follows: (1) (2) (3) (4)

i: argument e: text of Ibn al-Kammād e: computed, using equation [10] the difference: T(ext)–C(omp.), in minutes

The value 10;24 for r in equation [10] is made plausible by the two other models for trepidation associated with Azarquiel and his followers. Samsó (1992), pp. 235–236, describes the third model of Azarquiel and shows how it yields a maximum value, pmax very close to 10;24° (10;23,29°), based on the equations: [11] δ = r · sin (i), [12] sin (p) = sin (δ)/sin (23;33°)

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andalusian astronomy table 10

Trepidation according to Ibn al-Bannāʾ and Ibn al-Raqqām

(1)

(2)

(3)

(4)

10 20 30 40 50 60 70 80 90

1;48 3;32 5;11 6;40 7;57 8;59 9;46 10;14 10;24

1;47,47 3;32;23 5;10,43 6;39,48 7;56,55 8;59,39 9;45,59 10;14,25 10;24

1;47,48 3;32,24 5;10,40 6;39,40 7;56,42 8;59,20 9;45,35 10;13,56 10;23,29

where p is the amount of precession corresponding to an argument i, r is 4;7,58°, and 23;33° is the value for the obliquity. Moreover, Neugebauer (1962), p. 184, presents a formula based on Azarquiel’s first model: [13] sin (p) = sin (pmax) · sin (i), If we substitute 10;24° for Pmax in equation [13], we find a set of values that agree very well with those preserved by the late 13th-century astronomers Ibn al-Bannāʾ (ms Escorial Ar. 909, f. 22v) and Ibn al-Raqqām (ms Kandilli 249, f. 66v) who only tabulated these entries to degrees and minutes. However, since the agreement is equally good using equations [11] and [12] on the one hand, and equation [13] on the other, we cannot decide which procedure was used for computing this table. Table 10 displays these computations, where the columns are the following: (1) (2) (3) (4)

i: argument e: text of Ibn al-Bannāʾ and Ibn al-Raqqām e: computed, using equation [13] e: computed, using equations [11] and [12]

c Mean Motion Tables ( ff. 28r–34v) The headings of the tables indicate that they were intended for the meridian of Córdoba and calculated for Arabic years, months, etc. We have computed all mean motions from the corresponding tabulated values for ah 720, except for

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that of the lunar anomaly. The method is simply to subtract the corresponding radix from each value for ah720, taking into account the full rotations, and then to divide the result by the number of days elapsed from the epoch of the Hijra calendar. Sun (f. 28r) The tabulated values for the mean motions are given to seconds. They do not agree with either al-Khwārizmī (Suter (1914), p. 115, table 4), or al-Battānī (Nallino, ii, p. 20). The entry for the radix is 1s 6;35,9°, and the tabulated value for ah720 is 7s 16;13,19°. The difference between these two values is 6s 9;38,10° = 189;38,10°. Now, in 720 Arabic years, the Sun has completed 698 full rotations; therefore, the are length traveled by the Sun in 720 Ar. y. (= 255,144 days) is 189;38,10 + (698 · 360) degrees, and the daily mean motion of the Sun resulting is 0;59,8,9,21,15, … °/d, which yields a year-length of 365;15,36,34, … d. The value deduced from the table is sidereal. It differs from the daily mean motion embedded in the Toledan Tables, and attributed to Azarquiel (0;59,8,11,28,27, …°/d) by an amount which is exactly equal to Azarquiel’s value for the daily motion of the solar apogee (0;0,0,2,7,10,39, … °/d, cf. Toomer (1969), p. 319). The radix given here (1s 6;35,9°) corresponds to the solar centrum (the distance from a sidereally fixed apogee). To obtain the longitude of the Sun at epoch, add the given radix to the longitude of the apogee (2s 16;45,21°, f. 35r); the result is 3s 23;20,30°, a value which is close to, but not identical with, those of Azarquiel or al-Khwārizmī/Maslama (see Toomer (1968), p. 44). Solar Apogee (f. 28v) The daily mean motion of the solar apogee resulting from the tabulated value for ah720 (0s 2;24,24°) is 0;0,0,2,2,14,46, … °/d. This implies a progress of 1o in about 299 Arabic years or in about 290 Julian years, a value which differs from the daily motion of the apogee used by Azarquiel (0;0,0,2,7,10,39, … °/d), which corresponds to a progress of 1° in about 279 Julian years. A parameter very similar to that of Azarquiel is also found in the works of Ibn Isḥāq, Ibn al-Bannāʾ and Abū l-Ḥasan cAlī b. Abī cAlī al-Qusanṭaynī (Millás (1950), pp. 352–353; Samsó (1992), p. 212). The recomputations show that Ibn al-Kammād used Azarquiel’s value for the mean motion of the Sun, but that he incorporated a different parameter for the mean motion of the apogee which does not appear in any known text prior to this one.

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Moon (ff. 29r–v) The daily mean motion of the Moon in longitude resulting from the tabulated value for 720ah (10s 8;37,1°) is 13;10,34,52,46, …°/d. This is exactly alKhwārizmī’s value (cf. Neugebauer (1962), pp. 42, 92), and very nearly that in the Toledan Tables (cf. Toomer (1968), p. 44). The daily mean motion of the Moon in anomaly resulting from the tabulated value for 900 a.r. (9s 13;12°) is 13;3,53,56,19, …°/d. This is very nearly the value in the Toledan Tables (cf. Toomer (1968), p. 44), which differs from that of al-Khwārizmī (cf. Neugebauer (1962), p. 92). The tabulated values for the mean motion in longitude agree almost exactly with those in al-Khwārizmī (Suter (1914), pp. 117–119, table 6–8), and exhibit very small differences with those in the Toledan Tables (Toomer (1968), p. 48). However, the tabulated values for the mean motion in anomaly are in agreement with those of al-Battānī (Nallino, ii, p. 20), and in the Toledan Tables (Toomer (1968), p. 49), except for the fact that the motion in anomaly is given to seconds there. Planets (ff. 30r–34v) For Saturn, Jupiter, Mars and Mercury, the tables give entries for the mean motion of the centrum (i.e., the distance from a fixed apogee) and anomaly. For Venus, the entries display the mean motion in anomaly only, and for the lunar node, the entries are the complement in 360° of the mean motion of the ascending node in longitude. All entries are given to minutes. Those for the yearly progress of the centrum of Mercury agree with those for the Sun (f. 28r), except for the fact that the latter are given to seconds. The values for the radices are for the Hijra (see table 11, below), and differ from those in the zij of al-Khwārizmī/Maslama and those in the Toledan Tables. The mean motions in anomaly of the superior planets are not generally tabulated in other sets of astronomical tables (and this is also the case for almost all copies of the Toledan Tables). The recomputed values obtained here can be compared, however, with those in one copy of the Toledan Tables (Oxford, Bodleian Library, ms Laud. Misc. 644, cf. Toomer (1968), p. 45): 0;57,8°/d (Saturn), 0;54,9°/d (Jupiter) and 0;27,41°/d (Mars). The same values are explicitly found in the work of Ḥabash (Debarnot (1987), p. 44), where the value for Mars is 0;27,42°/d. All parameters computed from the tabulated entries for the mean motions show close, although not perfect, agreement with those derived from the Toledan Tables, the differences never being greater than 0;0,0,1°.

214 table 11

chapter 7 Summary of the mean motions and radices

Mean motion (°) Solar longitude Solar apogee Vernal point Lunar longitude Lunar anomaly Double elongation Lunar node Saturn (longitude) Saturn (anomaly) Jupiter (longitude) Jupiter (anomaly) Mars (longitude) Mars (anomaly) Venus (anomaly) Mercury (longitude) Mercury (anomaly)

0;59,8,9,21,15, … 0;0,0,2,2,14,46, … 0;0,0,54,56,57, … 13;10,34,52,46 … 13;3,53,56,19, … 0;3,10,46,41, … 0;2,0,25,36, … 0;57, 7,44,57, … 0;4,59,6,43, … 0;54,9,3,37, … 0;31,26,31,40, … 0;27,41,40,34, … 0;36,59,29,21, … 0;59,8, 11,23, … 3;6,24, 7, 19, …

Radix (°) 1s 2s 0s 4s 3s 0s 4s 7s 11s 5s 4s 3s 8s 1s 9s 2s

6;35, 9 * 16;45,21 ** 3;51,11 0;34,42 *** 18;11 14;33 6;30 26;52 * 27;48 21;58 * 23; 2 1;46 * 22;14 15;21 4;58 * 14; 1

* The values of the radices for the Sun and the planets correspond to their centrum, i.e., their distance from apogee. Note that the ms does not provide this information for Venus. ** This value for the longitude of the solar apogee is given on f. 35r. However, in the table for the mean motion of the solar apogee (f. 28v) one finds 0s 0;0,0 opposite “Radix”. *** The radix for the Moon is called its “centrum”, here meaning longitude.

d

Spherical Astronomy

Declination (f. 35v) This is a table giving the declination of the Sun for each integer degree; the maximum entry is 23;33°. This value for the obliquity of the ecliptic is associated with the zij al-Mumtaḥan (Vernet (1956), p. 515). The entries in this table agree, except for minor differences, with those in the Almanac of Azarquiel (Millás (1950), p. 174), where the argument is given only at intervals of 3°. The value found in the Toledan Tables, and attributed to Azarquiel, is 23;33,30° (Toomer (1968), p. 30).

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Daylight (f. 47v) The heading of the table mentions Córdoba, but it does not specify a value for its latitude. The geographical table on f. 54v gives its latitude, φ, as 38;30°, the most prevalent value for it at the time. This table gives the half-length of daylight as a function of the solar longitude. The maximum entry represents half of the longest daylight (m/2), and it is 7;21 h for Cancer 0°. This value follows from the formula: [14] tan (φ) = –cos(m/2) · cot (ε), where φ = 38;30° and ε = 23;33°. Normed Right Ascension (ff. 48v–49r) The same table of right ascension, beginning with Capricorn 0°, is found in alBattānī (Nallino, ii, pp. 61–64), for an obliquity (ε) of 23;35°. It is also found in the Toledan Tables (Toomer (1968), p. 34, table 17) and, with copying errors in an abridged version, in the Almanac of Azarquiel (Millás (1950), pp. 220– 221, tables 69–70). The table differs from that in the Handy Tables (Stahlman (1959), pp. 206–209, table 1) and in al-Khwārizmī’s zij (Suter (1914), pp. 171–173, tables 59–59b), both calculated for higher values of the obliquity. Oblique Ascension for Córdoba (ff. 49v–51r) The mean values between the rising times of Aries and Virgo, of Taurus and Leo, and of Gemini and Cancer, are respectively 27;50°, 29;54,30° and 32;15,30°. They are almost identical with those derived from the Toledan Tables for the seven climates: 27;50°, 29;54° and 32;16°, which are the Ptolemaic values for the right ascensions (cf. Toomer (1968), p. 42). They are also embedded in the zij of al-Khwārizmī, but differ from those in the zij of al-Battānī. When recomputing the entries, close, although not exact, agreement is obtained with φ = 38;30° and ε = 23;51°. With the two other values of the obliquity found in our text (23;33° and 23;35°), the agreement is worse. Another column in this table represents the length of the seasonal hours. The entry for Cancer 0° is 18;24°. Now 18;24 · 12/15 = 14;43 h, which gives a half daylight of 7;21,30 h, and this is quite close to the value 7;21 h for Cancer 0° (see the table for the half-length of daylight as a function of the solar longitude on f. 47v). A similar table for Salé, a place in North Africa near Rabat, whose latitude is given here as 33°, is found on ff. 59v–61r.

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e

Latitude Moon (f. 35v) The maximum value in the table for the latitude of the Moon is 5;0°. The same value for the inclination of the lunar orb is found in the Almagest v, 8, and in many other texts, including the Toledan Tables and the Almanac of Azarquiel (cf. Millás (1950), p. 173, where the argument is given only at intervals of 6°). For another table for the lunar latitude in al-Muqtabis, see Section iii. g.

Planets (f. 45r–v) This table for the latitude of the superior planets is the same as the one in the Almagest xiii, 5, and in al-Battānī (Nallino, ii, p. 140 (columns 1–3) and p. 141 (column 4)). The pattern of this table differs greatly from the corresponding one in the Handy Tables. Toomer listed some mss associated with the Toledan Tables that contain such a table, but concluded that it is not part of the original Toledan Tables (Toomer (1968), p. 72). In contrast to the superior planets, the latitude table for the inferior planets does not conform to the pattern of the Almagest, the zij of al-Battānī, or the tables associated with the Toledan Tables. Rather, this table reproduces, with variant readings, the entries which are multiples of 6° in the Handy Tables (Stahlman (1959), pp. 331–334, tables 49–50, where the entries are given at 3-degree intervals). In particular, the maximum values for the mean latitude of Mercury (3;52°) agree in both sets of tables, but those for the mean latitude of Venus differ (8;35° in our text and 8;51° in the Handy Tables). However, canon 16 (f. 10va) gives 8;36° and 4; 18° as the values for the maximum latitude of Venus and Mercury. Kennedy (1956a), p. 173, reports maximum values for Venus (8;56°) and Mercury (4;18°), and associates the zij al-Mumtaḥan and Ibn Hibintā with them. Nevertheless, the outstanding feature here is the juxtaposition of different Ptolemaic tabular material: the Almagest for the superior planets, and the Handy Tables for the inferior planets. The source for such a mixed approach has not been determined. Note that in the tables associated with al-Muqtabis no values are given for the longitudes of the planetary nodes, although ms 10023 (f. 66r) has a list of them. f

Equations Moon (ff. 36r–37r) The same table is found in the Almagest v, 8, in the Handy Tables, as well as in many medieval tables, such as the zij of Yaḥyā ben Abī Manṣūr (Salam and Kennedy, pp. 495–496), the zij of al-Battānī (Nallino, ii, pp. 78–83) and

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the Toledan Tables (Toomer (1968), pp. 58–59). The table lists columns for the equation for mean to true apogee, an interpolation function, the increment in the equation of center, and the equation of center. There is no column here for the lunar latitude, which is tabulated separately (see f. 35v). The entries for the equation of anomaly agree with those in al-Battānī’s zij, but differ slightly from those in the Toledan Tables (e.g., in our table the maximum of 5; 1,0° is reached at 95°, while in the Toledan Tables the value is 5;0,59°, and it occurs at 94–95°). On the other hand, in our table the order of the columns is the same as in the Toledan Tables, and differs from that in al-Battānī’s zij. Planets (ff. 37v–44v) The tables for the equations of the five planets are essentially those found in the Almagest xi, 11, in the Handy Tables, as well as in the zij of al-Battānī (Nallino, ii, pp. 108–137) and the Toledan Tables (Toomer (1968), pp. 60–68), among many others. Although in most respects the zij of al-Battānī and the Toledan Tables agree for the planetary equations, there are some differences between them: for instance, there is a column for the planetary stations in the Toledan Tables, which is neither in al-Battānī’s zij nor in our table. All we can deduce from these tables is that Ibn al-Kammād accepted the Ptolemaic tradition, as displayed in the Handy Tables, followed by most Muslim astronomers. It is worth noting that in the tables of Ibn al-Kammād the maximum solar equation (1;52,44°: f. 35r) differs from the maximum equation of center for Venus (1;59°). This value for Venus is not that of the Almagest, but follows al-Battānī, etc. (cf. Goldstein and Sawyer). This indicates that Ibn al-Kammād’s contribution was restricted to solar theory and that he did not introduce any changes in planetary theory. Values for the sidereally fixed apogees appear above the tables for the equations on ff. 37v–38v (Saturn), 39r–40r (Jupiter), 40v–41v (Mars), 42r–43r (Venus), and 43v–44v (Mercury). The apogee for Venus is that ascribed to the Sun on f. 35r. Saturn Jupiter Mars Venus Mercury

238;38,30° 158;21, 0° 119;41, 0° 76;45,21° 198;21, 0°

g Stations ( f. 46r) Only four values are given for each planet: the positions of the first and the second stationary points for arguments of 0° and 180° (see table 12). The tabulated

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chapter 7 Planetary stations

1st st. at apogee 2nd st. at apogee 1st st. at perigee 2ndt st at perigee

Saturn

Jupiter

Mars

Venus

Mercury

3s22;44 8s 7;16 3s25;30 8s 4;30

4s 4; 5 7s25;55 4s 7;11 7s22;49

5s 7;28 6s22;32 5s19;15 6s10;45

5s15;51 6s14; 9 5s18;21 6s11;39

4s27;14 7s 2;46 4s24;42 7s 5;18

values for the same argument add up correctly to 360°. In all cases, they coincide with those found in the Toledan Tables (Toomer (1968), pp. 60–68), and the zij of al-Khwārizmī (Suter (1914), pp. 138–167, tables 27–56), both zijes displaying tables for each integer degree of the argument. Nearly the same values are also found in the zij of al-Battānī (Nallino, ii, pp. 138–139), which have a Ptolemaic origin. However, Almagest xii.8 and the Handy Tables (Stahlman (1959), pp. 335–339, tables 51–55) have slightly different tables for the stations: in the latter the argument was modified, as well as the interval for the calculation of the entries (6° in the Almagest, 3° in the Handy Tables). h Equation of Time ( f. 46r) Beneath the table we read: “Mediatus solis in radice posita ad directionem dierum cum noctibus: 10.23.24.50. a puncto capitis arietis”. This value agrees with that appearing in canon 11 (f. 7rb), and seems to correspond to the argument (in signs and degrees) for the minimum entry in the table. Hence it is to be understood as 10s 23;24,50°. The entries in this table, in time-degrees, rounded to the nearest integer, may have been taken from the more precise values given in the zij of al-Battānī (Nallino, ii, pp. 61–64) or in the Toledan Tables (Toomer (1968), pp. 34–35). i

Trigonometry Functions Related to the Sine (f. 46v) We adopt the convention that Sin θ = 60 sin θ, and similarly other capitalized trigonometric functions are normed for r = 60 (cf. Kennedy (1956a), p. 139). Three functions are given for each integer degree: Sin θ, Cos θ, and Vers θ = r – Cos θ. The Almanac of Azarquiel (Millás (1950), p. 229) has a table with three functions (sine, cosine, and versine), but given at intervals of 3°. Except for copying errors, the entries agree in both tables.

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Cotangent Function (f. 48r) The entries in this table represent the length of a shadow (s) projected by a gnomon of 12 units as a function of the altitude of the Sun (h): [15] s = 12 · cotan (h), in the tradition of al-Khwārizmī (Suter (1914), p. 174, table 60) and al-Battānī (Nallino, ii, p. 60). j Star Table ( f. 47r) This is a list of 30 stars and it displays the following information for each star: magnitude, name, ecliptic coordinates (longitude and latitude), and the planets associated with it (for astrological purposes). Kunitzsch (1966), pp. 99–102, described this list under his type xv: the ecliptic longitude of each star is derived from that in Ptolemy’s star catalogue by adding 6;38°, thus indicating that the epoch of this star list is the Hijra. The type defined by Kunitzsch only includes two versions of this list: the other version is uniquely represented by ms Vienna 5311, f. 129v. Although the stars in both versions are the same, the coordinates for them do not always agree, thus suggesting a common Arabic ancestor. Kunitzsch also reports close similarity with the star list of Abū l-Ḥasan cAlī al-Marrākushī (ca. 1262). We have found additional copies of this list in some copies of the Tables of Barcelona, e.g., ms Vatican Heb. 356, f. 65b, where the names of stars are given in Hebrew. k Excess of Revolution In the first table on f. 54v, the “excess of revolution” is given in degrees and minutes, for 1, 2, 3, …, 10, 20, …, 100 years. The entry for 1 year is 92;36°, which seems to be an isolated error. To be coherent with all other entries in the table, one should read 93;36°, leading to a year-length of 365;15,36 days = 365d 6;14,24 h. The entry for 100 years is 0;5°, and the resulting length of the solar year is 365;15,36,0,30 days. In the second table on f. 54v, the “excess of revolution” is given in hours and minutes of time, for 1, 2, 3, …, 10, 20, …, 100 years. The entry for 1 year is 6;14 h, which corroborates the emendation above. The entry for 100 years is 23;58 h, and the resulting length of the solar year is 365;15,35,58,30 days, a value close, but not equal, to the parameter derived from the previous table. These values for the solar year are to be compared with that appearing in the canons (f. 2va): 365;15,36,19,34,12 days (cf. Millás (1942), p. 236), as well as with the information given on f. 65v of this manuscript: “Length of the solar year according to Ibn al-Kammād:

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365;15,36,19,35,32 days”. The daily mean motion of the Sun resulting from the first value is 0;59,8,9,23,44, 53 °/d (the second value yields a very similar parameter: 0;59,8,9,23,44,40 °/d), which agrees quite well with the value derived above from the table for the mean solar anomaly on f. 28r. On f. 57v of this manuscript there is an analogous table attributed to Azarquiel, and for a sidereal year of 365;15,24d. There follows a list of the various lengths of the solar year associated with Ibn al-Kammād: 365;15,36, 0,30d 365;15,35,58,30d 365;15,36,19,34,12d 365;15,36,19,35,32d

(computed from the first table: f. 54v) (computed from the second table: f. 54v) (mentioned in canon 1: f. 2va) (attributed to Ibn al-Kammād: f. 65v)

l Geographical Table ( f. 54v) This is a list of 30 places: for each of them we are given its longitude and latitude, in degrees and minutes. The prime meridian used here is located west of the shore of the Western Ocean. It thus differs from that in the Toledan Tables, where the shore of the Western Ocean seems to have been used for most longitudes (cf. Toomer (1968), p. 136). For a general discussion of the prime meridian in Islamic sources, see Kennedy and Kennedy, p. xi. The entry for the latitude of Córdoba is 38;30°, and that for its longitude 27;0°, which is the same value given in canon 9 (f. 6ra): “longitudo a circulo occidentis ex centro Erin est gradus 27″. The same values for the longitude and latitude of Córdoba are found in some other Islamic sources, notably in a work by Abū l-Ḥasan cAlī al-Marrākushī (see Kennedy and Kennedy, p. 95). We note that in the Toledan Tables, the longitude of Córdoba is given as 9;20° and its latitude as 38;30° (Toomer (1968), p. 134). m

Miscellaneous Tabula directionis arcus luminis et transitus (f. 48r)

This table has 3 columns: (1) “gradus longitudinis”, (2) “directio arcus luminis”, and (3) “minuta diuersitatis transitus”. Col. 1 lists degrees at 3° intervals from 3° to 90° (with some copying errors); col. 2 has entries in degrees and minutes from 0; 14° to 1507;0°; and col. 3 has entries in minutes and seconds from 0;1,23° to 0;23,33°. The structure of this table is similar to that of some

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tables in the Almanac of Azarquiel (Millás (1950), p. 226), and most of the entries are the same (the entries in col. 2 agree with those in the corresponding column in the Almanac of Azarquiel for arguments from 3° to 84°, but for roundings and copying errors: note that the column in the Almanac displays degrees, minutes, and seconds, rather than degrees and minutes). But the headings for the columns in the Almanac of Azarquiel are different from those here, and we have not succeeded in determining the purpose for which our table (or the corresponding table in the Almanac of Azarquiel) was computed.

v

Other Tables in ms Madrid 10023

a. Just after the last table associated with al-Muqtabis, there are two tables related to Azarquiel’s solar theory: (f. 55r) “Tabula motus centri circuli exeuntis centrum in longitudine longiori et propinquiori a centro terre”; (ff. 55v– 56r) “Tabula directionis composite centri circuli solis exeuntis centrum de diuersitate centri eiusdem in longitudine propinquiori et longiori a circulo diuersitatis motus centri morantis tempus”. Toomer (1969), p. 325, has reproduced an excerpt and has explained the two tables. b. (f. 56v) “Tabula uisuum lunarium post occasum solis in climatibus septem” c. (f. 57r) “Tabula eclipsis lune et quot digiti eclipsantur ex ea et hore dimidii temporis eclipsis” d. One of the tables on f. 57v has already been analysed above (see Section iv, k), in connection with a table in al-Muqtabis (f. 54v) giving the length of the solar year. For another table on f. 57v, an eclipse table, see the comments in Section iii, d, concerning lunar eclipses in al-Muqtabis (f. 52v). Folio 57v has still another table for the maximum values for latitudes in the seven climates. e. There follow two tables that are clearly related to table 1 in the zij of alKhwārizmī: (f. 58r) “Tabula cuius est inter annos gentium et alios annos preter illos ad inuicem”; (f. 58v) “Numeri dimissi per 28, 28 secundum annos romanorum et egyptiacumi”. According to Millás (1942), p. 245, they derive from Maslama rather than from al-Khwārizmī. f. (f. 58v) “Circuitus planetarum magni in sectis et divinationibus”. These are the “Mighty Years” of the planets: see table 13, where the entries of this table are compared with those of Abū Macshar as preserved by al-Sijzī (Pingree (1968a), p. 64). g. (f. 58v) “Tabula dierum prouenientium in retrogradationibus planetarum et directionibus eorum”

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h. (f. 59r) “Tabula partis cordarum supereminentium et partis almudarat solis et planetarum” i. Folios 59v to 61v present tables for the city of Salé: see the comments in Section iv, d, on the table in al-Muqtabis for the oblique ascension for Córdoba (ff. 49v–51r). j. The following table is certainly a part of al-Kawr ʿalā al-Dawr: (ff. 62v– 64r) “Tabula extractionis annorum quantitatis durationis creature in uentre matris per longitudinem lune a gradu occidentis”. It deals with astrological obstetrics and has been discussed by Vernet (1949), pp. 273–300. k. (f. 64v) “Tabula circuituum annorum planetarum in natiuitatibus” l. There follows information on houses, exaltations, triplicities, and signs, presented in tabular form (f. 65r), but every other entry has been left blank. The information given agrees with that in al-Khwārizmī/Maslama (Suter (1914), p. 231, table 116). The table with the heading, “Tabula terminorum egyptiorum” (f. 65r) is the same as the fourth sub-table in al-Khwārizmī/Maslama (Suter (1914), p. 231, table 116), but some of the entries are blank here. m. (f. 65v) “Mediatus cursus solis in descensu eius ad quartas circuli secundum probationem huius canonis”. The data are for ah 480 (1087–1088 ad). n. (f. 65v) “Tabula terminorum ciuium Babillonie ueteris qui sunt magistri ymaginum”. Two parameters are given in this table: the length of the solar year “according to Ibn al-Kammād” (365;15,36,19,35,32 days, cf. our comments in Section iv, k, on the table in al-Muqtabis on f. 54v) and his length of the lunar month (29;31,50,5,1 days). o. (f. 66r) “Tabula extracta per misilme de eo quod confirmatum extitit per ciues huius artis yspanenses super diuisionem Yspanie per signa duodecim 12 12 12 12″. Millás (1942), p. 256, suggested that “misilme” stands for “Maslama”. p. (f. 66r) “Residuum ascensionum ad reuoluciones annorum solarium secundum Muhad Arcadius”. Millás (1942), p. 256, identified Muhad Arcadius with Abū cAbd Allāh Muḥammad ben Yūsuf ben Aḥmad Ibn Mucādh alJayyānī, from Jaén. The value given for one year, 93;2,15°, corresponds to 365;15,30,22 days. The same value is found in al-Khwārizmī/Maslama (cf. Neugebauer (1962), p. 132, and Goldstein (1967), pp. 143, 242). In fact, this table attributed to Ibn Mucādh reproduces, with some scribal errors, the table of conversion for the “years of Nativity” given in units of time-degrees in al-Khwārizmī/Maslama (Suter (1914), p. 230, table 115; cf. Neugebauer (1962), p. 131). q. (f. 66r) “Tabula uisuum planetarum et absconsionum eorum sub radiis solis” r. (f. 66r) “Capita draconum planetarum in anno quingentesimo et quinquagesimo ab annis seductionis”. Note that ah550 corresponds to 1155–1156ad.

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Planet

“Mighty years” of the planets

ms Madrid 10023 (f. 58v)

Abū Macshar

1461 1151 480 420 * 625 * 567 * 684 *

1461 1151 480 520 265 427 284

Sun Venus Mercury Moon Saturn Jupiter Mars

* Note that several of the entries in ms Madrid 10023 are corrupt.

See our comments in Section iv, e, on the table of al-Muqtabis for the latitude of the planets (ff. 45r–v).

Note Added in Proof We are grateful to Àngel Mestres (University of Barcelona) for calling our attention to the following passage in ms Hyderabad, Andra Pradesh State Library 298 (no foliation): chap. 35 “Abū l-cAbbās al-Kammād said in a horoscope he drew in Córdoba in the year 510 Hijra …” (= 1116–1117ad).

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Goldstein, B.R. 1964, “On the Theory of Trepidation”, Centaurus 10: 232–247. Goldstein, B.R. 1967, Ibn al-Muthannā’s Commentary on the Astronomical Tables of al-Khwārizmī, New Haven and London. Goldstein, B.R. 1985, “Scientific Traditions in Late Medieval Jewish Communities”, in G. Dahan (ed.), Les Juifs au regard de l’histoire, Paris, pp. 235–247. Goldstein, B.R. 1992, “Lunar Velocity in the Ptolemaic Tradition”, in P.M. Harman and A.E. Shapiro (eds.), An Investigation of Difficult Things: Essays on Newton and the History of Exact Sciences, Cambridge, pp. 3–17. Goldstein, B.R., and Sawyer, F.W. iii, 1977, “Remarks on Ptolemy’s Equant Model in Islamic Astronomy” in Y. Maeyama and W.G. Saltzer (eds.) Prismata, Wiesbaden, pp. 165–181. Goldstein, B.R., Chabás, J., and Mancha, J.L. 1994, “Planetary and Lunar Velocities in the Castilian Alfonsine Tables”, Proceedings of the American Philosophical Society, 138: 61–95. Kennedy, E.S. 1956a, “A Survey of Islamic Astronomical Tables”, Transactions of the American Philosophical Society ns 46. Kennedy, E.S. 1956b, “Parallax Theory in Islamic Astronomy”, Isis 57: 33–53; reprinted in Kennedy, E.S., Studies in the Islamic Exact Sciences, Beirut (1983), pp. 164–184. Kennedy, E.S. 1986, The Verified Astronomical Tables for the Caliph al-Maʾmūn by Yaḥyā b. Abī Manṣūr. A photographic reproduction of ms Escorial Ar. 927, with an introduction. Frankfurt. Kennedy, E.S., and Destombes, M. 1966, “lntroduction to Kitāb al-cAmal biʾl-Asturlāb”, Osmania Oriental Pub., Hyderabad-Dn. (1966); reprinted in Kennedy, E.S., Studies in the Islamic Exact Sciences, Beirut (1983), pp. 405–447. Kennedy, E.S., and Faris, N. 1970, “The Solar Eclipse Technique of Yaḥyā b. Abī Manṣūr”, Journal for the History of Astronomy 1: 20–38; reprinted in Kennedy, E.S., Studies in the Islamic Exact Sciences, Beirut (1983), pp. 185–203. Kennedy, E.S., and Kennedy, M.H. 1987, Geographical Coordinates of Localities from Islamic Sources, Frankfurt. Kennedy, E.S., and Ukashah, W. 1969, “Al-Khwārizmī’s Planetary Latitude Tables”, Centaurus 14: 86–96; reprinted in Kennedy, E.S., Studies in the Islamic Exact Sciences, Beirut (1983), pp. 125–135. Kunitzsch, P. 1966, Typen van Sternverzeichnissen in astronomischen Handschriften des zehnten bis vierzehnten Jahrhunderts, Wiesbaden. Mercier, R. 1976–1977, “Studies in the Medieval Conception of Precession”, Archives internationales d’histoire des seiences 26: 197–220, and 27: 33–71. Millás, J.M. 1942, Las traducciones orientales en los manuscritos de la Biblioteca Catedral de Toledo, Madrid. Millás, J.M. 1947, El libro de los fundamentos de las Tablas astronómicas, Madrid–Barcelona.

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Millás, J.M. 1950, Estudios sobre Azarquiel, Madrid–Granada (1943–1950). Millás, J.M. 1962, Las Tablas Astronómicas del Rey Don Pedro el Ceremonioso, Madrid– Barcelona. Morelon, R. 1987, Thābit ibn Qurra. Oeuvres d’astronomie, Paris. Nallino, C.A. 1899–1907, Al-Battānī sive Albatenii Opus Astronomicum, 3 Vols. Milan. Neugebauer, O. 1962, The Astronomical Tables of al-Khwārizmī. Copenhagen. North, J.D. 1967, “Medieval Star Catalogues and the Movement of the Eighth Sphere”, Archives internationales d’histoire des sciences 17:73–83. North, J.D. 1976, Richard of Wallingford: An edition of his writings with introductions, English translation and commentary, 3 Vols. Oxford. Pingree, D. 1968a, The Thousands of Abū Macshar. London. Pingree, D. 1968b, “The Fragments of the Works of Yacqūb ibn Ṭāriq”, Journal of Near Eastern Studies 27: 97–125. Pingree, D. 1970, “The Fragments of the Works of al-Fazārī”, Journal of Near Eastern Studies 29: 103–123. Salam, H., and Kennedy, E.S. 1967, “Solar and Lunar Tables in Early Islamic Astronomy”, Journal of the American Oriental Society 87: 492–497. As-Saleh, J.A. 1970, “Solar and Lunar Distances and Apparent Velocities in the Astronomical Tables of Ḥabash al-Ḥāsib”, Al-Abhath 23: 129–177; reprinted in Kennedy, E.S., Studies in the Islamic Exact Sciences, Beirut (1983), pp. 204–252. Samsó, J. 1987, “Al-Zarqāl, Alfonso x and Peter of Aragon on the Solar Equation”, in David A. King and George Saliba (eds.), From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E.S. Kennedy, Annals of the New York Academy of Sciences 500: 467–476. Samsó, J. 1992, Las Ciencias de los Antiguos en al-Andalus. Madrid. Sédillot, J.-J., and Sédillot, L.-A. 1834, Traité des instruments astronomiques des Arabes. Paris; reprinted Frankfurt 1984. Stahlman, W.D. 1959, The Astronomical Tables of Codex Vaticanus Graecus 1291. Brown Univ. Ph.D. Thesis. Suter, H. 1914, Die astronomischen Tafeln des Muḥammad ibn Mūsā al-Khwārizmī. Copenhagen. Toomer, G.J. 1968, “A Survey of the Toledan Tables”, Osiris 15: 5–174. Toomer, G.J. 1969, “A History of Errors”, Centaurus 14: 306–336. Toomer, G.J. 1987, “The Solar Theory of Az-Zarqāl: An Epilogue” in David A. King and George Saliba (eds.), From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E.S. Kennedy, Annals of the New York Academy of Sciences 500: 513–519. Vernet, J. 1949, “Un tractact d’obstetricia astrològica”, Boletín de la Real Academia de Buenas Letras de Barcelona 22: 69–96; reprinted in Vernet, J., Estudios sobre Historia de la Ciencia Medieval, Barcelona–Bellaterra (1979), pp. 273–300.

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Vernet, J. 1956, “Las Tabulae Probatae”, Homenaje a Millás Vallicrosa, Barcelona, ii: 501– 522; reprinted in Vernet, J., Estudios sobre Historia de la Ciencia Medieval, Barcelona– Bellaterra (1979), pp. 191–212.

chapter 8

Early Alfonsine Astronomy in Paris: The Tables of John Vimond (1320)* It has been clear for many years that medieval European astronomy in Latin was heavily dependent on sources from the Iberian peninsula, primarily in Arabic, but also in Hebrew, Castilian, and Catalan. The Castilian Alfonsine Tables, compiled by Judah ben Moses ha-Cohen and Isaac ben Sid under the patronage of Alfonso x (d. 1284), were an important vehicle for the transmission of this body of knowledge to astronomers north of the Pyrenees, but the details of this transmission remain elusive, in part because only the canons to these tables survive (see Chabás and Goldstein 2003a). In this paper we build on our preliminary studies of a figure who previously had barely been mentioned in the recent literature on medieval astronomy (Chabás and Goldstein 2003a, pp. 267–277, and 2003b). John Vimond was active in Paris ca. 1320 and, as we shall see, his tables have much in common with the Parisian Alfonsine Tables (produced by a group in Paris, notably John of Murs and John of Lignères), but differ from them in many significant ways. As far as we can tell, there is no evidence for any interaction between Vimond and his better known Parisian contemporaries and in our view the best hypothesis is that they all depended on Castilian sources. As a result of our analysis, we are persuaded that Vimond’s tables are an intelligent reworking of previous astronomical material in the Iberian peninsula to a greater extent than is the case for the Toledan Tables (compiled in Toledo about two centuries before the Castilian Alfonsine Tables). It is most likely that Vimond’s principal source was the Castilian version of the Alfonsine Tables. Paris, Bibliothèque nationale de France, ms lat. 7286c is a 14th-century manuscript containing an unusual set of tables (ff. 1r–8v) as well as the canons and tables of 1322 by John of Lignères (ff. 9r–58r). In a brief text at the end of the first set of tables they are attributed to John Vimond (Iohannes Vimundus), an astronomer who compiled them “for the use of students at the University of Paris” (f. 8v):

* Suhayl 4 (2004), 207–294.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_010

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Et in hoc terminatur opus Iohannis Vimundi baiocensis dyocesis de disposicionibus planetarum et stellarum fixarum; et cum istis sequitur de hiis que per ipsum ordinantur ad conversionem temporum verorum et equalium sociatorum, et de disposicionibus eclipsalibus solis et lune sibi pertinentibus, et de aliis disposicionibus ipsorum et aliorum corporum celestium, ad utilitatem scolarium universitatis parisiensis et omnium aliorum. Here ends the work by John Vimond of the diocese of Bayeux on the dispositions of the planets and the fixed stars; (…) and on the dispositions of solar and lunar eclipses and [other syzygies] corresponding to them, and on the other dispositions of these and other celestial bodies, for the use of students at the University of Paris and all others. The complete set of Vimond’s tables are uniquely extant in this manuscript, and no canons for them have been identified. They are a coherent set of tables with all the elements needed to compute the positions of the celestial bodies, much in the tradition of the Arabic zijes and their derivatives. The exact date of composition of Vimond’s tables is not given in the text, but they were probably produced shortly before 1320. In the paragraph preceding his tables, Vimond tells us that they were compiled for Paris with 1320 as epoch (f. 1r: see below) and this date is confirmed by recomputation. These tables also include a calendar with the dates of syzygies: this strongly suggests that they were constructed prior to the year of the calendar because the astronomical information would no longer be of any use after the year had passed. However, the calendar poses special problems which will be discussed below. Vimond’s only other known work is a short treatise on the construction of an astronomical instrument, extant in Erfurt, ms ca 2° 377 (ff. 21r–22r), beginning Planicelium vero componitur ex eis que sunt diversorum operum …, and ending Explicit tractatus johannis vimundi … in a manuscript containing various works by other Parisian astronomers such as John of Murs and John of Lignères (Thorndike and Kibre 1963, col. 1050; Saby 1987, pp. 471, 474). John Vimond and his works were seldom mentioned by his contemporaries. However, in Vatican, Biblioteca Apostolica, ms Ottob. lat. 1826, we are told that John of Spira (14th century), the author of a commentary on John of Lignères’s canons (Thorndike and Kibre 1963, col. 204), composed his own canons to several of Vimond’s tables (for a description of this manuscript, especially ff. 148–153, see F.S. Pedersen 2002, p. 177). This manuscript includes a text that begins on f. 148ra ascribed to a certain M.J.C., Canon tabulae sequentis quae intitulatur tabula motus diversi solis et lunae in una hora et semidiametrorum secundum tabulas Alfonsi, at the end of which John Vimond is

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mentioned. On the other hand, Vimond is not mentioned in Madrid, Biblioteca Nacional, ms 4238, a manuscript containing a few tables that can be attributed to him, as well as a copy of the Parisian Alfonsine Tables computed for Morella (in the province of Valencia) for the years 1396 and 1400 (Chabás 2000). As far as we can tell, John of Murs and John of Lignères do not refer to Vimond at all in any of their numerous works, but it seems implausible that they did not know him or his work which was addressed to the students at the University of Paris. Indeed, there were not so many competent astronomers working in Paris around 1320 and both Vimond and Murs came from the same region, Normandy, from places about 70km apart, Bayeux and Lisieux, respectively. We would expect Vimond to be well known and frequently cited by practitioners of astronomy, for he is named as one of the outstanding astronomers of his time by Simon de Phares in his Recueil des plus celebres astrologues (1494–1498), a chronologically ordered list with comments, edited by Boudet (1997–1999, 1:467). In fact, Vimond is mentioned before John of Lignères, John of Saxony, John of Janua, and John of Murs: Maistre Jehan Vymond fut a Paris, homme moult singulier et grant astrologien, lequel eut en ce temps grant cours pour la science des estoilles. Entre ses euvres, fist une verifficacion de la conjunction des lu[mi]naires, aussi des eclipses et estoilles fixes pour plusieurs ans. Cestui predist les grans vens qui furent en son temps et fist plusieurs beaulx jugemens, dont il acquist grant loz et renommee en France et fut moult devost en Nostre Seigneur. Master John Vimond lived in Paris, a most singular man and a great astrologer, who had at that time much prestige because of (his knowledge of) the science of the stars. Among his works is a verification of the conjunction of the luminaries, as well as eclipses and the fixed stars, for many years. He predicted the great winds which took place in his time and made many fine judgments for which he acquired great praise and renown in France and he was most devoted to our Lord. The “verifficacion de la conjunction des lu[mi]naires” refers to Vimond’s tables. These tables are arranged very differently from those of his Parisian contemporaries and are based, in part, on parameters that probably came from the Castilian Alfonsine Tables or a tradition closely associated with them. Of special interest is the proper motion of the solar and planetary apogees, a feature previously unknown in medieval tables produced outside Spain and North Africa.

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We are convinced that Vimond’s tables provide an indication of the arrival in Paris of new astronomical material coming from Castile, in the sense that they propose new approaches to replace those based on the Toledan Tables and developed at the end of the 13th century by astronomers working in Paris such as Peter Nightingale, Geoffreoy of Meaux, and William of St.-Cloud. Further, we believe that Vimond’s tables are prior to, and independent of, the tabular work developed in the early 14th century, which we call the Parisian Alfonsine Tables, by the group of Parisian astronomers that included John of Murs and John of Lignères, which were also based on Castilian sources. Vimond’s tables and the Parisian Alfonsine Tables have many parameters in common both for mean motions and equations. In principle, it is possible that one set of tables depended on the other, but the differences between them suggest to us that it is far more likely that they depended on a common source. Moreover, if Vimond composed his tables prior to 1320, he did so before any datable text of the Parisian Alfonsine Tables. A description and analysis of Vimond’s tables follow. f. 1r The first numerical information given in this set of tables is the “radix for mean conjunctions of the Sun and the Moon”: 13;54,54d. In modern terminology, the initial time for a set of tables is called its “epoch” whereas its “radices” are the positions of the Sun, Moon, planets, etc., at that time. The medieval convention, however, is to use “radix” for both the time and the position. We are convinced that the author refers to the time, in Paris, of the mean conjunction on March 10, 1320. The year and the place are mentioned by Vimond himself in a short paragraph following the numerical value of the radix (f. 1r): Et est intelligendum quod ista radix mediarum coniunccionum sit immediate post 19 secunda diei que consistunt immediate post (…) lucis beati Mathie composite procedendo ab ortu solis usque ad occasum scilicet anno domini nostri Ihesu Christi 1320 secundum numeracionem annorum romanorum qui incipiunt ex inicio diei circoncisionis domini nostri Ihesu Christi et existentis ad longitudinem civitatis Parisius que distat a medio mundi per 49 g et 30 min ita quod illa civitas est in parte occidentali et etiam distat ab illo medio per 8 min et 15s diei equalis. Note that this radix for the mean conjunctions comes immediately after 19 seconds of a day that fall immediately after the (…) [space for one word; illegible] (day)light of Saint Matthew, proceeding from sunrise to

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sunset, namely, in the year of our Lord Jesus Christ 1320 according to the count of Roman years which start from the beginning of the Day of the Circumcision of our Lord Jesus, for the longitude of the city of Paris which is distant from the middle of the world by 49 degrees and 30 minutes because that city is in the western direction and distant from that middle by 8 minutes and 15 seconds of an equinoctial day. Madrid, Biblioteca Nacional, ms 4238, f. 66v, has a short text which is very similar to the paragraph on f. 1r in Paris, Bibliothèque nationale de France, ms lat. 7286c: Radix coniunccionum: dies 13 m 54 2ª 54 Radix opposicionum: dies 14 m 45 2ª 55 Hec radix est post 19 2ª diei que sunt post meridiem diei Mathie anno 1320° secundum romanos. Nota quod annus in meridie diei mathie 25 dies bisexti erit semper ultimus dies anni. Iste tabule radicum sunt facte Parisius ad meridiem cuius cenith distat ab equinocciali 49 g 30 m vel 8 m 15 2ª diei. The numerical datum, 0;0,19d (= 0;7,36h), represents the equation of time for that day. In the Madrid version, the “radix of the opposition”, 14;45,55d, is half the length of a mean synodic month which is about 29;31,50d, and this is the entry for the first opposition in Vimond’s calendar (see below). It is clear that, according to the version of this text in the Paris manuscript, the civil day in the calendar includes the period of daylight, that is, the time from sunrise to sunset, in contrast to the astronomical day that goes from noon to the following noon. Both values given for the longitude of Paris from Arin, called “the middle of the world”, are equivalent. Arin, a corruption of Ujjain (a city in India), was thought to be halfway between the eastern and western limits of the world (Neugebauer 1962, p. 11, n. 2). The distance from Arin to Toledo was taken to be 61;30° and, since Paris was generally said to be 0;48h or 12° to the east of Toledo, its longitude from Arin is 49;30°, as in the passage above (Millás 1943–1950, p. 49; Kremer and Dobrzycki 1998, p. 194; and F.S. Pedersen 2002, p. 431). Moreover, when a day is taken to be 360°, it follows that 49;30° corresponds to 0;8,15d, for 49;30°/360° = 0;8,15. The expression in the Madrid manuscript, “Parisius ad meridiem cuius cenith distat ab equinocciali” is a corrupt version of the better reading in the Paris manuscript, for it would imply that 49;30° is the latitude of Paris, but then its equivalence to 0;8,15d would become meaningless. We also note that, according to this text, Vimond’s tables were computed for Paris whereas in the early 1320s other Parisian astronomers who recast the Alfonsine Tables computed them for Toledo, as is the case for the

232 table 1

chapter 8 Mean conjunctions ( f. 1r)

[years]

[excess] d

[years]

[excess] d

[years]

[excess] d

1 2 3 4 8 12 16 20 24

18;53,52 8;15,53 27; 9,45 15;31,46 1;31,43 17; 3,29 3; 3,25 18;35,12 4;35, 8

28 32 36 40 44 48 52 56 60

20; 6,54 6; 6,50 21;38,37 7;38,33 23;10,19 9;10,15 24;42, 2 10;41,58 26;13,44

64 68 72 76 152 304 608 1216 2432

12;13,41 27;45,27 13;45,23 29;17,10 29; 2,29 28;33, 8 27;34,26 25;47, 1 21;43,12

tables with epoch 1321 by John of Murs (see, e.g., Lisbon, Biblioteca de Ajuda, ms 52-xii-35). According to our computations based on the Parisian Alfonsine Tables, the mean conjunction on March 10, 1320 took place in Toledo at 9;10h, civil time (i.e., counting from midnight) which, with a correction of 0;48h, is 9;58h in Paris (civil time), that is, 2;2h before noon. Thus, the radix for the tables on f. 1r (as well as the radices for the planetary tables, as will be seen later) is the time of the first mean conjunction in March 1320 (March 10, 1320, at 9;58 a.m., Paris, or March 9, 1320, 21;58h, Paris, counting from noon). Indeed, the sexagesimal part of the “radix” (0;54,54d) is exactly the sum of 12h and 9;58h. The integer part of the “radix”, as will be explained later in reference to the annual calendar presented on this same folio, is counted from the epoch of the calendar, almost 14 days before the mean conjunction of March 10, 1320, that is, February 25, 1320 or February 24b, 1320, where 24b represents the second day called February 24 in a leap year (such that the last day of February is always day 28 both in ordinary and leap years).

f. 1r Table 1: Mean Conjunctions The entries in this table give the instant of the first mean conjunction after a certain number of years. We are given entries for 1, 2, 3, and 4 years; for multiples of 4 years up to 76 (= 19 · 4) years; and for 152, 304, 608, 1216, and 2432 years. The entries represent the excess of days after an integer number, n, of synodic

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months have elapsed (where n = 13 for year 1, …, and 30,081 for year 2432). Madrid, Biblioteca Nacional, ms 4238, f. 66v, reproduces this table except for the last row for 2432 years, which is missing. The value for the mean synodic month derived from year 2432 is 29;31,50,7, 44,35d ±0;0,0,0,0,4d. Thus, for year 1: 13 · 29;31,50,7,44,35d – 365d = 18;53,52d, in agreement with the tabulated value. In the Parisian Alfonsine Tables, the mean synodic month is 29;31,50,7,37,27,8,25d: this value is found, for example, in Lisbon, Biblioteca de Ajuda, ms 52-xii-35, f. 16v, containing the tables for epoch 1321 by John of Murs. So Vimond’s parameter is very similar to, but not identical with, the parameter in the Parisian Alfonsine Tables.

Vimond

Parisian Alf. t.

Mean synodic month 29;31,50,7,44,35d 29;31,50,7,37,27,8,25d

f. 1r Table 2: Annual Calendar with Syzygies This annual calendar begins on the day of Saint Matthew (February 24) and lists the dates associated with several saints, as well as the dates and times of 25 consecutive mean syzygies. The practice of adding the extra day in a leap year after Feb. 24 goes back to the Roman calendar as revised by Julius Caesar, when the additional day followed Feb. 24 and was called bis-sextus ante calendas martias (the sixth day before the calends of March). In a leap year February lasted 29 days, but the last day was numbered “28”, for the 24th was assigned to two consecutive days. This is what is intended in Vimond’s calendar where the year begins on that very day. We know of no other calendar in the late 13th century or early 14th century beginning on Feb. 24; in particular, the calendars composed by Geoffreoy of Meaux and William of St.-Cloud do not begin on that day (Chabás and Goldstein 2003a, pp. 245–247). It is worth noting that Vimond’s calendar which lists mean syzygies together with saints’ days is in the tradition of these two astronomers who were active in Paris shortly before him: they displayed planetary data in calendars and depended on the Toledan Tables for their computations. We also note that the feast of St. Matthew is mentioned in the canons to the Parisian Alfonsine Tables by John of Saxony as the last day in a leap year (see Poulle 1984, p. 36, line 41). Vimond offers no explanation for basing his calendar and his tables on syzygies; we can only conjecture that he was being faithful to some unknown source.

234 table 2

chapter 8 Annual calendar with syzygies ( f. 1r)

(1) [Saint’s day / No. syzygy]

Romanus Perpetua virgo 1 Opposition Gregorius papa … 2 Conjunction … 3 Opposition … 4 Conjunction … 5 Opposition … 6 Conjunction … 7 Opposition … 8 Conjunction … 9 Opposition … 10 Conjunction … 11 Opposition … 12 Conjunction … 13 Opposition … 14 Conjunction … 15 Opposition … 16 Conjunction

(2) [date]

(3) [time since epoch] d

February 28 March 7 March 10 12

4 11 14;45,55 16

March 25

29;31,50

April 9

44;17,45

April 24

59; 3,40

May 8

73;49,35

May 23

88;35,30

June 7

103;21,25

June 22

118; 7,21

July 6

132;53,16

July 21

147;39,11

August 5

162;25, 6

August 20

177;11, 1

September 3

191;56,56

September 18

206;42,51

October 3

221;28,46

October 18

236;14,41

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(1) [Saint’s day / No. syzygy]

… 17 Opposition … 18 Conjunction … 19 Opposition … 20 Conjunction … 21 Opposition Circoncisio domini Ihesu Christi initium anni … 22 Conjunction … 23 Opposition … 24 Conjunction Iuliana virgo Petrus ad cathedram 25 Opposition

(2) [date]

(3) [time since epoch] d

November 2

251; 0,36

November 14

265;46,31

November 31

280;32,26

December 16

295;18,21

December 31

310; 4,16

[January 1]

311

January 14

324;50,11

January 29

339;36, 6

February 13 16 22 February 28

354;22, 2 357 363 369; 7,57

In Table 2, columns 1 and 2 have no heading, but column 3 has the heading “days, minutes, and seconds”. In the manuscript the name of the month is usually given in col. 2, and occasionally in col. 1 which has about 90 entries such as: Annunciatio Domini, Dyonisius, Lucas Evangelista, Innocentes, etc. The syzygies are numbered from 1 to 25, and they are transcribed above. The numbers in column 3 are integers when a saint’s day is meant and indicate the number of days that elapsed since the epoch (day 1) of the calendar, that is, February 25, 1320 (Julian) or what we have called February 24b, 1320. Vimond seems to use here civil days (from midnight to midnight) rather than astronomical days (from noon to noon), which makes sense in a calendar. When a conjunction or an opposition is indicated, we would expect the number in column 3 to refer to the accumulated time from the radix (the conjunction on March

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10, 1320) in multiples of half a mean synodic month, i.e. 14;45,55d but, in fact, we are given the accumulated time from the mean syzygy—an opposition— immediately preceding the radix, which occurred on February 24, 1320. If this was the author’s intention, it is not clear how the user of these tables was to take account of the radix given at the beginning of them. Moreover, despite the coherence of the arithmetic in this calendar, something is seriously wrong with it, for we find the word oppositio next to March 10, when a conjunction took place, and the word coniunccio next to March 25, when an opposition occurred. The same pattern is followed throughout the calendar. There is an “explanatory” note on f. 1r concerning the calendar, but we were unable to make sense of it. Year 1324 might be considered as an alternative date for the calendar for, according to computations with the Parisian Alfonsine Tables, a mean opposition occurred on March 10 (counting from noon) or on March 11 (counting from midnight). This would conform with the character of the syzygy mentioned in the calendar, and the computations associated with this date yield results that are quite close to (but not exactly the same as) the information given in the text. Indeed, in our preliminary discussion of these tables, this near agreement mislead us to think that 1324 was the radix of Vimond’s tables (Chabás and Goldstein 2003a, p. 270). However, as indicated previously, year 1320 is specifically mentioned, and it fits much better with the radix of mean conjunctions and with the radices for planetary positions displayed on ff. 1v and 4r. f. 1v Radices for the argument of solar anomaly, the argument of lunar anomaly (henceforth, solar and lunar anomaly, respectively), the solar apogee, and the lunar ascending node: Solar anomaly 8s 26;14,33° Lunar anomaly 1s 3; 6,14° Solar apogee 2s 29;56,15° Ascending node 10s 13;14,43° Note the use of zodiacal signs of 30°, a characteristic of all tables in this set. A short text below these parameters explains that the radices for the motion of the solar apogee and the ascending node are counted from the beginning of Aries on the 9th sphere, indicating that tropical coordinates are used here. These radices were calculated for March 10, 1320, at the time of the mean conjunction of the Sun and the Moon. According to the Parisian Alfonsine Tables the solar apogee for March 10, 1320 is 89;23,50°, a value which differs by about

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half a degree from the entry in the text. Both values in turn differ from the solar apogee for 1320 in the tables for 1322 by John of Lignères (89;24,22°) that is found by adding two values given on f. 9v of this same manuscript: the solar apogee (81;7,15,39°) and the motion of the 8th sphere at that time (8;17,6,48°). The same result, 89;24,22°, can also be found in another copy of John of Lignères’s tables, Erfurt, ms ca q 362, f. 21ra. For the rest of the radices, recomputation with the Parisian Alfonsine Tables for the epoch, March 9, 1320, at 21;10h in Toledo (counting from noon), yields results which are very close to the values in the text, especially for the Moon:

Vimond Solar anomaly 266;14,33° Lunar anomaly 33; 6,14° Solar apogee 89;56,15° Ascending node 313;14,43°

Parisian Alf. t. 266;47, 0° 33; 6,28° 89;23,50° 312;54,39°

The solar longitude is the sum of the solar anomaly and the solar apogee:

Vimond Solar longitude 356;10,48°

Parisian Alf. t. 356;10,50°

and again the agreement is very good. Since this is the time of a mean conjunction, the mean lunar longitude will be equal to the mean solar longitude. According to the Parisian Alfonsine Tables, the mean lunar longitude at this epoch was 356;11,3°, i.e., it differed from the mean solar longitude by only 0;0,13° (note that the Moon travels this distance in about 20 seconds of time which is below the accuracy of 1 minute for the time of mean conjunction). Hence the absence of a radix for lunar mean motion simply reflects the fact for the epoch of Vimond’s tables the mean longitude of the Moon is the same as the mean longitude of the Sun. The agreement for the radix of lunar anomaly to the minute is particularly impressive since the motion in lunar anomaly is about 0;30°/h. Lunar anomaly is not subject to precession and it is independent of solar motion. (We use the term precession for a constant motion of the eighth sphere, and trepidation for a

238 table 3

chapter 8 Yearly radices ( f. 1v)

s

Year 1 (°)

Year 2 (°)

s

s

Year 3 (°)

…

Solar anomaly 0 18;22, 3 0 7;37,47 0 25;59,49 Lunar anomaly 11 5;37, 8 9 15;25,15 8 21; 2,22 Solar apogee 0 0; 1,12 0 0; 2,18 0 0; 3,30 Ascending node 11 9;43,23 10 20;58,27 10 0;40,50

…

Solar anomaly Lunar anomaly Solar apogee Ascending node

Year 8 s (°)

…

0 1;24,48 1 5;51,58 0 0; 9, 7 6 25;27,27

Year 608 s (°)

…

0 20; 6, 6 4 8;22, 2 0 11;32,36 4 29;26,44

variable motion of the eighth sphere.) So, even though Vimond and the authors of the Parisian version of the Alfonsine Tables differ on matters of definition and made slight changes in mean motions, it is unlikely that either of them would change the motion in anomaly significantly from what it had been in their common source.

f. 1v Table 3: Yearly Radices This table displays the radices for the solar anomaly, the lunar anomaly, the solar apogee, and the lunar ascending node for 1, 2, 3, and 4 years; for multiples of 4 years up to 76 years; and then for 152, 304, 608, 1216, and 2432 years, as in Table 1. A selection of the entries is displayed in Table 3. The entries for year 1 represent the progress made by the Sun, the Moon, the solar apogee, and the lunar node in a year of 13 mean syzygies of the same kind (henceforth “lunations”) of 29;31,50,7,44,35d. To be sure, the difference between the solar anomalies for year 2 and year 1 is 349;15,44°, meaning that year 2 contains 12 lunations, for 349;15,44°/ (29;31,50 … · 0;59, 8, …) = 12, whereas the difference between year 3 and year 2 is 378;22,2° (the same value as that

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for year 1), indicating that year 3 contains 13 lunations (378;22,2°/ (29;31,50 … · 0;59, 8, …) = 13), as is the case with year 1. With this procedure, we see that 50 is the total number of lunations in the first 4 years, 99 in the first 8 years, …, 7,521 in the first 608 years, and so on. That is, for Vimond 1 year is equivalent to 13 mean lunations, 2 years is equivalent to 25 mean lunations, etc. Where possible, we have derived the associated mean motions from the entries for year 608, because those for years 1216 and 2432 are not completely legible in the manuscript. The mean motion in solar anomaly resulting from the entry for year 608 (0s 20;6,6°), that is, after 7,521 lunations, and the length of the synodic month obtained before (29;31,50,7,44,35d), is 0;59,8,8,23,30°/d, for (608 · 360° + 20;6,6°)/(7521 · 29;31,50,7,44,35d) = 0;59,8,8,23,30°/d. This daily mean motion implies a year length of 365;15,42,32d which is sidereal. In the Parisian Alfonsine Tables, however, the length of a sidereal year is variable, and the fixed length of the tropical year is 365;14,33,9,57, … d (= 360°/0;59,8,19,37,19,13,56°/d). Similarly, the mean motion in lunar anomaly can be computed from the entry corresponding to 7,521 lunations (year 608: 4s 8;22,2°), for 7521 lunations corresponds to 8060 complete revolutions in anomaly with an excess of about 120° (computed with approximate values for the appropriate parameters). Hence, with the data in the text, the mean motion in lunar anomaly is (8060 · 360° + 128;22,2°)/(7521 · 29;31,50,7,44,35d) = 13;3,53,57,27,11°/d, in very good agreement with the corresponding value in the Parisian Alfonsine Tables (13;3,53,57,30,21°/d); the difference only accumulates to 1° in well over 10,000 years. As for the motion of the solar apogee derived from the entry corresponding to 7,521 lunations (year 608: 0s 11;32,36°), we find 0;0,0,11,13,35°/d. By the same reasoning, the mean motion of the lunar ascending node resulting from the entry of year 8 in the table (6s 25;27,27°) is –0;3,10,18,6,48°/d, in contrast to the value found in the Parisian Alfonsine Tables (–0;3,10,38,7,14,49,10°/d). In this case, the entry in the manuscript for 608 years is corrupt. We note that Vimond’s value for the motion of the solar apogee includes precession as well as its proper motion for, if we add the value for the mean

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Vimond Year length 365;15,42,32d Solar anomaly 0;59, 8, 8,23,30°/d Lunar anomaly 13; 3,53,57,27,11 Solar apogee 0; 0, 0,11,13,35 Ascending node –0; 3,10,18, 6,48

Parisian Alf. t. 365;14,33, 9,57d 0;59, 8,19,37,19°/d 13; 3,53,57,30,21 – –0; 3,10,38, 7,14

motion in solar anomaly (which is sidereal) to the motion of the solar apogee, we find 0;59,8,19,37,4°/d, in close agreement with the corresponding value of the mean motion in solar longitude (tropical) in the Parisian Alfonsine Tables. In the Almagest, the planetary apogees are sidereally fixed whereas the solar apogee is tropically fixed. In the 9th century astronomers in Baghdad fixed the solar apogee sidereally so that it too was subject to precession (or trepidation). But in the 11th century Azarquiel realized that the solar apogee had a proper motion in addition to precession, and fixed its amount as 1° in 279 Julian years or about 0;0,0,2°/d (Chabás and Goldstein 1994, p. 28). In one Andalusian tradition, this proper motion of the solar apogee was applied to the planetary apogees as well (see Samsó and Millás 1998, p. 269; cf. Mestres 1996, pp. 394–395). If we take al-Battānī’s value for precession of 1° in 66 years or about 0;0,0,9°/d and add it to the proper motion of the solar apogee, the result is about 0;0,0,11°/d. There is no hint of this proper motion for either the solar apogee or the planetary apogees in the Parisian Alfonsine Tables where these apogees are all sidereally fixed and, instead of precession, the Parisian Alfonsine Tables have tables for trepidation; hence, there is nothing in those tables with which to compare directly the motion of the solar apogee in Vimond’s tables. We see, then, that the parameters in Vimond’s tables are not identical with those in the Parisian Alfonsine Tables, and some of these parameters (e.g., the length of the solar year) are defined differently.

ff. 1v–2r Table 4: Monthly Radices This table displays the radices for the solar anomaly, the lunar anomaly, the solar apogee, and the lunar ascending node for 25 consecutive syzygies after the corresponding integer numbers of semi-lunations have elapsed. An excerpt is shown in Table 4.

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early alfonsine astronomy in paris table 4

Monthly radices ( ff. 1v–2r)

s

Syzygy 1 (°)

s

Syzygy 2 (°)

Solar anomaly 0 14;33, 9 0 29; 6,19 Lunar anomaly 6 12;54,30 0 25;49, 1 Solar apogee 0 0; 0, 3 0 0; 0, 6 Ascending node 11 29;13,10 11 28;26,20

… s

Syzygy 25 (°)

0 3;48,53 4 22;42,27 0 0; 1, 9 11 10;29,13

The entries represent the progress made by the Sun, the Moon, the solar apogee, and the lunar node in 1, 2, …, 25 mean semi-lunations of 29;31,50,7,44, 35d/ 2 = 14;45,55,3,52,17d. The entries in this table agree with those in Table 3, for in each case the value for 26 consecutive semi-lunations (the sum of the entries for Syzygy 1 and Sygygy 25 in Table 4) equals the value for 13 lunations (year 1 in Table 3).

f. 2r Table 5: Sun This table in 5 columns is original in presentation. Column 1 gives the argument (argumentum) at 3°-intervals in signs and degrees from 0s 3° to 12s 0°; this is the mean solar anomaly. Column 2 displays the true solar anomaly (motus completus) in signs, degrees, and minutes. Column 3 (motus gradus) displays the increment in true anomaly per degree of the argument. Column 4 gives the solar velocity, in units of minutes and seconds of arc in a minute of a day (minutum diei), i.e., in a sixtieth of a day. Column 5 displays the time (also called argumentum), in days, with sexagesimal fractions of a day, that the Sun takes to complete the arc indicated in column 1. To obtain an entry in column 5 multiply the corresponding entry in column 1 by the daily mean motion in solar anomaly; the entry for 360° (365;15,42d) represents the length of the sidereal year, in good agreement with the value deduced from 99 mean synodic months in Table 3. As shown in Table 5a, the difference between the argument (col. 1) and the true anomaly (col. 2) represents the solar equation, with a maximum of 2;10° as in the Parisian Alfonsine Tables. To emphasize the solar equation, we have added a third column for the differences between entries in columns ii and i, labeled ii – i.

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table 5

Sun ( f. 2r)

(1) argum. s (°)

(2) motus c. s (°)

0 0 … 2 3 3 3 3 … 5 6 6 … 8 8 8 9 9 … 11 12

0 0

3 6

(5) argum. d

2;54 5;47

57;51 57;51

0;57 0;57

3; 2,38 6; 5,16

27 0 3 6 9

2 24;51 2 27;50 3 0;50 3 3;50 3 6;51

59;47 59;59 60; 3 60;16 60;19

0;59 0;59 0;59 0;59 0;59

88;16,18 91;18,55 94;21,33 97;24,11 100;26,49

27 0 3

5 26;53 6 0; 0 6 3; 7

62;24 62;24 62;22

1; 1 1; 1 1; 1

179;35,13 182;37,51 185;40,29

21 24 27 0 3

8 23; 9 8 26;10 8 29;10 9 2;10 9 5; 9

60;16 60; 3 59;59 59;47 59;43

0;59 0;59 0;59 0;59 0;59

264;48,53 267;51,31 270;54, 9 273;56,46 276;59,24

27 0

11 27; 6 12 0; 0

57;51 57;51

0;57 0;57

362;13, 4 365;15,42

table 5a

The solar equation embedded in Table 5

i argumentum s (º) 0 0 … 2 3

(3) (4) motus g. min. diei min. min.

ii motus completus s (º)

ii – i (°)

3 6

0 0

2;54 5;47

–0; 6 –0;13

27 0

2 2

24;51 27;50

–2; 9 –2;10

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early alfonsine astronomy in paris

i argumentum s (º) 3 3 3 … 5 6 6 … 8 8 8 9 9 … 11 12

ii motus completus s (º)

ii – i (°)

3 6 9

3 3 3

0;50 3;50 6;51

–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

The entries for the solar equation are not explicit in Vimond’s table; they can be graphed as a smooth curve but they do not allow us to decide which specific table for the solar equation he used. The reason is that Vimond’s entries are only given to minutes in contrast to most other tables in which the maximum equation is 2;10,0° where entries are given to seconds, and rounding those values produces Vimond’s entries.

ff. 2v–3r Table 6: Moon This table has the same format as Table 5. An excerpt is displayed in Table 6. Column 1 gives the argument (argumentum) at 1°-intervals in signs and degrees from 0s 1° to 6s 0° and its complement in 360° from 6s 0° to 11s 29°. For columns 2, 3, and 4, one enters with the mean argument of lunar anomaly, whereas for column 5 one enters with the argument of lunar latitude. Column 2 displays the lunar equation of center (motus completus) in degrees and minutes with a maximum of 4;56° as in the Parisian Alfonsine Tables.

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table 6

Moon ( ff. 2v–3r)

(1) (2) (3) (4) (5) argumentum motus c. motus min. min. diei latitud. s (°) s (°) (°) sec. min. (°) 0 0 … 2 3 3 3 3 3 3 3 3 3 … 5 6

1 2

11 11

29 28

0; 5 0;10

5 5

12; 9 12; 9

0; 5,13 0;10,27

29 0 1 2 3 4 5 6 7 8

9 9 8 8 8 8 8 8 8 8

1 0 29 28 27 26 25 24 23 22

4;54 4;55 4;55 4;56 4;56 4;56 4;56 4;56 4;56 4;55

0 0 0 0 0 0 0 0 0 0

13; 4 13; 5 13; 6 13; 8 13; 9 13; 9 13;11 13;13 13;14 13;15

4;59,58 4; 0, 0* 4;59,58 4;59,50 4;59,35 4;59,15 4;58,51 4;58,21 4;57,45 4;57, 4

29 0

6 6

1 0

0; 6 0; 0

6 6

14;25 14;25

0; 5,13 0; 0, 0

* Sic, instead of 5;0,0.

Column 3 (motus minuti) displays the line-by-line differences in column 2 divided by 60 (for purposes of interpolation). Column 4 gives the lunar velocity, in minutes and seconds, in a minute of a day (minutum diei). The minimum corresponds to 0;30,23°/h and the maximum to 0;36,3°/h: for a comparison with other tables for lunar velocity, see Goldstein 1996. Column 5 displays the lunar latitude, with a maximum of 5;0,0° as in the Parisian Alfonsine Tables and the Almagest. It is surprising that the expression motus completus is used here for the lunar equation of center, whereas in Table 5 it was used for the true solar anomaly; clearly, it has a range of meanings and cannot be translated by a single expression.

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early alfonsine astronomy in paris table 7(1) True syzygies ( f. 3r)

* (º) 1 2 3 4 5 6 7

11 (º)

[diff.]

12 (º)

[diff.]

13 (º)

[diff.]

14 (º)

1;31 3; 2 4;33 6; 4 7;35 9; 6 10;36

7 15 23 31 38 45 53

1;24 2;47 4;10 5;33 6;57 8;20 9;43

7 13 19 25 32 38 45

1;17 2;34 3;51 5; 8 6;25 7;42 8;58

5 11 16 22 28 33 38

1;12 2;23 3;35 4;46 5;57 7; 9 8;20

* In the ms, gradus velocitatis appears above this column but it refers to the headings of the other columns, labeled: 11, 12, 13, 14.

f. 3r Table 7: True Syzygies There are two subtables for computing the time from mean to true syzygy: see Tables 7(1) and 7(2). The first subtable is a double-argument table where, on analogy with the other subtable, the vertical argument seems to be the elongation between the Sun and the Moon (at 1°-intervals from 1° to 7°) and the horizontal argument, the velocity in elongation (i.e., the difference between the lunar and the solar velocities) in degrees per minute of a day (only four values for the velocity in elongation are given: 11, 12, 13, and 14). An entry, e, in this subtable was derived by means of the formula (expressed in modern notation) e = 16;40 · η / [vm(t) – vs(t)] where η is the true elongation at mean conjunction (or the result after subtracting 180° at mean opposition), and the velocity in elongation, vm(t) – vs(t), is the difference between the daily velocities of the Moon and the Sun at the time of mean syzygy. We cannot give a satisfactory explanation for the factor 16;40 (= 100/6) or for the headings of the columns indicating that the entries are in degrees and minutes (rather than in units of time). Between these four columns, one finds the differences, in minutes (but labeled “seconds”), between two consecutive entries in the same row, to facilitate interpolation. The second subtable is also a double-argument table giving the time in days as a function of the elongation (at intervals of 0;1° from 0;1° to 1°, or 60 minutes)

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table 7(2) True syzygies ( f. 3r)

min.*

11 min.

[diff.] sec.

12 min.

[diff.] sec.

13 min.

[diff.] sec.

14 min.

1 2 … 9

5;27 10;55

27 55

5; 0 10; 0

23 46

4;37 9;14

20 40

4;17 8;34

49; 5

245

45; 0

208

41;32

178

38;34

days

min.

days

min.

days

min.

days

0;55

5

0;50

4

0;46

3

0;43

5;22 5;27

27 27

4;55 5; 0

23 24

4;32 4;37

19 20

4;13 4;17

10 … 59 60

* In the ms, gradus velocitatis appears above this column but it refers to the headings of the other columns, labeled: 11, 12, 13, 14.

and the velocity in elongation in degrees per minute of a day (again, only 4 values for the velocity in elongation are given: 11, 12, 13, and 14). Between these four columns, one finds the differences, in minutes of a day, between successive entries in the same row, to facilitate interpolation. Some selected rows of this subtable are displayed in Table 7(2). The entries in this subtable were computed by means of the formula (expressed in modern notation) Δt = –η / [vm(t) – vs(t)] where Δt is the time interval between mean and true syzygy, η is the true elongation, and the velocity in elongation, vm(t) – vs(t), is the difference between the daily velocities of the Moon and the Sun at the time of mean syzygy. This approach to the problem of finding true syzygy was followed by a number of medieval astronomers and differs from that presented by Ptolemy in Almagest vi.4 (Chabás and Goldstein 1997, pp. 93–96; cf. Kremer 2003, pp. 305–329). Madrid, Biblioteca Nacional, ms 4238, f. 67r, reproduces both subtables except that the last row of Table 7(2) corresponds to the argument of 9 min.

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early alfonsine astronomy in paris table 8

Correction of the lunar position for each day between syzygies ( ff. 3v–4r)

s

(°)

Day 1 motus c. min s (°)

0 0 1 … 2 … 2 … 5 6 6 … 7 … 8 … 9 … 11 11 12

12 24 6

0 0 0

10;57 10; 7 9;24

11;53 12; 1 12;11

0 24

0

8;10

13;18

18 0 12

0 0 0

13;37 14;45 15;47

14;50 14;45 14;35

18 12

0

18;11

Day 2 motus c. min s (°) 0 0 0

22;50 22; 8 21;35

11;55 12; 6 12;19

0

21; 4

12;53

0 0 1

28;27 29;30 0;22

14;52 14;41 14;26

1

1;38

12;52

0 0 0

13;46 12;48 11;51

Day 14 motus c. min s (°) 6 6 6

5;37 6;41 7;38

14;37 14;26 14;10

6

9;27

13;56

6 6 6

5;25 4;27 3;30

11;49 11;51 11;55

5

29;30

13;28

6 6 6

2;17 3;27 4;29

14;44 14;46 14;45

13; 7

6 6 18 0

…

11;49 11;48 11;49

0 0 0

25;35 24;36 23;40

11;41 11;42 11;47

ff. 3v–4r Table 8: Correction of the Lunar Position for Each Day between Syzygies This double-argument table displays two columns for each day, from day 1 to day 14. The days in the horizontal argument refer to the time from conjunction to opposition. The vertical argument is given at intervals of 12°, from 0s 12° to 12s 0°. The heading calls it elongatio lune ab auge epicicli and it represents the mean lunar anomaly at mean syzygy. For each day, the first column gives the increment in lunar longitude, here called motus completus, in signs and degrees, to be added to the mean lunar longitude at the preceding mean syzygy, whereas the second column displays one sixtieth of the differences between

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successive entries in the same row, here called motus ad minutum diei, and given in arc-minutes. The entries in the second column thus represent the true lunar velocity in a minute of a day for that particular day. As mentioned above, John of Spira composed canons to some of Vimond’s tables. In particular, the canon in Vatican, Biblioteca Apostolica, ms Ottob. lat. 1826, ff. 152v–153r, describes the use of Table 8, here entitled Tabula veri loci lune ad dies datos post mediam coniunccionem vel opposicionem solis et lune. The canon ends with an explicit reference to John Vimond working in Paris: Explicit canon tabule sequentis que est una tabularum quas composuit Magister Johannes Vimondi. Iste autem canon est undecimus canonum quos composuit magister Johannes de Spira supra tabulas predicti magistri Johannis Parisius. On ff. 153v–155v we find a copy of Table 8, but in this case the entries in the second column (the true lunar velocity in a minute of a day) are given to one sexagesimal place. We know of only a few similar tables for the same purpose, but the entries in them differ from those given by Vimond. Erfurt, ms ca 2° 388, is a 15th-century manuscript which, according to Poulle (1973), contains one of the rare copies of John of Lignères Tabule magne. On ff. 30r–32v, there is an expanded version of Table 8, with the same structure and the same columns. In this case, the horizontal argument runs from day 1 to day 15 and the column for velocity gives entries in minutes and seconds per hour which result from the entries in Vimond’s Table 8 by multiplying them by 2;30 (= 60/24) for conversion from arc to time. Another example is furnished by Levi ben Gerson (d. 1344) who compiled a double-argument table, based on his own model, for finding the lunar position between syzygies as a function of the number of days since syzygy from 1 to 14 and the mean lunar anomaly at 10°-intervals from 0° to 350° (Goldstein 1974, pp. 148–149, 246–254). Yet another such table is found in an anonymous zij in Hebrew for year 1400: this double-argument table shares the same structure, but the anomaly is given at intervals of the daily increment in mean lunar anomaly from day 0 to day 27 (cf. Goldstein 2003, p. 166). The zij of Judah ben Verga (ca. 1470) also includes a table with the same structure (Goldstein 2001, pp. 247, 269–270).

early alfonsine astronomy in paris

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f. 4r Radices for the Planets In a small table, the text gives the following values for the radices of the planets: Mercury 11s 16; 6,10° Venus 3s 3;46,55° Mars 3s 15; 9,42° Jupiter 6s 4; 8, 5° Saturn 9s 14; 0,46° When recomputed for the instant of the mean conjunction on March 10, 1320, these radices that depend on the mean longitudes or mean arguments of anomaly (henceforth, simply “anomaly”) confirm the use of this date as epoch. In the case of the superior planets the radix can be represented by the following formula: Rx(planet) = λ̄ 0 – a(Sun) where λ̄ 0 is the mean longitude of the planet at epoch, and a(Sun) is the apogee of the Sun at that time. According to the Parisian version of the Alfonsine Tables, the mean motions for the superior planets on that day, in Toledo at 9;10 a.m. (= 9;58 a.m. in Paris), counting from midnight, are: Saturn 13;57, 1° Jupiter 274; 4,20° Mars 195; 5,57° If we subtract the value of the solar apogee for this epoch (89;56,15°) given by Vimond (f. 2r), we obtain: Saturn 284; 0,46° Jupiter 184; 8, 5° Mars 105; 9,42° in perfect agreement with the radices given in the text. Note that using the standard Alfonsine value for the solar apogee at that time (89;23,50°) yields no agreement, confirming the author’s preference for his value, 89;56,15°. The reason for subtracting the solar apogee is that for Vimond the planetary apogees partake in the motion of the solar apogee.

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For Venus and Mercury, Vimond’s radices can be obtained by adding the planet’s anomaly and the solar longitude and subtracting from the sum the value for the solar apogee at epoch. For Venus we compute according to the Parisian Alfonsine Tables at Vimond’s epoch: Rx(Venus) = t · v + ᾱ0(Venus) + λ̄ 0(Sun) – a(Sun) = 24754;52,55d · 0;36,59,27,23,59,31°/d + 45;45,55,19° + 356;10,50° – 89;56,15° = 93;46,58° where t, the time from epoch Alfonso to epoch Vimond, is 24754;52,55d; v, the mean motion in anomaly for Venus, is 0;36,59,27,23,59,31°/d; ᾱ0(Venus), the radix for Venus’s mean anomaly at Alfonso’s time, is 45;45,55,19°; λ̄ 0(Sun), the mean longitude of the Sun at Vimond’s epoch, is 356;10,50°; and a(Sun), the solar apogee at Vimond’s epoch, is 89;56,15°. This result, 93;46,58°, differs from the radix in Vimond’s text by only 0;0,3°. For Mercury, we compute according to the Parisian Alfonsine Tables at Vimond’s epoch, as for Venus, where λ̄ 0(Sun) – a(Sun) = 266;14,35°: Rx(Mercury) = 24754;52,55d · 3;6,24,7,42,40,52°/d + 213;48,38,56° + 266; 14,35° = 346;6,15° whereas Vimond’s text has 11s 16;6,10° (= 346;6,10°), in excellent agreement with our recomputation. A short text below these radices tells us that we should add two quantities, the radix for the planet and the solar apogee. For Vimond the solar apogee and each of the planetary apogees share the same motion; hence the difference between them is always the same. In particular, since Venus’s apogee is always the same as that of the Sun, nothing is given for Venus. The text then displays values for each planet of the distance of its apogee from the solar apogee: Saturn Jupiter Mars Venus Mercury

5s 12° = 162° 2s 22° = 82° 1s 14° = 44° – 3s 29° = 119°

These values agree closely with those of Ibn Isḥāq (early 13th century) [Mestres 1996, p. 395]. They are used as shifts in subsequent tables for the planets, and can be derived from the radices used in the Parisian Alfonsine Tables by

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early alfonsine astronomy in paris Yearly radices ( f. 4r –v)

table 9

s Mercury Venus Mars Jupiter Saturn

Year 1 (°)

s

Year 2 (°)

4 11; 1,24 4 21;11,55 8 15; 2,47 3 12;46,53 6 1;10, 7 0 26;51,41 1 1;53,32 2 1;19,53 0 12;50,22 0 24;41,28

… s

Year 8 (°)

…

2 23;56,47 0 3;48,53 3 1;58,32 8 2;52,21 3 7;46,36

subtracting the solar apogee for the time of Alfonso from the radix of the apogee for each planet (see, e.g., the editio princeps of the Alfonsine Tables printed by Ratdolt (1483), c8–d1; note that the signs used there are physical signs of 60°): Saturn Jupiter Mars Venus Mercury

4, 2;35,20,41° – 1,20;37,0° = 161;58,20° 2,42;48,38,41° – 1,20;37,0° = 82;11,38° 2, 4;23,51,41° – 1,20;37,0° = 43;46,51° 1,20;37, 0° – 1,20;37,0° = 0° 3,19;51,11,41° – 1,20;37,0° = 119;14,11°

These results, when rounded to the nearest degree, are in perfect agreement with Vimond’s data. Therefore, the conclusion is that Vimond started with the same planetary apogees as those in the Parisian Alfonsine Tables.

f. 4r–v Table 9: Yearly Radices Table 9 displays selected entries. This table displays the radices for the five planets for 1, 2, 3, and 4 years; for multiples of 4 years up to 76 years; and then for 152, 304, 608, 1216, and 2432 years, as in Table 3. As was the case for the radices for the Sun and the Moon, 1 year is equivalent to 13 mean lunations, 2 years is equivalent to 25 mean lunations, …, 8 years is equivalent to 99 mean lunations, etc. The mean daily motion in longitude resulting from the entries for year 8 (computed in the same way that was used for finding the mean motions in Table 3) are shown below under the heading “Vimond”. If we add the daily motion of the apogees (0;0,0,11,13,35°/d), as we did in the case of the Sun, we

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obtain the entries displayed in the second column, in good agreement with the values for the mean motions in longitude in the Parisian Alfonsine Tables (see Ratdolt 1483).

Saturn Jupiter Mars Venus Merc.

Vimond

Including the motion of the apogee

Parisian Alf. t.

0; 2, 0,24, 3,56°/d 0; 4,59, 4, 1,19 0;31,26,27,26,34 1;36, 7,35,47,21 4; 5,32,16, 5,55

0; 2, 0,35,17,31°/d 0; 4,59,15,14,54 0;31,26,38,40, 9 1;36, 7,47, 0,56 4; 5,32,27,19,30

0; 2, 0,35,17,40°/d 0; 4,59,15,27, 7 0;31,26,38,40, 5 1;36, 7,47, 1,19 4; 5,32,27,20, 0

It is most unusual for the mean motions of Venus and Mercury to be the sum of their mean motions in anomaly and the solar mean motion, but there can be no doubt that this is what Vimond did, as is confirmed by the note on f. 4vb. In fact, we know of no other medieval astronomer writing in Latin who presented the mean motions of the inferior planets in this way. For purposes of comparison, the entries for Venus and Mercury under “Parisian Alfonsine Tables” are the sum of the mean motions in anomaly and the solar mean motion: for Venus 0;36,59,27,24,0°/d and 0;59,8,19,37,19°/d, and for Mercury 3;6,24,7,42,41°/d and 0;59,8,19,37,19°/d. Note that in Ptolemy’s models the solar mean motion is also the mean argument of center for Venus and Mercury.

ff. 4v–5r Table 10: Monthly Radices Table 10 displays selected entries. This table displays the radices for the five planets for 25 consecutive syzygies. The entries in this table are based on the same motions as those embedded in the previous table. As was the case for the monthly radices in Table 4, for each planet the entries for Syzygy 1 and Syzygy 25 add up to the entry corresponding to Year 1 in the previous table (except for 1″ for Mercury, Mars, and Jupiter). For Venus and Mercury the mean motions extracted from Table 9 give exact agreement, confirming the interpretation given above. Thus, in the cases of Venus and Mercury one has obtained the sum of the solar anomaly and their mean anomalies, respectively, at any syzygy (see Figure 8.1). This quantity is not the argument in the table of equations (see Tables 12 and 15, below), and it is

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early alfonsine astronomy in paris table 10

Monthly radices ( ff. 4v–5r)

s Mercury Venus Mars Jupiter Saturn

Syzygy 1 (°)

s

Syzygy 2 (°)

2 0;25,26 4 0;50,53 0 23;39,20 1 17;18,41 0 7;44,14 0 15;28,28 0 1;13,36 0 2;27,12 0 0;29,38 0 0;59,16

… s 2 7 6 1 0

Syzygy 25 (°) 10;35,57 21;23,27 13;25,52 0;29,57* 12;20,44

* Sic, instead of 1s 0;39,57°.

figure 8.1 A geometric interpretation of Vimond’s tables for the mean motion for Venus

not clear that there is any advantage to this method as against computing the mean anomaly directly.

Tables 11 (Mercury, f. 5r), 14 (Venus, f. 5v), 17 (Mars, f. 6r), 20 (Jupiter, f. 6v), and 23 (Saturn, f. 7r): Equation of Center and First Station Tables 11 and 12 are to be used together to compute the true longitude of a planet from its mean longitude. In most zijes in the Ptolemaic tradition, there is only one such table for each planet, but Vimond has separated those functions that depend on the mean argument of center from those that depend on the mean anomaly and put them in different tables. A similar idea is already found in the zij of Ibn Isḥāq, described in Mestres 1996. Ibn Isḥāq’s parameters for the maximum equations of center for Mars and Mercury are those of al-Battānī,

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but for Saturn, Jupiter, and Venus they are not; rather, they are 5;48° for Saturn, 5;41° for Jupiter, and 1;51° for Venus. “The tables for planetary equations (…) are divided into two groups: the first group contains the tables for the equation of centre and the interpolation function. (…) The second group (two tables for each planet) contains the tables for the equations of anomaly at apogee and perigee and for the middle position” (Mestres 1999, p. 234). So, the arrangement of Vimond’s tables bears a similarity to an Andalusian/Maghribi tradition that is not otherwise attested in Latin. However, it is not uncommon to find later sets of tables associated with the Parisian Alfonsine Tables where the planetary equations are split into two tables for each planet: see, e.g., Erfurt, ms ca q 362, ff. 28r–36r, where the entries are displayed at intervals of 1° and the radices are given for Paris (1320) as well as for London and Brugge (1366). Besides offering two tables for the equations of each planet, Vimond’s tables give additional information arranged in a presentation which is certainly peculiar, as explained below. table 11

Equation of center and first station of Mercury ( f. 5r)

s

(1) (°)

s

(2) (°)

(3) min

(4) min

(5) min

s

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

6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18

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

8;51 14;57 21; 0 27; 2 3; 0 8;57 14;51 20;44 26;34 2;23 8;12 13;59 19;45 25;30 1;15 7; 0 12;45 18;30

61; 0 60;30 60;20 59;40 59;30 59; 0 58;50 58;20 58;10 58;10 57;50 57;40 57;30 57;30 57;30 57;30 57;30 57;30

60 60 59 59 59 58 58 57 57 57 57 57 57 57 57 57 57 57

60 59 58 57 54 51 48 44 40 35 29 24 19 14 10 6 4 2

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

(6) (°) 24;30 24;30 24;32 24;35 24;38 24;44 24;50 24;56 25; 7 25;20 25;37 25;53 26; 9 26;24 26;38 26;50 27; 2 27; 9

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early alfonsine astronomy in paris

(1)

(2)

s

(°)

s

(°)

(3) min

3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 10

24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24

3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 10 10 10 10 10

24;15 29;59 5;40 11;25 17;10 22;55 28;40 4;25 10:10 15:56 21;44 27;33 3;22 9;13 15; 7 21; 2 26;59 2;58 9; 1 15; 5 21;11 27;21 3;32 9;44 15;57 21;11 28;26 4;44 11; 2 17;21 23;44 0; 3 6;22 12;41 19; 0 25;18

57; 0 57;10 57;30 57;30 57;30 57;30 57;30 57;30 57;40 58; 0 58;10 58;10 58;30 59; 0 59;10 59;30 59;50 60;30 60;40 61; 0 61;40 61;50 62; 0 62;10 62;20 62;30 62;40 63; 0 63;10 63;10 63;20 63;10 63;10 63;10 63; 0 62;50

(4) min

(5) min

s

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

1 0 1 3 5 7 11 16 21 26 31 37 41 45 49 52 55 57 59 60 60 60 59 59 58 57 56 55 54 54 53 53 53 54 54 55

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 4 4 4 4 4 4

(6) (°) 27;13 27;14 27;12 27; 7 26;59 26;46 26;34 26;19 26; 4 25;48 25;30 25;16 25; 3 24;54 24;48 24;42 24;37 24;34 24;31 24;30 24;29 24;29 24;29 24;30 24;32 24;34 24;36 24;38 24;39 24;40 24;41 24;42 24;41 24;40 24;39 24;37

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table 11

Equation of center and first station of Mercury ( f. 5r) (cont.)

s

(1) (°)

s

(2) (°)

(3) min

(4) min

(5) min

s

11 11 11 11 11 12

0 6 12 18 24 0

11 11 11 11 11 12

1;35 7;51 14; 7 20;20 26;32 2;43

62;40 62;40 62;10 62; 0 61;50 61;20

62 62 61 61 61 60

56 57 58 59 60 60

4 4 4 4 4 4

table 14

(6) (°) 24;35 24;33 24;31 24;30 24;29 24;29

Equation of center and first station of Venus ( f. 5v)

(1)

(2)

s

(°)

s

(°)

(3) min

(4) min

(5) min

s

(6) (°)

0 0 … 2 3 3 3 … 8 8 9 9 … 11 12

6 12

0 0

5;47 11;34

57;50 57;50

57 57

0 0

5 5

15;52 15;54

24 0 6 12

2 2 3 3

21;51 27;50 4;50 * 9;52

59;50 60; 0 60;20 60;30

59 59 59 60

27 31 33 36

5 5 5 5

17; 2 17;11 17;17 17;23

18 24 0 6

8 8 9 9

20; 8 26;10 2;10 8; 9

60;20 60; 0 59;50 59;30

59 59 59 59

36 33 31 27

5 5 5 5

17;23 17;17 17;11 17; 2

24 0

11 12

24;13 0; 0

57;50 57;50

57 57

0 0

5 5

15;52 15;50

* Sic, instead of 3s 3;50°.

The table for the equation of center of each of the five planets has six columns. Column 1 gives the argument (argumentum) at 6°-intervals in signs and degrees from 0s 6° to 12s 0°. Column 2 displays the entry in col. 1 corrected for the equation of center (motus completus), in signs, degrees, and minutes. The author

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early alfonsine astronomy in paris table 17

Equation of center and first station of Mars ( f. 6r)

s

(1) (°)

s

(2) (°)

(3) min

(4) min

(5) min

s

(6) (°)

0 0 … 1 1 … 4 … 7 7 … 10 … 11 12

6 12

0 0

12;31 17;36

50;50 50; 0

26 26

6 4

5 5

8;41 8;21

12 18

1 1

12;22 17;16

49; 0 49;10

26 25

0 0

5 5

7;29 7;31

18

4

6;36

60;30

31

32

5

13;46

12 18

7 7

11;33 18;54

73;30 73;10

38 38

60 59

5 5

19;14 19;13

12

10

23;24

58;50

30

31

5

13;36

24 0

12 12

2;13 7;24

51;50 51;10

27 26

10 8

5 5

9;31 9; 6

follows here the same pattern as that for the true solar anomaly (see Table 5). Column 3 (motus gradus) gives the increment of the true argument per degree of the argument, in minutes and seconds. Most entries in this column are generated by dividing by 6 the differences between two successive entries in col. 2 and thus were probably intended for interpolation in col. 2. Column 4 (motus diei) displays the velocity in minutes of arc per day, and the range of values for each planet is the same as in the column labeled motus centri or motus puncti (that only depends on the argument of center) in the table for planetary velocities associated with the Toledan Tables and the Castilian Alfonsine Tables (Chabás and Goldstein 2003a, pp. 170–182); for the other component of the planetary velocity, see Tables 12, 15, 18, 21, and 23, col. 4, below. So, the entries in this column are only one component of the planet’s velocity. Column 5 is intended to provide minutes of interpolation and is headed diametri (perhaps to distinguish these “linear” minutes from minutes of an hour, minutes of a day, and minutes of a degree). Finally, column 6 lists the first station in signs, degrees, and minutes. But for a shift of the entries, the equations of center for Mercury, Mars, and Saturn that can be derived from cols. 1 and 2 are basically the same (with

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table 20

Equation of center and first station of Jupiter ( f. 6v)

s

(1) (°)

s

(2) (°)

(3) min

(4) min

(5) min

s

(6) (°)

0 0 … 2 2 2 3 … 5 … 8 8 8 9 … 11 11 12

6 12

0 0

11;43 17;31

58; 0 57;20

5 5

23 20

4 4

5;19 5; 9

12 18 24 0

2 2 2 2

12;59 18;23 23;48 29;12

44;10 44; 0 44; 0 44;10

4 4 4 4

0 0 0 0

4 4 4 4

4; 4; 4; 4;

24

5

18; 3

60;10

4

32

4

5;44

12 18 24 0

8 8 8 9

10;55 17;33 14;14 0;52

66;20 66;30 66;20 66;20

6 6 6 6

59 60 60 60

4 4 4 4

7;10 7;11 7;11 7;10

18 24 0

11 11 12

23;57 29;56 0;51 *

59;50 59;10 58;40

5 5 5

32 29 26

4 4 4

5;47 5;39 5;29

6 5 5 6

* Sic, instead of 5;51.

minor variants) as in the zij of al-Battānī (Nallino 1903–1907, 2:110–137) and the Toledan Tables (Toomer 1968, pp. 60–68; F.S. Pedersen 2002, pp. 1259– 1308). The maximum value for Mercury (3;2°) occurs at about 0s 24° and 7s 6°, that of Mars (11;24°) at 4s 18° and 10s 12°, and that of Saturn (6;31°) at 2s 12° and 8s 12°. However, for the other two planets the entries differ systematically from those in the above-mentioned zijes: for Venus the maximum value is 2;10° at 3s 0° and 3s 6°, and 8s 24° and 9s 0°; and for Jupiter the maximum value is 5;57° at 5s 24° and 11s 18°. The entries for Mercury, Mars, Jupiter, and Saturn are shifted by about 119°, 44°, 82°, and 162°, respectively, in relation to those in the zij of al-Battānī and the Toledan Tables. No such shift appears in the table for Venus. As mentioned above, these shifts result from the difference between the apogee of each of the planets and that of the Sun. Because of these shifts, for the superior planets one enters these tables in col. 1 directly

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early alfonsine astronomy in paris table 23

Equation of center and first station of Saturn ( f. 7r)

s

(1) (°)

s

(2) (°)

(3) min

(4) min

(5) min

s

(6) (°)

0 0 … 2 … 5 5 5 … 8 … 11 11 11 12

6 12

0 0

8;46 15;24

66;20 66; 0

2 2

57 55

3 3

25;22 25;19

12

2

18;31

59;20

2

31

3

24;11

6 12 18

5 5 5

6;39 12; 0 17;21

53;30 53;30 53;40

2 2 2

0 0 0

3 3 3

22;45 22;44 22;45

12

8

5;29

60;10

2

31

3

24;11

6 12 24 0

11 11 11 12

12; 0 18;43 25;25 2; 7

67;10 67;10 67; 0 66;30

2 2 2 2

60 60 59 58

3 3 3 3

25;30 25;28 25;27 25;25

with their mean motions for a given syzygy (the radix plus the motion in years and semi-lunations); for Venus and Mercury one enters with the solar anomaly for a given syzygy. Clearly, Vimond intended to make this table more “user-friendly” than the standard version of the table for the equation of center. Vimond has a double motion of the solar apogee: precession and proper motion. The planetary apogees are fixed with respect to the solar apogee (i.e., they are subject to both precession and the proper motion of the solar apogee). If we add the solar apogee (about 90°) to the values for the shifts listed above, we find that the planetary apogees are 209° for Mercury, 90° for Venus, 135° for Mars, 172° for Jupiter, and 252° for Saturn. In the Toledan Tables, the apogees of the Sun and of Venus are both 77;50° (Toomer 1968, p. 45), that is, about 12° less than 90°. Adding this difference to the planetary apogees in the Toledan Tables rounded to degrees, we find the following:

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Apogees v (from the shifts) v (from the radices) tt + 12° Mercury Venus Mars Jupiter Saturn

table 11a

209° 90° 135° 172° 252°

209° 90° 134° 172° 252°

210° 90° 134° 176° 252°

Maximum values for the equation of center

al-Battānī Toledan t. Vimond Parisian Alf. t. Mercury Venus Mars Jupiter Saturn

3; 2° 1;59° 11;24° 5;15° 6;31°

3; 2° 1;59° 11;24° 5;15° 6;31°

3; 2° 2;10° 11;24° 5;57° 6;31°

3; 2° 2;10° 11;24° 5;57° 6;31°

The agreement of Vimond’s data with the apogees in the Toledan Tables shows that Vimond has included the motion of the solar apogee in the motions of the planetary apogees, thus following a theory for which there was no previous evidence outside al-Andalus and the Maghrib (Samsó and Millás 1998, pp. 268–270). We know of no other set of planetary equation tables arranged in this way. See also Table 27 (equation of access and recess), below, for yet another shift in Vimond’s tables. The maximum values for the equation of center in Vimond’s planetary tables are the same as in the editio princeps of the Alfonsine Tables (see Table 11a). Despite their agreement for the values of the maximum equations, the structure of Vimond’s tables is very different from that of the Parisian Alfonsine Tables and would seem to be independent of it. Moreover, it is significant that the maximum equation of center for Jupiter in both cases is 5;57°, for this value is not known in any text prior to the Parisian Alfonsine Tables, indicating a strong connection between the tables of Vimond and the work of his Parisian contemporaries. The origin or derivation of this parameter for Jupiter is not described in any extant text, and it is likely that this value was simply taken

early alfonsine astronomy in paris

261

from an earlier work: the most reasonable candidate is the Alfonsine Tables as they existed in Castile. For all planets, except Mercury, an entry, c, in column 5 can be computed, but for shifts, from the modern formula c = 60 (1 – cos κ̄)/2, where κ̄ is the mean argument of center. The same approach is found in Levi’s lunar theory (Goldstein 1974, table 35, col. ii: see p. 54). The entries for Mercury in col. 5 do not follow the same pattern as that for the rest of the planets. The entries can be recomputed, approximately, according to the following formula: [1]

c5(κ̄) = [d – r(κ̄)] / [d – d]

where d is the maximum distance of the center of the epicycle from the observer, d is the minimum distance, and r(κ̄) is the distance as a function of the mean argument of center, κ̄. A similar formula for interpolation was already used by Ḥabash in the 9th century (as-Saleh 1970, pp. 137–138). In Ptolemy’s model for Mercury d is 69 for argument 0°, d is 55;34 for an argument close to 120°, and r(180°) is 57 (O. Pedersen 1974, pp. 313–324). Hence, formula [1] can be replaced by [2] c5(κ̄) = [69 – r(κ̄)] / 13;26. In general, the computation of r(κ̄) is a difficult and lengthy procedure, and it is likely that Vimond (or his source) used approximations (if this, indeed, was the formula he had in mind). We computed the distances from the observer to the center of Mercury’s epicycle according to formulas in modern terms given by O. Pedersen (1974, p. 320, equations 10.34 and 10.35), and then used them in equation [2], above. A comparison of our results for c5(κ̄) with the entries in Vimond’s table is displayed in Table 11b. Col. ii has the values for c5(κ̄) that depend on the distances computed according to the formulas given by O. Pedersen and equ. [2], above; col. iii has the arguments in Vimond’s table (with the shift); and col. iv has the entries in Vimond’s Table 11, col. 5. Although the agreement is not exact between col. ii and col. iv, the trend is clear. Vimond’s value for 180° (col. iii), 37, has the poorest agreement, but this entry should probably be corrected to 36, judging from the surrounding values.

262

figure 8.2 Vimond’s equation of center, col. 5, for Venus, Mars, Jupiter, and Saturn

figure 8.3 Vimond’s equation of center, col. 5, for Mercury

chapter 8

early alfonsine astronomy in paris table 11b

263

A comparison of column 5 for Mercury with recomputation

i κ̄

ii c5(κ̄ )

0 30 54 60 66 90 120 150 180

0; 0 10;50 29;24 34;15 38;53 53;33 60; 0 56;38 53;34

iii iv κ̄ (Vimond) c5(κ̄ ): Vimond 120 150 174 180 186 210 240 270 300

0 11 31 37 41 55 60 56 53

It may be of interest that in Copernicus’s table for the equations of Mercury (Copernicus 1543, ff. 177v–178r), his col. 4 (for interpolation) shows the same trend as Vimond’s col. 5. We are convinced that column 5 in Vimond’s tables for the equation of center is intended to be used for interpolation with column 5 in the tables for the equation of anomaly, and this is analogous to Copernicus’s use of his col. 4 (see below). Indeed, Vimond’s col. 5 serves much the same purpose as col. 8 in Ptolemy’s tables for the planetary equations (Almagest, xi.11) but, since the definitions for the columns that yield the equation of anomaly are different, so is the function for interpolation. Moreover, in contrast to the geometric methods in the Almagest used for computing the coefficients of interpolation for each of the four planets (Venus, Mars, Jupiter and Saturn), Vimond has approximated the results that would be derived from the geometry of the models by introducing a single trigonometric function in those cases. In Almagest, xi.11 (Toomer 1984, pp. 549–553), col. 8 in the planetary equation tables is intended for interpolation as a function of κ̄, the mean argument of center, and the entries are given to minutes and seconds (for Ptolemy’s method of computation and a graph of the entries in his col. 8, see Neugebauer 1975, pp. 184–186, 1267). A similar set of values, given only to minutes, is found in al-Battānī’s zij in the tables for the planetary equations, col. iv (Nallino 1903– 1907, 2:110–137), and in corresponding tables in the Parisian Alfonsine Tables, col. 3 (Ratdolt 1483, e7r–g5v). As for the entries for the first station of each planet, they are essentially the same as in previous tables of the same kind (Almagest, Handy Tables,

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al-Khwārizmī, al-Battānī, and the Toledan Tables) with the same shifts that we noted above.

Tables 12 (Mercury, f. 5r), 15 (Venus, f. 5v), 18 (Mars, f. 6r–v), 21 (Jupiter, f. 7r), and 24 (Saturn, f. 7v): Equation of Anomaly The tables for the equation of anomaly for each of the five planets have seven columns. Table 12 displays a selection of values for the equation of anomaly for Mercury. Column 1 gives the mean argument of anomaly (argumentum) at 6°-intervals (at 3°-intervals for Mars and Venus) from 0s 6° to 6s 0° and its complement in 360° from 6s 0° to 11s 24°. Column 2 displays the correction due to the argument of anomaly at maximum distance (motus completus) in degrees and minutes and represents the difference between the equation of anomaly and the correction for maximum distance (cf. Almagest, xi.11, columns 6 and 5; and Neugebauer 1975, pp. 183–184). The only other text of which we are aware that treats the equation of anomaly in this way is the zij of Ibn al-Bannāʾ (d. 1321) where this presentation is applied in his tables for Saturn and Jupiter but not in those for the other planets (see Samsó and Millás 1998, pp. 278–285). The extremal values in col. 2 that appear in the text are shown below; they are followed by the corresponding entries for col. vi and col. v in the zij of al-Battānī (Nallino 1903–1907, 2:109–137):

Mercury Venus Mars Jupiter Saturn

Vimond

al-Battānī

19; 1° 44;49° 36;44° 10;34° 5;53°

(= 21;59° – 2;58°) at 3s 18° (= 45;59° – 1;10°) at 4s 15° (= 40;58° – 4;14°) at 4s 6° (= 11; 3° – 0;29°) at 3s 12° (= 6;12° – 0;19°) at 3s 0° and (= 6;13° – 0;20°) at 3s 6°

These corrections agree with those that follow from the Almagest as well as the zij of al-Battānī, the Toledan Tables, and the editio princeps of the Alfonsine Tables (with minor variants: 40;59° rather than 40;58° for Mars; 6;12° and 0;19° correspond to 3s 0° rather than 3s 1° for Saturn), and this means that Ptolemy’s eccentricities underlie them even though, in the case of Venus and Jupiter,

265

early alfonsine astronomy in paris table 12

Equation of anomaly for Mercury ( f. 5r)

s

(°)

(1) s

(°)

(2) (°)

(3) min

(4) min

(5) min

(6) sec

(7) sec

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

6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0

11 11 11 11 11 10 10 10 10 10 9 9 9 9 9 9 8 8 8 8 7 7 7 7 7 6 6 6 6 6

24 18 12 6 0 24 18 12 6 0 24 18 12 6 0 24 18 12 6 0 24 18 12 6 0 24 18 12 6 0

1;28 2;56 4;24 5;50 7;15 8;37 9;58 11;15 12;30 13;39 14;44 15;44 16;38 17;25 18; 4 18;34 18;53 19; 1 18;56 18;39 18; 4 17;12 16; 4 14;36 12;49 10;42 8;18 5;42 2;53 0; 0

15 15 14 14 14 13 13 12 11 10 10 9 8 7 5 3 1 1 3 6 9 11 15 18 21 24 26 28 29 29

45 45 44 44 42 42 40 39 36 34 31 28 25 19 15 9 3 3 9 18 27 35 45 55 66 74 81 87 89 89

0;18 0;33 0;48 1; 3 1;18 1;33 1;48 2; 0 2;18 2;35 2;53 3;10 4;14 4;30 4;45 4;57 5; 5 5;10 5;13 5; 6 4;55 4;29 4;55 4;12 4;29 3;55 3;12 2;15 1;10 0; 0

2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 3 2 1 1 0 2 4 6 7 9 11 12 12

8 8 8 8 8 8 8 8 9 9 9 8 8 8 8 8 8 6 4 3 1 1 6 13 18 22 29 34 36 36

the eccentricities were modified for computing the equation of center (cf. North 1976, 3:196). Similarly, in the tables of Ibn al-Bannāʾ the eccentricities underlying the equations of anomaly are taken from the Almagest, but his

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maximum equations of center for Venus and Jupiter are not those of either Ptolemy or of Vimond (Samsó and Millás 1998, p. 276). Column 3 (motus gradus) gives the increment of the motus completus in col. 2 per degree of the argument in minutes: in most cases the entry results from taking the difference between successive entries in col. 2 and dividing that difference by 6 (or by 3 for Mars and Venus); the purpose of this column is facilitate interpolation. Column 4 (motus diei) displays the velocity in minutes of arc per day, and the range of values for each planet is the same as in the column labeled motus argumenti (that only depends on the argument of anomaly) in the table for planetary velocities; see the comments to the Castilian Alfonsine Tables, chapter 27 (Chabás and Goldstein 2003a, pp. 170–182). So, an entry in this column is the second component of the planet’s velocity and it complements the first component already displayed in Tables 11, 14, 17, 20, and 23, above. The entries in column 5 (minutum diametri), in minutes and seconds, actually represent degrees and minutes, and result from adding the correction for maximum distance to the correction for minimum distance (columns c5 and c7 in Almagest, xi.11). For the extremal values in col. 5 in the text see below; they are followed by the corresponding entries for col. v and col. vii in the zij of al-Battānī (Nallino 1903–1907, 2:109–137):

Mercury Venus Mars Jupiter Saturn

Vimond

al-Battānī

5;13° 3;34° 13;37° 1; 3° 0;46°

4;56° * (= 1;42° + 1;52°) at 5s 12° (= 5;34° + 8; 3°) at 5s 9° (= 0;30° + 0;33°) at 3s 24° (= 0;21° + 0;25°) at 3s 12°

* In al-Battānī’s zij for 3s 24° we find 3;4° + 1;52° = 4;56°, whereas for 3s 24° in Vimond’s table we find 3;12° + 2;1° = 5;13°, al-Battānī’s maximum which occurs at 4s 10°–4s 12°.

In the absence of instructions by Vimond it is not easy to decide how the correction to the planet’s mean longitude is to be computed, but it seems likely that one component of this correction is to be computed by adding an entry in col. 2 to an interpolation factor times an entry in column 5, as is the case with the tables of Ibn al-Bannāʾ for Saturn and Jupiter. The most likely candidate for this interpolation factor is col. 5 in Table 11, for it depends on the argument of

267

early alfonsine astronomy in paris table 15

Equation of anomaly for Venus ( f. 5v)

(The entries from 5s 18° to 6s 12° are given at 2°-intervals, rather than at 3°-intervals as in the rest of the table.) (1) s

(°)

s

(°)

(2) (°)

(3) min

(4) min

(5) min

(6) sec

(7) sec

0 0 … 4 4 4 … 5 5 5 … 5 6

3 6

11 11

27 24

1;15 2;30

25 25

15 15

0; 2 0; 3

0 1

0 0

12 15 18

7 7 7

18 15 12

44;44 44;49 44;44

2 2 6

1 1 4

2;18 2;25 2;32

2 2 3

1 1 2

12 15 18

6 6 6

18 15 12

33;25 29;43 25;25

74 89 104

46 55 64

3;34 3;27 3;14

2 4 7

1 2 4

28 0

6 6

2 0

4;48 0; 0

144 144

89 89

0;45 0; 0

22 22

14 14

Equation of anomaly for Mars ( f. 6r –v)

table 18

(The entries from 5s 18° to 6s 12° are given at 2°-intervals, rather than at 3°-intervals as in the rest of the table.) (1) s

(°)

s

(°)

(2) (°)

(3) min

(4) min

(5) min

(6) sec

(7) sec

0 0 … 4 4 4 … 5 5 5 …

3 6

11 11

27 24

1; 8 2;16

23 23

11 11

0; 8 0;17

3 3

1 1

3 6 9

7 7 7

27 24 21

36;40 36;44 36;43

1 0 3

1 0 1

8;53 9;19 9;46

9 9 9

4 4 4

6 9 12

6 6 6

24 21 18

28;15 25;56 23;17

46 53 62

21 25 29

13;30 13;37 13;19

0 6 13

0 2 6

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table 18

Equation of anomaly for Mars ( f. 6r –v) (cont.)

s

(°)

(1) s

(°)

(2) (°)

(3) min

(4) min

(5) min

(6) sec

(7) sec

5 6

28 0

6 6

2 0

3; 1 0; 0

90 90

42 42

2;29 0; 0

74 74

35 35

table 21

Equation of anomaly for Jupiter ( f. 7r)

s

(°)

(1) s

(°)

(2) (°)

(3) min

(4) min

(5) min

(6) sec

(7) sec

0 0 … 3 3 3 3 4 … 5 6

6 12

11 11

24 18

0;57 1;52

9 9

8 8

0; 4 0; 8

1 1

1 1

6 12 18 24 0

8 8 8 8 8

24 18 12 6 0

10;33 10;34 10;29 10;15 9;54

0 1 2 3 5

0 1 2 3 4

0;59 1; 1 1; 2 1; 3 1; 2

0 0 0 0 0

0 0 0 0 0

24 0

6 6

6 0

1;21 0; 0

13 13

12 12

0; 9 0; 0

1 1

1 1

table 24

Equation of anomaly for Saturn ( f. 7v)

(1) s

(°)

s

(°)

(2) (°)

(3) min

(4) min

(5) min

(6) sec

(7) sec

0 0 … 2 3 3 3 …

6 12

11 11

24 18

0;34 1; 7

5 5

5 5

0; 3 0; 7

1 1

1 1

24 0 6 12

9 9 8 8

6 0 24 18

5;46 5;53 5;53 5;51

1 0 0 1

1 0 0 1

0;41 0;42 0;44 0;46

0 0 0 0

0 0 0 0

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early alfonsine astronomy in paris

(1) s

(°)

s

(°)

(2) (°)

5 6

24 0

6 6

6 0

0;42 0; 0

(3) min

(4) min

(5) min

(6) sec

(7) sec

7 7

7 7

0; 7 0; 0

1 1

1 1

figure 8.4 Ptolemy’s model for the three superior planets and Venus (not to scale)

center as it should (see Samsó and Millás 1998). Column 6 (motus gradus) seems to be the increment per degree of argument of the entries in col. 5: in many cases the entry in col. 6 results from taking the difference between successive entries in col. 5 and dividing it by 6 (or by 3 for Mars and Venus), and it is for purposes of interpolation. The entries in col. 6 are given in seconds. The entries in column 7 (motus diei) are also given in seconds; they are probably associated with those in the preceding column, for in all cases columns 6 and 7 have their extremal values for the same arguments, but we have failed to identify their specific purpose. Figure 8.4 displays Ptolemy’s model for the three superior planets and Venus. o is the observer, d is the center of the deferent circle rac, and e is the equant point, such that the eccentricity, e = od = de. a is the planet’s apogee, and κ̄ = angle aec, the mean argument of center, is measured from it to the center of the epicycle about point e. Angle gcp is the mean argument of anomaly, ᾱ, and the planet is at point p. Angle hcg is the equation of center and it is also applied to correct the mean argument of anomaly to yield the true argument of anomaly, α = angle hcp. In the case of the superior planets, cp, the direction from the

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center of the epicycle to the planet, is always parallel to οs̄, the direction from the observer to the mean Sun. In the case of Venus, ec is parallel to the direction from the observer to the mean Sun. The goal is to find the direction from the observer to the planet, i.e., angle rop is the longitude of the planet, and r is in the direction to Aries 0°. The mean argument of center, κ̄, is angle aec, and the true argument of anomaly, α, is angle hcp. With these arguments, κ̄ and α, we can determine the equation of anomaly, c(α), with Vimond’s tables and compare the result with computations based on the Parisian Alfonsine Tables. According to our understanding of Vimond’s procedure, c(α) = c2(α) + c5(κ̄) · c5(α), where ci refers to the i-th column in the table. Note that c5(κ̄) is taken from the table for the equation of center (with the shifts), and c5(α) is taken from the table for the equation of anomaly. For instance, for Venus, when κ̄ = 120° and α = 135° c(α) = c2(135°) + c5(120°) · c5(135°) c(α) = 44;49° + 0;45 · 2;25° c(α) = 46;38°. With the same arguments for Venus in the Parisian Alfonsine Tables, we find c(α) = c5(α) + c3(κ̄) · c6(α) c(α) = c5(135°) + c3(120°) · c6(135°) c(α) = 45;59° + 0;31 · 1;15° c(α) = 46;38° and this is exactly what resulted from Vimond’s tables. In the tables for the planetary equations in Almagest xi.11 and its derivatives in al-Battānī and in the Parisian Alfonsine Tables (among others), the rules for computing the equation of anomaly require careful attention to algebraic signs. Vimond simplified the rules for this computation, making his tables more “user-friendly”. A similar procedure is described by Copernicus for using his planetary tables in De revolutionibus, v.23, to compute the equation of anomaly (Copernicus 1543, ff. 173v–179r; cf. Swerdlow and Neugebauer 1984, p. 453).

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Tables 13 (Mercury, f. 5r), 16 (Venus, f. 6r), 19 (Mars, f. 6v), 22 (Jupiter, f. 7r), and 25 (Saturn, f. 7v): Planetary Latitudes The tables for the planetary latitudes, both for the superior and the inferior planets, are in the style of Almagest xiii.5, the zij of al-Battānī (Nallino 1903– 1907, 2:140–141), and some tables associated with the Toledan Tables (Toomer 1968, pp. 71–72; F.S. Pedersen 2002, pp. 1322–1326), as opposed to those in the Handy Tables and those in the zij of al-Khwārizmī. The table for the planetary latitudes of Mercury has seven columns; the table for Venus lacks the seventh; and the tables for the superior planets have only five columns (i.e., cols. 1, 3, 4, 5, and 6). In all cases column 1 displays the argument (argumentum) at 12°-intervals from 0s 12° to 12s 0°. Column 2 is only found in the tables for the inferior planets and the entries are given in minutes. The heading is radix meridionalis in the case of Mercury and radix septentrionalis in that for Venus. This column is for determining the deviation, otherwise called the third component of latitude, that is, the inclination of the plane of the deferent with respect to the ecliptic. The entries for deviation can be derived from: β3 = –0;45 · c5 for Mercury β3 = +0;10 · c5 for Venus where c5 is the column for the minutes of proportion in the table for planetary latitude in Almagest xiii.5 (given there in minutes and seconds). As will be seen, column 5 for Venus in Table 16, given only to minutes, corresponds to c5 in Almagest xiii.5. It is noteworthy that column 2 for Mercury is shifted downwards about 119° whereas there is no shift in the case of Venus. This is exactly the same feature we noticed in the tables for the equation of center and the amount of the shift is the same. The column for deviation is certainly not a common feature in medieval tables (for a survey of the few that have them, see Goldstein and Chabás 2004), and Vimond’s is the earliest set of tables in the West we know to display such columns. For the inferior planets, columns 3 and 5 (diametri) give the minutes of proportion for the inclination and the slant, respectively. We note that columns 3 and 5 for Mercury also exhibit a shift of less than 120°, and that no shifts appear in the case of Venus. We also note that column 5 for Venus lists the rounded values in the column for the sixtieths found in the corresponding table in the Almagest xiii.5, the zij of al-Battānī, etc. For the superior planets, columns 3 and 5 give the minutes of proportion for the northern and southern latitudes, respectively, of the planets. Only half of

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table 13

(1)

Latitude of Mercury ( f. 5r)

s

(°)

(2) min

(3) min

(4) min *

(5) min

(6) min *

(7) sec

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 12

12 24 6 18 0 12 24 6 18 0 12 24 6 18 0 12 24 6 18 0 12 24 6 18 0 12 24 6 18 0

13 4 5 14 23 31 37 41 44 45 43 40 36 29 22 13 4 5 14 23 31 37 41 44 45 43 40 36 29 22

57 60 60 57 51 44 34 23 11 0 13 25 36 45 52 57 60 60 52 51 44 34 23 11 1 13 25 36 45 52

1;44 1;40 1;39 1;16 0;59 0;38 0;16 0;15 0;48 1;25 2; 6 2;47 3;26 3;54 4; 5 3;54 3;26 2;47 2; 6 1;25 0;48 0;15 0;16 0;38 0;59 1;16 1;30 1;40 1;44 1;46

17 5 7 19 31 41 49 55 59 60 58 54 48 39 29 17 5 7 19 31 41 49 55 59 60 58 54 48 39 29

0;12 0;44 1; 6 1;26 1;44 2; 0 2;14 2;25 2;29 2;29 2;20 2; 0 1;29 0;48 0; 0 0;48 1;29 2; 0 2;20 2;29 2;29 2;25 2;10 2; 0 1;44 1;26 1; 6 0;44 0;12 0; 0

1 4 7 9 10 12 13 14 15 15 14 13 9 5 0 5 9 12 14 15 15 14 13 12 10 9 7 4 1 0

* Despite the headings, these columns display degrees and minutes.

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early alfonsine astronomy in paris table 16

(1)

Latitude of Venus ( f. 6r)

s

(°)

(2) min

(3) min

(4) min *

(5) min

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 12

12 24 6 18 0 12 24 6 18 0 12 24 6 18 0 12 24 6 18 0 12 24 6 18 0 12 24 6 18 0

10 9 8 7 5 3 1 1 3 5 7 8 9 10 10 10 9 8 7 5 3 1 1 3 5 7 8 9 10 10

12 24 35 44 52 57 60 60 52 44 35 24 12 0 12 24 35 44 52 57 60 60 57 52 44 44 35 24 12 0

1; 1 0;59 0;55 0;46 0;35 0;29 0;18 0;10 0;32 0;59 1;38 2;23 3;44 5;13 7;12 5;13 3;44 2;23 1;38 0;59 0;32 0;10 0;19 0;29 0;35 0;46 0;55 0;59 1; 1 1; 3

59 55 48 40 30 18 6 6 18 30 40 48 55 59 60 59 55 48 40 30 18 6 6 18 30 40 48 55 59 60

(6) min * 0;16 0;33 0;49 1; 5 1;20 1;35 1;50 2; 3 2;15 2;25 2;30 2;28 2;12 1;27 0; 0 1;12 2;28 2;30 2;25 2;15 2; 3 1;50 1;35 1;20 1; 5 0; 5 ** 0;49 0;33 0;16 0; 0

* Despite the headings, these columns display degrees and minutes. ** Sic.

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table 19

Latitude of Mars ( f. 6v)

s

(1) (°)

(3) min

(4) min *

(5) min

(6) min *

0 0 1 1 … 4 4 … 6 … 10 10 11 11 12

12 24 6 18

50 55 59 59

0; 9 0;13 0;16 0;21

m m m m

0; 4 0; 6 0; 9 0;15

12 24

8 s

2; 1 2;34

m 10

2;10 2;56

0

s

4;21

43

7;30

12 24 6 18 0

s 10 22 33 43

0;21 0;16 0;13 0; 9 0; 6

2 [blank] m m m

0;15 0; 9 0; 6 0; 4 0; 2

* Despite the headings, these columns display degrees and minutes.

the columns are filled with numbers, the others have capital letters indicating “North” [s] and “South” [m]. Column 3 is shifted about 45° (Mars), about 100° (Jupiter), and about 110° (Saturn) in relation to the corresponding columns in the Almagest, whereas the shifts for column 5 are increased by 180° in each case. These shifts are totally consistent with those found for the equation of center (about 44°, 82°, and 162° for Mars, Jupiter, and Saturn, respectively). Indeed, subtracting these numbers for each planet, we find 0° (Mars), about –20° (Jupiter), and +50° (Saturn), in perfect agreement with the differences given by Ptolemy in Almagest xiii.6 between the northern limits on the deferent and the apogees of each superior planet, respectively. Thus, it is quite clear that the compiler of Vimond’s tables, whether Vimond or not, had a good understanding of this difficult issue as it is presented in the Almagest. Columns 4 and 6 display the inclination (declinatio minuti diametri) and the slant (reflexio minuti diametri) for the inferior planets, and the entries are given in degrees and minutes, despite the headings, which read “minutes and seconds”. For the superior planets, these two columns display the northern and

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early alfonsine astronomy in paris table 22

(1)

Latitude of Jupiter ( f. 7r)

s

(°)

(3) min

(4) min *

(5) min

(6) min *

0 0 … 3 3 … 6 6 6 … 9 9 … 11 12

12 24

10 12

1; 8 1; 9

0 m

1; 6 1; 7

6 18

60 60

1;33 1;39

m m

1;33 1;39

0 12 24

12 0 s

2; 5 2; 3 2; 0

m 0 12

2; 8 2; 6 2; 3

6 18

s s

1;27 1;21

60 60

1;26 1;21

18 0

s s

1; 8 1; 6

34 12

1; 6 1; 5

* Despite the headings, these columns display degrees and minutes.

southern limits (both labeled latitudo minuti diametri) and are given in degrees and minutes. The extremal values of columns 4 and 6 in the text are shown below: Mercury Venus Mars Jupiter Saturn

4; 5° (for 6s 0°) 7;12° (for 6s 0°) 4;21° (for 6s 0°) 2; 5° (for 6s 0°) 3; 2° (for 6s 0°)

2;29° (for 3s 18°–4s 0° and 8s 0°–8s 12°) 2;30° (for 4s 12° and 7s 6°) 7;30° (for 6s 0°) 2; 8° (for 6s 0°) 3; 5° (for 6s 0°)

These extremal values in Vimond’s tables agree with those in the Toledan Tables with two exceptions, one of which is a trivial variant for Mercury. But, as far as we know, the maximum value for the inclination of Venus in Vimond’s table is not attested in any other previous text. It is probably significant that this value later appeared in the editio princeps of the Alfonsine Tables (1483), as indicated in Table 13a.

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table 25

Latitude of Saturn ( f. 7v)

s

(1) (°)

(3) min

(4) min *

(5) min

(6) min *

0 0 1 … 3 … 5 6 6 6 … 9 … 11 12

12 24 6

[blank] 2 14

2; 5 2; 7 2;10

9 [blank] s

2; 3 2; 4 2; 7

18

60

2;39

s

2;39

18 0 12 24

33 22 10 n

3; 1 3; 2 3; 1 2;59

s s s 2

3; 3; 3; 3;

18

n

2;21

60

2;21

18 0

n n

2; 5 2; 3

33 22

2; 3 2; 2

3 5 3 0

* Despite the headings, these columns display degrees and minutes. table 13a Extremal planetary latitudes

Almagest al-Battānī Toledan t. Vimond Paris. Alf. t. Mercury Venus Mars Jupiter Saturn

4; 5° –2;30° 6;22° –2;30° 4;21° –7; 7° 2; 4° –2; 8° 3; 2° –3; 5°

4; 5° –2;30° 6;22° –2;30° 4;21° –7; 7° 2; 4° –2; 8° 3; 2° –3; 5°

4; 5° –2;30° 7;24° –2;30° 4;21° –7;30° 2; 5° –2; 8° 3; 2° –3; 5°

4; 5° –2;29° 7;12° –2;30° 4;21° –7;30° 2; 5° –2; 8° 3; 2° –3; 5°

4; 5° –2;30° 7;12° –2;30° 4;21° –7;30° 2; 8° –2; 8° 3; 3° –3; 5°

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early alfonsine astronomy in paris table 26

Yearly radices ( f. 7v)

s years

76

mean motion 0 argument 0 years

s years

0;33,32 mean motion 0 3;54,46 argument 1 152

mean motion 0 argument 0 years

(º)

years

1; 7, 2 mean motion 0 7;49,16 argument 2 304

years

(º) 608 4;28, 3 1;16,21 1216 8;56, 4 2;32,27 2432

mean motion 0 2;14, 3 mean motion 0 17;52, 5 argument 0 15;38,18 argument 4 5; 4,38

For the inferior planets, between columns 2 and 3 and between columns 5 and 6 we are also given some indications (“North” and “South”) to help the user. Column 7 appears only in Table 13 (Mercury), and it seems to be outside the general framework of the table. Its entries are given in seconds and result from dividing the corresponding entries in column 6 by 10. This probably corresponds to the instructions given by Ptolemy in Almagest xiii.6: to compute the true minutes of proportion for the slant, add 1⁄10 when the argument lies between 90° and 270°, or subtract 1⁄10 when the argument lies between 0° and 90° or 270° and 360°. Whether tabulated or not, these instructions are rarely found in the medieval Latin literature on the planets (Goldstein and Chabás 2004).

f. 7v Table 26: Yearly Radices This table displays the radices for the mean motion (motus) and argument (argumentum) of the fixed stars for intervals of 76, 152, 304, 608, 1216, and 2432 years. Vimond does not give a radix for a specific year but perhaps this information was in the canons that we have not found. As we shall argue (see

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Table 27, below), it is likely that the epoch of this table was also 1320 or a date close to it, that is, the epoch is consistent with our dating of the other radices. In 76 years the value in the text for the mean motion of the fixed stars is 0;33,32° and in 2432 years it is 17;52,5°, corresponding to 0;0,0,4,20,56°/d and 0;0,0,4,20,42°/d, respectively. These values are equivalent to 48,954 years and 48,999 years, respectively, to complete one revolution, or 1° in about 136 years, as in the linear term in the standard Alfonsine model for trepidation which is based on one revolution in exactly 49,000 years. These differences in the periods depend on the seconds in the entries in Vimond’s table and have no astronomical significance. However, they indicate that Vimond is not using the standard table for mean motion of the apogees and the fixed stars in the Parisian Alfonsine Tables (Ratdolt 1483, f. d4v). In 76 years the value in the text for the mean motion of the argument for the fixed stars is 3;54,46° and in 2432 years it is 4s 5;4,38°, corresponding to 0;0,0,30,26,47°/d and 0;0,0,30,24,52°/d, respectively. These values are equivalent to 6,992 years and 7,000 years, respectively, to complete one revolution. The periodic term in the standard Alfonsine model for trepidation is based on one revolution in exactly 7,000 years, and it corresponds to 0;0,0,30,24,49°/d. These differences have no astronomical significance, but indicate that, once again, Vimond is not using the standard table for mean motion of access and recess in the Parisian Alfonsine Tables (Ratdolt 1483, f. d4r). In fact, an entry for the mean motion of the argument is 7 times the corresponding entry for the mean motion of the linear term. As in the Parisian Alfonsine Tables, Vimond separates two terms for trepidation: a linear term which corresponds to the difference between the calendar year of 365;15 days and a fixed tropical year, and a periodic term which corresponds to the difference between a variable sidereal year and the calendar year of 365;15 days. But in his other tables Vimond has used a fixed sidereal year: we are unable to account for this inconsistency. To be sure, Vimond’s canons may have explained what he intended.

f. 7v Table 27: Motion of the Fixed Stars The argument is given at 6°-intervals from 0s 6° to 12s 0° and the equation of access and recess (here called motus) is given in degrees and rounded to minutes. In Table 27, below, the editors have supplied a minus sign in a few entries, where appropriate. The table has a maximum of 17;17° for argument 204° and a minimum of –0;43° for argument 24°. These extremal values are 18° apart (= 17;17° + 0;43°); hence the amplitude of the sinusoidal curve corresponding to

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early alfonsine astronomy in paris table 27

Motion of the fixed stars ( f. 7v)

argumentum (º) 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6

6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0

motus (º) –0;19 –0;33 –0;41 –0;43 –0;39 –0;29 –0;13 0; 8 0;35 1; 6 1;43 2;24 3; 8 3;56 4;47 5;40 6;35 7;30 8;26 9;22 10;18 11;12 12; 4 12;54 13;41 14;24 15; 4 15;39 16;15 16;34

argumentum (º)

motus (º)

6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 10 10 10 10 11 12 11 11 11 11 12

16;53 17; 7 17;15 17;17 17;13 17; 3 16;47 16;26 15;59 15;28 14;51 14;10 13;26 12;38 11;47 10;54 9;58 9; 4 8; 8 7;11 6;16 5;22 4;30 3;40 2;53 2;10 1;30 0;55 0;19 0; 0

6 12 18 24 0 6 12 18 14 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0 6 12 18 24 0

Vimond’s table is 9°. This is indeed the characteristic parameter of the table for the equation of access and recess in the Parisian Alfonsine Tables, whose maximum is 9° for argument 90°.

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Comparison of the entries in both tables shows that the curve representing Vimond’s table is the same as that used by other Parisian astronomers of his time but shifted in two ways: 247° on the x-axis and –8;17° on the y-axis. In fact, the entries in Vimond’s table can be derived from those in the Parisian Alfonsine Tables by taking an argument and its corresponding equation in the latter (where they are given to seconds) and then adding 113° to the argument and 8;17° to the equation. Vimond’s table begins at a point that in the Parisian Alfonsine Tables corresponds to a value of the equation of –8;17° and an argument of about 247°. The value for the equation of access and recess that Vimond thought correct for his time was 8;17°, and he shifted the curve (i.e., the entries in the table) accordingly; indeed, calculation of the periodic term in trepidation with the parameters for 1320 in the Parisian Alfonsine Tables yields 8;17° exactly: 1320 · 0;3,5,8,34,17°/y ≈ 67;53° radix Incarnation 359;13 Total

67; 6

and 9° sin 67;6° = 8;17°. Note that 67;6° + 180° = 247;6° or about 247°, and 360° – 247° = 113° which is the phase angle of the shift introduced by Vimond. This table establishes a strong connection between Vimond and the Parisian Alfonsine Tables, for this theory of trepidation is not found in any previous text. But again, since the mean motions are different (see Table 26), we see no reason to assume that Vimond based his theory on the Parisian Alfonsine Tables. Rather, Vimond may have depended on an Andalusian or Castilian tradition that was closely related to (but distinct from) the Castilian Alfonsine Tables, for there is no hint of phase shifts in the Castilian canons.

f. 8r–v Table 28: Fixed Stars This table displays the longitude, the latitude, and the magnitude of 225 stars and nebulae but, in general, their names are omitted. The list is too long to be related to an astronomical instrument, and the absence of star names makes us wonder what purpose it was intended to serve. Both coordinates are given

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to minutes. The stars are divided into three groups, in turn divided into several subgroups according to the associated planets, a feature which is certainly not common. Group i has 137 stars that belong to the zodiacal constellations arranged in 52 subgroups, group ii has 44 stars in northern constellations (19 subgroups), and group iii has 44 stars in southern constellations (19 subgroups); the total number of subgroups is thus 90. We note the balanced representation of the stars on both sides of the zodiac. We have found the same table in an early 14th-century copy, Cambridge, Gonville and Caius College, ms 141/191, pp. 377–382 (for an excerpt, see F.S. Pedersen 2002, pp. 1507–1508); as well as in Segovia, Catedral, ms 84, ff. 46r–51v; and in Paris, BnF, ms 7482; ff. 61v–69v. There are some cases where an entry in one copy does not agree with the value in, or derived from, Ptolemy’s treatises in contrast to the other copy, but there are also examples where entries in both copies do not agree with those in Ptolemy. On the other hand, in all cases where there is a blank entry in one copy, it is filled in the other copy. In the Paris copy only 18 star names are given whereas in the Cambridge copy this number is reduced to 15. The star names in these copies are generally not identical, and they are not always ascribed to the same stars. For instance, the names “almalak” and “almalac” are attributed, respectively, to the star in the 20th subgroup (ms Paris) and to the first star of the 8th subgroup (ms Cambridge). The star list does not bear a general title in the Paris copy but the Cambridge copy reads tabula de dispositionibus stellarum fixarum existentibus ad terminum complementi radicis mediarum coniunctionum solis et lunae quae alibi signantur. Et primo de dispositionibus illarum stellarum quae sunt prope viam solis. (Here begins the table on the groups of the fixed stars as they were at the point of completion [the epoch?] of the radix of the mean conjunctions of the Sun and the Moon specified elsewhere. First come the groups of those stars close to the zodiac [lit.: the path of the Sun].) The first sentence serves as a general title for the table, and the second sentence is a heading for the groups in the zodiacal constellations, corresponding to the headings in both manuscripts for the groups in the northern and southern constellations. The expression “the radix of the mean conjunctions” seems to refer to the radix given on f. 1r, “13;54,54d”, which we identified with March 10, 1320. But we do not understand the expression “at the end of the complement”. Madrid, Biblioteca Nacional, ms 4238, ff. 65v–66v, reproduces the same star list except that the signs used here are of 60°, contrary to the other manuscripts containing this list. We are grateful to Paul Kunitzsch for information on two additional copies of the same star list: Erfurt, Universitätsbibliothek, ms Amplon. 2°395, ff. 104v–

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105v; and Munich, Bayerische Staatsbibliothek, ms Clm 26667, ff. 46v–47v (cf. Kunitzsch 1986a, p. 96, n. 10, and p. 98, n. 44). In both manuscripts the list is anonymous, but in the Erfurt copy a marginal note (in the same hand as the list) reads: Notandum istas stellarum tabulas fuisse equatas ad annum domini 1338 (f. 105v). As Kunitzsch suggested to us (in a private communication), this marginal note may have been added by the copyist and not belong to the original list; no date appears in the other three manuscripts. In fact, the list in the Erfurt ms has two extra stars: one is added to the northern constellations, in subgroup 7 (Bootes), and the other to the southern constellations, in subgroup 6 (Eridanus). We also note that in the list for the southern constellations the stars in subgroup 19 (Ara) are located in the Erfurt ms between subgroups 4 and 5 in the manuscripts in Paris and Cambridge (we have not seen the manuscript in Munich). Another special feature of the Erfurt ms is that the subgroups are not numbered; rather, most are given the name of a star belonging to them or even a generic name. But its main distinguishing characteristic is that the subgroups have no associated planets, in contrast to the copies in Paris and Cambridge. It may be of interest that the 5 manuscripts of which we are aware that contain this star list are spread all over Europe: 2 in Germany, 1 each in England, France, and Spain. The order and the grouping of the stars in this list is peculiar, for they do not follow the pattern of the catalogue in Ptolemy’s Almagest that was generally adopted in medieval star lists and catalogues. Rather, this list is organized according to Ptolemy’s Tetrabiblos, a handbook on astrology written by Ptolemy after the Almagest. It was translated several times from Arabic into Latin: in 1138 by Plato of Tivoli, in 1206 anonymously, and in 1256 via Castilian at the court of Alfonso x by Egidius de Tebaldis (Chabás and Goldstein 2003a, p. 232), and was known as the Quadripartitum. In Tetrabiblos i.9, Ptolemy grouped the stars into three main categories (zodiacal, northern, and southern constellations), following an order differing from that in the Almagest where the northern constellations precede the zodiacal constellations, and grouped the stars within each category according to their associated planets. As an example, we reproduce a passage of Tetrabiblos i.9 corresponding to the stars in the constellation of Aries (Robbins 1940, p. 47): The stars in the head of Aries, then, have an effect like the power of Mars and Saturn, mingled; those in the mouth like Mercury’s power and moderately like Saturn’s; those in the hind foot like that of Mars, and those in the tail like that of Venus.

early alfonsine astronomy in paris

283

As is readily seen, the order, the subgroups, and the planets associated with the stars in Aries in Vimond’s list perfectly match those in Ptolemy’s Tetrabiblos. And this is indeed the case for almost all stars in the 90 subgroups displayed in Vimond’s list. The star positions generally agree with those in Gerard of Cremona’s version of Ptolemy’s star catalogue in the Almagest with an increment in longitude of 17;52° for precession, a value otherwise unattested. If the rate of precession was taken to be 1° in 66 years, 17;52° would correspond to about 1179 years and, if we add it to 137ad (the date of the star catalogue in the Almagest), we get 1316ad. But it is not clear that this date had any significance for the author. We have compared this list to that in the Libro de las estrellas de la ochaua espera (Madrid, Universidad Complutense, ms 156; see also Rico Sinobas 1863–1867, vol. 1, pp. 5–145), also known as Libro de las xlviii figuras de la viii spera or even as Libro de las estrellas fixas. This is an adaptation of the star catalogue for 964 ad by the Persian astronomer al-Ṣūfī (903–986) which in turn depended on the star catalogue in Ptolemy’s Almagest (see Comes 1990). This work, where the total precession is 17;8°, was compiled in 1256 by Judah ben Moses ha-Cohen, one of the most distinguished collaborators of Alfonso x. The presentation of the star data in this Alfonsine text differs substantially from that of a typical star list although the data themselves are what one would expect, namely, for each star we are given its name, longitude, latitude, and magnitude. The associated planets are also given for each star, often adding an indication of their relative strength, showing that the Alfonsine Libro ultimately relied on Ptolemy’s Quadripartitum. However, after comparing the data in the Libro with those of Vimond, we see no evidence to suggest that the star list found among Vimond’s tables is systematically related to this Alfonsine book. As Kunitzsch informed us, there is a star list by John of Lignères containing data for 276 stars, but the longitudes are Alfonsine, i.e., Ptolemy’s values plus 17;8°: Paris, Bibliothèque nationale de France, ms lat. 10264, ff. 36v–38v, and Florence, Biblioteca Nazionale Centrale, ms Conv. soppr. j.4.20, fols. 214v–216r. This list was extracted from the star table that later appeared in the editio princeps of the Alfonsine Tables (1483), and sheds no additional light on the list included in Vimond’s tables. Moreover, in the course of examining the star names in the five manuscripts containing this list, Kunitzsch noticed that the author drew upon a variety of Latin sources, mainly the translations of the Tetrabiblos but also sources not in the Tetrabiblos tradition (some of which cannot be identified). Thus, Vimond’s list is dependent on Ptolemy in two ways: the choice of the stars, their order and grouping, as well as the associated planets, are borrowed from the Quadripartitum; and the numerical data are taken from the Latin version of the Almagest.

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In sum, we believe that the star list attributed to Vimond in the Paris ms, and that is anonymous in the Cambridge, Erfurt, Madrid, and Munich mss, derives from an unknown archetype; we know of no similar star list in Latin in the 14th century or in the previous Arabic literature with which to compare it. In Table 28 we present in the first 3 columns a complete transcription of the Paris copy with translations of the headings and the names of the associated planets in each case. For the latitudes “north” is indicated by an abbreviation of the term septentrionalis, and “south” by an abbreviation of meridionalis; we have replaced them with the modern designations + and –. Column iv gives the few star names found in the Paris copy, which were added in interstitial spaces within the table (some of the star names are partly hidden in the gutter of the manuscript and cannot be read completely); column v lists the modern star designation; column vi gives the standard number assigned to each of the 1028 stars in Ptolemy’s catalogue; column vii offers comparisons and comments, together with variants in the Cambridge copy; and column viii provides the identification of the star names.

+6; 0

Mars –5; 5 –1;30 –1;20

3 [Aries] 1 2;52 1 5;52 1 7;32

4 5 5

5

Mars, Saturn +7;20 3 +8;20 3 Mercury, Saturn +7;40 5

0 29;22

1 [Aries] 0 24;32 0 25;32 2 [Aries] 0 28;52

[Zodiacal constellations]

Associated planets ii iii iv Latitude Magn. Name (degrees)

Star list ( f. 8r –v)

[Constellation] i Longitude (sign, degrees)

table 28

μ Cet σ Ari ρ Ari

374 373 372

365

364

η Ari θ Ari

362 363

vi Number (P.-K.)

γ Ari β Ari

v Modern designation

C: +5;15, G: –5;15 C: +1;30 C: +1;20 G: –1;10, +1;10

C(iii): blank C(iv): flamai? C(iii): blank C(iv): hercules

vii Comparisons and comments

Unidentified: see β Gem, below

Unidentified

viii Identification of star names

early alfonsine astronomy in paris

285

ii

Star list ( f. 8r –v) (cont.)

iii

4 [Aries] Venus 1 9;12 +4;50 5 1 11;42 +1;40 4 1 13;12 +2;30 4 1 14;52 +1;50 4 5 [Taurus] Venus, Jupiter 1 17;32 –9;30 5 –8; 0 3 1 21;32 6 [Taurus, The Pleiades] Moon, Mars 1 20; 2 +4;30 5 1 20;22 +4;40 5 1 20;32 +5; 5 5 1 21;32 +5;20 5 7 [Taurus] Mars 2 0;32 –5;10 1

i

table 28

384 385 409 410 412 411

30(e) Tau λ Tau 19 Tau 23 Tau 27 Tau* bsc 1188* 393

368 369 370 371

vi

ε Ari δ Ari ζ Ari τ Ari

v

aldebaran? α Tau

iv

C(iv): aldebaran

C: +8; 0

C: Moon

vii

G, p. 89 n. 10, etc.

viii

286 chapter 8

3 4

3

Mars –3;30 5 –5; 0 5 +5; 0 5 –2;30 3 Mercury, Venus –1;30 4 –1;15 4 –3;30 4

–3; 0 –4; 0

1 29;42 2 3;32

9 [Taurus] 2 7;52 2 8;12 2 13;32 2 15; 2 10 [Gemini] 2 24;22 2 26; 2 2 28; 2

–5;50

1 28;42

Saturn, Mercury –5;45 3

8 [Taurus] 1 26;52

iii

ii

i

iv

397 396 230/400 398 437 438 439

η Gem μ Gem ν Gem

394 399

392

390

vi

106(l) Tau 104(m) Tau β Tau ζ Tau

ε Tau τ Tau

θ1 Tau

γ Tau

v

C: +1;30 C: +1;15 C: +3;30

C: +5; 0 G: 3 G + 17;52: 15;32

C: +3; 0 C: +4; 0 G: –4; 0, +4; 0

C: +5;50 G: –5;50, –0;50

C(iv): almalac

vii

If this is a corruption of Arabic al-malik (the king), it should designate α Leo (Regulus). See G, p. 101 n. 12.

viii

early alfonsine astronomy in paris

287

Mercury, Mars +1; 0 5 –7;30 4 Saturn, Mercury +11;50 4 +5;30 4 Moon, Mars n +0;40 meollef?

hercules?

[..]

14 [Cancer] 3 20;32 3 25; 2 15 [Cancer] 3 26;12 4 4;22 16 [Cancer] 3 28;12

α Gem

( )annai?

2

449

455 454

ι Cnc α Cnc gc 2632 Galaxy m 44

456 457

425

424

435

440 441

vi

μ Cnc β Cnc

β Gem

δ Gem

3

v γ Gem ξ Gem

iv

3 4

–7;30 –10;30 Saturn –5;30 Mars +9;40 Mars [..]6;15

2 29;52 3 2;32 11 [Gemini] 3 9;32 12 [Gemini] 3 11;12 13 [Gemini] 3 14;32

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

viii

C: 2. C(iv): mellef?

G: –5;30 P, f. 15va: meelef

C(ii): +6;15. C(iii): 2 R, p. 48: Herakles C(iv): almueredan K1959, p. 127: ε Vir is called almuredin

C: +7;30 C: +10;30

vii

288 chapter 8

Mars, Sun +2;40 4 +0;10 4 Saturn, Mars +9;30 3 +12; 0 3 Saturn, Mars +11; 0 3 +4;30 3 +7;30 2 Mars, Jupiter +0;10 1 Venus, Saturn +13;15 5 +13;40 2 +11;30 5 + 9;40 3 +12;50 1 Venus, Mercury +5;50 3 –3; 0 5

17 [Cancer] 3 28;12 3 29;12 18 [Leo] 4 12; 2 4 12; 2 19 [Leo] 4 18; 2 4 18;32 4 20; 2 20 [Leo] 4 20;22 21 [Leo] 4 29;12 5 2; 2 5 2;12 5 4;12 5 12;22 22 [Leo] 5 8;12 5 8;22

iii

ii

i

480 481 482 483 488 484 487

60(b) Leo δ Leo 81 Leo* θ Leo β Leo ι Leo υ Leo

??

466 468 467

ζ Leo η Leo γ Leo 469

465 464

ε Leo μ Leo

α Leo

452 453

vi

γ Cnc δ Cnc

v

almalak?

assinis?

iv

G: +11;50

C, G: +12;15

G: +8;30

G + 17;52: 12;12 C: Mercury

C(iv): asini G: –0;10

vii

G, p. 101 n. 12.

G

viii

early alfonsine astronomy in paris

289

+0;50 4 +1;15 4 Mercury, Mars +4;35 5 +5;40 5 +6; 0 3 +5;30 5 Mercury, Venus +1;10 3 +2;50 3 Saturn, Mercury +15;10 3 Venus, Mercury –2; 0 1 Mercury, Mars +7;30 4 +2;40 4 +0;30 4 +9;50 4

5 9;32 5 9;32 23 [Virgo] 5 14;12 5 14;52 5 17; 2 5 18; 2 24 [Virgo] 5 26; 7 6 1; 2 25 [Virgo] 6 0; 2 26 [Virgo] 6 14;32 27 [Virgo] 6 24;32 6 25;12 6 7;52 7 0;32

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

almure?

iv

518 519 521 522

509

ε Vir

ι Vir κ Vir λ Vir μ Vir

502 503

η Vir γ Vir

510

497 498 501 500

ν Vir ξ Vir β Vir π Vir

α Vir

486 485

vi

τ Leo σ Leo

v

C, G + 17;52: 27;52

C(iv): alcimech

G + 17;52: 16;52

vii

G: ascimech

viii

290 chapter 8

8 18;22

–16;40

3

Jupiter, Mercury +0;40 2 +8;30 2 Saturn, Mercury [..]1;15 [..] [..]1;40 [..] [..]3;45 [..] [..]4;30 [..] Mars, Saturn –1;40 3 –5; 0 3 +1;20 3 Mars, Jupiter –4; 0 2 Saturn, Venus –15; 0 4 –19;30 3 –18;50 3 –15;10 3

28 [Libra] 7 5;52 7 10; 2 29 [Libra] 7 9;12 7 11;52 7 15;22 7 20;52 30 [Scorpius] 7 23;32 7 23;32 7 24;12 31 [Scorpius] 8 0;29 32 [Scorpius] 8 5;52 8 11; 2 8 16; 2 8 15;52

iii

ii

i

iv

547 548 546 553 558 561 562 564

δ Sco π Sco β Sco α Sco μ1 + μ2 Sco η Sco θ Sco κ Sco

563

534 533 535 536

ν Lib ι Lib γ Lib θ Lib

ι1 Sco

529 531

vi

α Lib β Lib

v

G + 17;52: 6;42 C: +19;30 C: +18;50 G + 17;52: 16;52 C(ii): +16;10 C: +16;40

G + 17;52: 0;32

C: +5; 0

C(ii): +1;15. C(iii): 4 C(ii): +2;40. C(iii): 4 C(ii): +3;45. C(iii): 4 C(ii): +4;30. C(iii): 4

vii

viii

early alfonsine astronomy in paris

291

Saturn, Moon –6;30 3 –3;50 4 Jupiter, Mars +2; 7 4 –1;30 3 Mercury, Jupiter, Sun, Mars –7;45 n Jupiter, Mercury –6;45 3 –2;30 4 –2;30 5

35 [Sagittarius] 8 22;22 9 0;52 36 [Sagittarius] 8 24;32 8 26;52 37 [Sagittarius] 9 3; 2 38 [Sagittarius] 9 4;12 9 5;32 9 7;52

3

Mars, Moon –13;15 n

–13;20

34 [Scorpius] 8 19; 2

8 14;22

Mercury, Mars, Moon –23;30 4

iv

33 [Scorpius] 8 15;52

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

570 576 574 573 577 591 590 589

μ Sgr λ Sgr ν1 + ν2 Sgr ζ Sgr τ Sgr ψ Sgr

567

565

566

vi

γ Sgr φ Sgr

G Sco* + CGlo 6441

λ Sco

υ Sco

v

C: +2;30, G: –4;30 C: +2;30

C: Moon G: –0;45. C(iv): 2

C: +3;50

C: 2

C: [blank] C, G + 17;52: 14;52 C, G: –13;30 C, G + 17;52: 15;22 C(ii): +13;20

vii

viii

292 chapter 8

Jupiter, Saturn –18; 0 2 –23; 0 2 –13; 0 3

Venus, Saturn –5;50 5 –4;50 5

–4;50 5 –6;30 5 Mars, Venus +2;20 3 +5; 0 3 +1;30 6 +0;45 6 Mars, Mercury –8;40 4 –7;40 4 –6;50 4 –6; 0 5

39 [Sagittarius] 9 4;52 9 5;32 9 14;32

40 [Sagittarius] 9 16;32 9 16;32

9 16;42 9 17;32 41 [Capricornus] 9 25;12 9 25;12 9 26;42 9 26;52 42 [Capricornus] 9 29;32 10 4;32 10 8; 2 10 8;12

iii

ii

i

iv

597 600 601 603 607 605 612 613 614 615

α1 + α2 Cap β Cap ρ Cap ο Cap ω Cap 24(a) Cap ζ Cap 36(b) Cap

599 598

593 592 594

vi

ω Sgr 62(c) Sgr

59(b) Sgr 60(a) Sgr

α Sgr β1 + β2 Sgr η Sgr

v

C: +7;40 C: +6;50 C: +6; 0

C, G + 17;52: 16;22 G + 17;52: 15;32 C(ii): +4;50 C: +4;50 C: +6;30

C: +23; 0 G + 17;52: 24;32 C(ii): +13; 0

vii

viii

early alfonsine astronomy in paris

293

Saturn, Mercury +2;10 3 +2; 0 3 –0;20 4 –2;50 5 Saturn, Mercury +8;40 3 +8; 0 4 +8;50 2 +11;15 4 Mercury, Saturn –5; 0 4 –7;30 3

–5;40 5 Saturn, Jupiter –1; 0 4 –7;30 4 –0;30 4 –1;40 4

43 [Capricornus] 10 12;42 10 14;12 10 14;42 10 15;32 44 [Aquarius] 10 2;32 10 4; 2 10 14;22 10 24;12 45 [Aquarius] 10 19;12 10 19;32

10 22;32 46 [Aquarius] 11 5;32 11 6;52 11 7;52 11 8;12

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

a[n]phora

iv

653 656 654 655

83(h) Aqr ψ1 Aqr φ Aqr χ Aqr

647 646

τ Aqr δ Aqr 648

636 635 632 630

ε Aqr μ Aqr β Aqr α Aqr

53(f) Aqr

623 624 625 627

vi

γ Cap δ Cap 42(d) Cap λ Cap

v

C: +7;30, G: –8;30 C: +7;30 C: +1;40

G + 17;52: 29;12 G + 17;52: 29;32 C, G: +7;30 C: +5;40

G: +11;0. G: 3

G: +2;10, –2;10 G: +2; 0, –2; 0 G: +0;20 C, G: +2;50

vii

See note 1.

viii

294 chapter 8

Mercury, Saturn +9;15 4 +7;30 4 +9;20 4 Jupiter, Mercury +4;30 4 +2;30 4 Saturn, Mercury +6;20 4 +5;45 6 Jupiter, Venus +15;20 4 +17; 0 4 Saturn, Jupiter +14;20 4 +13; 0 4 +12; 0 4 Mars, Mercury –8;30 3

47 [Pisces] 11 9;32 11 12; 2 11 13;52 48 [Pisces] 11 13;52 11 17;32 49 [Pisces] 11 23;52 11 28;52 50 [Pisces] 0 17;12 0 20; 2 51 [Pisces] 0 13;32 0 14;12 0 15;32 52 [Pisces] 0 20;22

iii

ii

i

iv

674 675 676 679 680 681 682 706 705 702 703 704 692

κ Psc λ Psc ω Psc 41(d) Psc φ Psc υ Psc ψ1 Psc ψ2 Psc χ Psc* α Psc

vi

β Psc γ Psc 7(b) Psc

v

G + 17;52: 17;22

vii

viii

early alfonsine astronomy in paris

295

ii

Star list ( f. 8r –v) (cont.)

iii

iv

v

vi

vii

Saturn, Venus 72;50 2 74;50 2

Moon, Venus 53;30 2 55;40 2

54; 0 2 Saturn, Mars 84;50 3 88; 0 3 Saturn, Jupiter 69; 0 3 71;10 4 72; 0 4

1 [Ursa Minor] 4 5; 2 4 14; 2

2 [Ursa Maior] 5 0; 2 5 5;52

5 17;42 3 [Draco] 5 26;22 5 27;52 4 [Cepheus] 0 4;32 0 25;22 11 27;12

benezna

aliedin alforcami

[Northern constellations: all latitudes are positive]

35 67 68 78 77 79

ζ Dra η Dra α Cep β Cep η Cep

33 34

6 7

η UMa

ε UMa ζ UMa

β UMi γ UMi

G: 78; 0

G: 14; 0

G: 13;30 G: 15;40

C(iv): aliedim C(iv): alfoza

[Title:] Then follow the constellations (dispositio) of the other fixed stars in the northern part.

i

table 28

K1966, p. 42, no. 23: benenaz (η UMa)

See note 2. K1961, p. 58: al-farqadān (β+γ UMi)

viii

296 chapter 8

Saturn, Mars 53;30 4 54;10 3

59;50 3 60;20 4 Venus, Mercury 46;10 4 44;30 2 44;45 4 44;50 4 Mercury 28; 0 3 Venus, Mercury 62; 0 1 Saturn, Jupiter 23; 0 2 Mars, Mercury 30; 0 2 Mars, Mercury 22;30 1

5 [Hercules] 7 28; 2 7 29;42

8 1;52 8 3;12 6 [Corona Borealis] 6 29;32 7 2;32 7 5; 2 7 7; 2 7 [Bootes] 6 9;12 8 [Lyra] 9 5;12 9 [Perseus] 1 17;29 10 [Perseus] 1 22;42 11 [Auriga] 2 12;52

iii

ii

i

α Aur

222

197

α Per alhaioch

202

β Per

eiumezuz?

107

η Boo 149

112 111 115 116

β CrB α CrB γ CrB δ CrB

α Lyr

133 134

130 129

vi

π Her 69(e) Her

ε Her ζ Her

v

lilurah

alfeca

iv

C(iv): alhaioch

G + 17;52: 17;32

C(iv): lulurach

C(iv): alfeca

G: 13;30 G + 17;52: 21;42 G: 16;10 G: 19;50, 59;50

vii

G

Unidentified

G: allore

G

viii

early alfonsine astronomy in paris

297

Saturn, Venus 36; 0 3

Saturn, Mars 25;30 3 36;30 4 24; 0 3 16;30 4 Mars, Venus 39;10 6 39;20 4 Jupiter, Mars 29;10 2

Saturn, Mars 32; 0 3 33;50 3 32; 0 3 32;10 3

12 [Ophiuchus] 8 12;42

13 [Serpens] 7 12;12 7 12;42 7 14;12 7 16;32 14 [Sagitta] 9 24;32 9 28; 2 15 [Aquila] 9 21;42

16 [Delphinus] 10 6;22 10 8; 2 10 9;12 10 11;22

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

vultur

alhanue

iv

β Del α Del δ Del γ Del

304 305 306 307

288

282 281

ζ Sge γ Sge α Aql

271 270 272 273

234

vi

α Ser λ Ser ε Ser μ Ser

α Oph

v

G: 33;10

C: 19;10 C(iv): vultur

G: 25;20 C, G: 26;30

C(iv): alhanue

vii

G

K1966, p. 55, no. 33: alhaue, alhane

viii

298 chapter 8

358 359

G + 17;52: 21;42 C(ii): +16;20

G + 17;52: 10; 2 G: 12;30. C, G: 2

vii

1 [Piscis Austrinus] 10 9;42

Mars, Venus, Mercury 16;30 4

[Southern constellations: all latitudes are negative]

θ PsA

1020

[Title:] Then follow the constellations (dispositio) of the other stars in the southern part.

α Tri β Tri

349

345 347 348 346

η And μ And ν And β And

3 3

23; 0 Mercury 16;30 20;40

1 4;42 19 [Triangulum] 0 28;52 1 3;52

317

316

vi

β Peg

γ Peg

v

γ And

31; 0 2 Mars, Venus 15; 7 3 30; 0 3 32;30 3 26;20 3

11 20; 2 18 [Andromeda] 0 13;32 0 19;42 0 19;52 0 25;42

iv

3

Mars, Mercury 12;31 3

17 [Pegasus] 0 15; 2

iii

ii

i

viii

early alfonsine astronomy in paris

299

15; 0 4 14; 0 4 20;20 4 Saturn 20; 0 2 Mars, Mercury 17; 0 1 Jupiter, Saturn 31;30 1 24;10 2 24;50 2 25;40 2 Jupiter 53;30 1 [..] 31;50 4 Saturn, Mars 44;20 3 41;30 3

10 16; 2 10 16;42 10 18;32 2 [Cetus] 0 12;52 3 [Orion] 2 19;52 4 [Orion] 2 7;42 2 13;12 2 15;12 2 16; 2 5 [Eridanus] 0 18; 2 6 [Eridanus] 2 5;12 7 [Lepus] 2 12;42 2 13;22

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

iv 1019 1018 1012 725 735 768 759 760 761 805 772 813 812

ζ Cet α Ori β Ori δ Ori ε Ori ζ Ori θ Eri λ Eri β Lep α Lep

vi

η PsA λ PsA β PsA

v

C(i): 16; 2, G: 13;30 C: Saturn G + 17;52: 6;12

C, G + 17;52: 13; 2 G: 14;40

vii

viii

300 chapter 8

Venus 57;40 2 59;40 2 Jupiter, Mars 39;10 1 35; 0 4 Mercury, Mars 14; 0 4 16;10 1 Saturn, Jupiter 20;30 2 26;30 4 26; 0 4 Venus, Mercury 18; 0 4 18;30 4 19;30 4 Saturn, Mercury 19;40 3 14;50 3

8 [Canis Maior] 2 13;52 2 16;52 9 [Canis Maior] 3 5;32 3 7;32 10 [Canis Minor] 3 13;22 3 17; 2 11 [Hydra] 4 17;52 4 23;52 4 26;32 12 [Crater] 5 17;52 5 17;52 5 20;22 13 [Corvus] 6 2;12 6 6;22

iii

ii

i

iv

845 844 818 819 847 848 905 906 907 923 924 922 929 931

α CMa θ CMa β CMi α CMi α Hya κ Hya υ1 Hya δ Crt ζ Crt γ Crt ε Crv γ Crv

vi

α Col β Col

v

C: 6;12

G + 17;52: 24;52

G + 17;52: 12;52

C: 57;40

vii

viii

early alfonsine astronomy in paris

301

Saturn, Jupiter 69; 0 1 Mars, Venus 25;40 3 22;30 3 Venus, Jupiter 41;10 1

51;10 2 55;20 2 51;40 2 45;20 2 Venus, Mars 29;10 3 24;10 3 Saturn, Mercury 15;20 4 16; 0 4 17;10 4

14 [Argo] 3 5; 2 15 [Centaurus] 6 24; 2 7 3;32 16 [Centaurus] 6 26;13

6 27;52 6 29; 2 7 3;12 7 12; 2 17 [Lupus] 7 13;42 7 15;52 18 [Corona Australis] 9 4;22 9 4;42 9 5;12

iii

ii

Star list ( f. 8r –v) (cont.)

i

table 28

iv

973 972 1005 1004 1003

α Lup* β Lup γ Cr a α Cr a β Cr a

969

α Cen 965 968 966 970

939 940

ι Cen θ Cen

γ Cru α Cru β Cru β Cen

892

vi

α Car

v

G + 17;52: 4;52

G: 15;10

G: 24;50

C: 45;[..]

G + 17;52: 26;12 C: 26;12

C: 2

G: 29; 0

vii

viii

302 chapter 8

Jupiter, Mercury 4 30;20 33;20 4 34;10 4

19 [Ara] 8 8;32 8 12;52 8 13; 2

iii

ii

i

iv

ε1 Ara β Ara γ Ara

v

994 996 995

vi

G: 5

vii

viii

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Col. i. The number of the zodiacal sign is not repeated in col. vii where variants are listed; in all cases reported in that column only the degrees and minutes differed from the entry in the Paris manuscript. Col. iii: n means nebulous. Col. iv: In the manuscript the names of the stars are not presented in a column. Col. v: The entries in this column have been taken from Toomer 1984. * indicates that Kunitzsch 1986 and Kunitzsch 1991, pp. 187–200, give a different modern designation. Col. vi: These numbers are taken from Peters and Knobel 1915 (ultimately from Baily 1843), and they are also used in Kunitzsch 1986 and 1990. Col. vii: C refers to Cambridge, Gonville and Caius College, ms 141/191; in certain cases, it is followed by a column number in Roman numerals. G refers to Gerard of Cremona’s version of Ptolemy’s star catalogue (Kunitzsch 1990). We underline entries in Vimond’s table for which there is a variant reading. The entries for longitudes in both copies generally agree with those in G with an increment of 17;52° for precession; those cases where they differ have been noted. Col. viii: G refers to Gerard of Cremona’s version of Ptolemy’s star catalogue (Kunitzsch 1990); K1959 refers to Kunitzsch 1959; K1966 refers to Kunitzsch 1966; P refers to Plato of Tivoli’s Latin version of the Tetrabiblos (ed. 1493); and R refers to Robbins 1940. Note 1. We are informed by Kunitzsch that anphora is not a proper name; rather, it is a noun used in the description of the star’s position: “where the water flows out from the vessel”; Erfurt, Universitätsbibliothek, Amplon. 2°395, f. 105r, in decursu aque … ab anphora; p, f. 16va: In aque vero decursu collocate (without anphora). Note 2. As Kunitzsch informed us, aliedim, apparently renders the Arabic aljady (the kid), an old Arabic name for α UMi (Kunitzsch 1961, p. 62). It is uncertain where the compiler of this list might have found it. In the Tetrabiblos tradition this name never occurs.

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Acknowledgments We thank Paul Kunitzsch and Beatriz Porres for assistance with the Latin texts cited in this article, and Fritz S. Pedersen, John D. North, and Julio Samsó for detailed comments on a preliminary version of this paper.

References as-Saleh 1970. See Saleh, J.A. as- 1970. Baily, F. 1843. The Catalogues of Ptolemy, Ulugh Beigh, Tycho Brahé, Halley, Hevelius. Memoirs of the Royal Astronomical Society, vol. 13. London. bsc. See Hoffleit (ed.) 1964. Boudet, J.-P. 1997–1999. Le “Recueil des plus celebres astrologues” de Simon de Phares, 2 vols. Paris. Casulleras, J. and J. Samsó (eds.) 1996. From Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet. Barcelona. Chabás, J. 2000. “Astronomía alfonsí en Morella a finales del siglo xiv”, Cronos: Cuadernos Valencianos de Historia de la Medicina y de la Ciencia, 3:381–391. Chabás, J. and B.R. Goldstein 1994. “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for History of Exact Sciences, 48:1–41. Chabás, J. and B.R. Goldstein 1997. “Computational Astronomy: Five centuries of Finding True Syzygy”, Journal for the History of Astronomy, 28:93–105. Chabás, J. and B.R. Goldstein 2003a. The Alfonsine Tables of Toledo. Dordrecht. Chabás, J. and B.R. Goldstein 2003b. “John Vimond and the Alfonsine Trepidation Model”, Journal for the History of Astronomy, 34:163–170. Comes, M. 1990. “Al-Ṣūfī como fuente del libro de la ‘Ochaua Espera’ de Alfonso x”, in Comes, Mielgo, and Samsó (eds.) 1990, pp. 11–113. Comes, M., H. Mielgo, and J. Samsó (eds.) 1990. “Ochava espera” y “astrofísica”. Barcelona. Copernicus, N. 1543. De revolutionibus. Nuremberg. Goldstein, B.R. 1974. The Astronomical Tables of Levi ben Gerson. Transactions of the Connecticut Academy of Arts and Sciences, 45. New Haven. Goldstein, B.R. 1996. “Lunar Velocity in the Middle Ages: A Comparative Study”, in Casulleras and Samsó (eds.) 1996, pp. 181–194. Goldstein, B.R. 2001. “The Astronomical Tables of Judah ben Verga”, Suhayl, 2:227–289. Goldstein, B.R. 2003. “An Anonymous Zij in Hebrew for 1400ad: A Preliminary Report”, Archive for History of the Exact Sciences, 57:151–171. Goldstein, B.R. and J. Chabás 2004. “Ptolemy, Bianchini, and Copernicus: Tables for Planetary Latitudes”, Archive for History of Exact Sciences, 58:453–473.

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Hoffleit, D. (ed.) 1964. Catalogue of Bright Stars. Yale University Observatory. New Haven. King, D.A. and M.H. Kennedy (eds.) 1983. Studies in the Islamic Exact Sciences. Beirut. Kremer, R.L. 2003. “Wenzel Faber’s Table for Finding True Syzygy”, Centaurus, 45:305– 329. Kremer, R.L. and J. Dobrzycki 1998. “Alfonsine meridians: Tradition versus experience in astronomical practice c. 1500”, Journal for the History of Astronomy, 29:187–199. Kunitzsch, P. 1961. Untersuchungen zur Sternnomenklatur der Araber. Wiesbaden. Kunitzsch, P. 1966. Typen von Sternverzeichnissen in astronomischen Handschriften des zehnten bis vierzehnten Jahrhunderts. Wiesbaden. Kunitzsch, P. 1986a. “The star catalogue commonly appended to the Alfonsine Tables”, Journal for the History of Astronomy, 17:89–98. Kunitzsch, P. (ed. and tr.) 1986b. Claudius Ptolemäus, Der Sternkatalog des Almagest: Die arabisch-mittelalterliche Tradition. i: Die arabischen Übersetzungen. Wiesbaden. Kunitzsch, P. (ed.) 1990. Claudius Ptolemäus, Der Sternkatalog des Almagest: Die arabisch-mittelalterliche Tradition. ii: Die lateinische Übersetzung Gerhards von Cremona. Wiesbaden. Kunitzsch, P. 1991. Claudius Ptolemäus, Der Sternkatalog des Almagest: Die arabischmittelalterliche Tradition. iii: Gesamtkonkordanz der Stern-koordinaten. Wiesbaden. Mestres, A. 1996. “Maghribī Astronomy in the 13th Century: a Description of Manuscript Hyderabad Andra Pradesh State Library 298”, in Casulleras and Samsó (eds.) 1996, pp. 383–443. Mestres, A. 1999. Materials Andalusins en el Zīj d’Ibn Isḥāq al-Tūnisī. Doctoral Thesis, University of Barcelona. Millás, J.M. 1943–1950. Estudios sobre Azarquiel. Madrid–Granada. Nallino, C.A. 1903–1907. Al-Battānī sive Albatenii Opus Astronomicum, 2 vols. Milan. Neugebauer, O. 1962. The Astronomical Tables of al-Khwārizmī. Copenhagen. Neugebauer, O. 1975. A History of Ancient Mathematical Astronomy. Berlin. North, J.D. 1976. Richard of Wallingford, 3 vols. Oxford. Pedersen, F.S. 2002. The Toledan Tables: A review of the manuscripts and the textual versions with an edition. Copenhagen. Pedersen, O. 1974. A Survey of the Almagest. Odense. Peters, C.H.F. and E.B. Knobel 1915. Ptolemy’s Catalogue of Stars: A Revision of the Almagest. Washington. Poulle, E. 1973. “John of Lignères”, in The Dictionary of Scientific Biography, 7:122–128. New York. Poulle, E. 1984. Les tables alphonsines avec les canons de Jean de Saxe. Paris. Ptolemy. Quadripartitum. See G. Salio (ed.) 1493. Ratdolt, E. (ed.) 1483. Tabule astronomice illustrissimi Alfontij regis castelle. Venice.

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Rico Sinobas, M. 1863–1867. Libros del Saber de Astronomía del Rey D. Alfonso x de Castilla, 5 vols. Madrid. Robbins, F.E. (ed. and trans.) 1940. Ptolemy: Tetrabiblos. London. Saby, M.-M. 1987. Les canons de Jean de Lignères sur les tables astro-nomiques de 1321. Unpublished thesis: Ecole Nationale des Chartes, Paris. 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. Saleh, J.A. as- 1970. “Solar and Lunar Distances and Apparent Velocities in the Astronomical Tables of Ḥabash al-Ḥāsib”, Al-Abhath, 23:129–176. Reprinted in King and Kennedy (eds.) 1983. Salio, G. (ed.) 1493. Liber quadripartiti Ptholemei; … Venice. Samsó, J. and E. Millás 1998. “The Computation of Planetary Longitudes in the Zīj of Ibn al-Bannāʾ”, Arabic Sciences and Philosophy, 8:259–286. Sédillot, J.-J. and L.-A. Sédillot 1834. Traité des instruments astronomiques des Arabes. Paris. Reprinted Frankfurt a/M (1984). Swerdlow, N.M. and O. Neugebauer 1984. Mathematical Astronomy in Copenricus’s De revolutionibus. New York and Berlin. Thorndike, L. and P. Kibre 1963. A catalogue of incipits of mediaeval scientific writings in Latin. London. Toomer, G.J. 1968. “A Survey of the Toledan Tables”, Osiris, 15:5–174. Toomer, G.J. 1984. Ptolemy’s Almagest. New York and Berlin.

chapter 9

John of Murs’s Tables of 1321*

… To John D. North (1934–2008), in memoriam

∵ John of Murs (fl. 1320–1340), a scholar active in Paris in the first half of the fourteenth century, was a key figure in the history of astronomy in addition to making contributions to music and mathematics. Indeed, his work on astronomy played a decisive role in the transmission of scientific ideas in the late Middle Ages.1 He is largely responsible for the introduction of Alfonsine astronomy into the Parisian milieu, notably a set of astronomical tables originally elaborated in Toledo by the astronomers in the service of King Alfonso of Castile (d. 1284).

* Journal for the History of Astronomy, 40 (2009), 297–320. 1 L. Gushee, “New Sources for the Biography of Johannes de Muris”, Journal of the American Musicological Society, xxii (1969), 3–26; E. Poulle, “John of Murs”, in The Dictionary of Scientific Biography (16 vols, New York, 1970–1980), vii (1973), 128–133; G. Beaujouan, “Observations et calculs astronomiques de Jean de Murs (1321–1344)”, in Proceedings of the xivth International Congress of the History of Science (Tokyo–Kyoto 1974) (Tokyo, 1975), ii, 27–30, reprinted in idem, Par raison des nombres: L’art du calcul et les savoirs scientifiques médiévaux (Aldershot, 1991), Essay vii; J.D. North, “The Alfonsine Tables in England”, in Y. Maeyama and W.G. Salzer (eds), Prismata: Festschrift für Willy Hartner (Wiesbaden, 1977), 269–301; G. l’Huiller, “Aspects nouveaux de la biographie de Jean de Murs”, Archives d’histoire doctrinale et littéraire du moyen âge, xlvii (1980), 272–276; C. Schabel, “John of Murs and Firmin of Beauval’s Letter and Treatise on the Calendar Reform for Clement vi”, Cahiers de l’Institut du moyen-âge grec et latin, lxvi (1996), 187–215; J. Chabás and B.R. Goldstein, The Alfonsine Tables of Toledo (Dordrecht and Boston, 2003); M. Lejbowicz, “Présentation de Jean de Murs ‘observateur et calculateur sagace et laborieux’ ”, in C. Grelland (ed.), Méthodes et statut des sciences à la fin du Moyen Âge (Villeneuve d’ Ascq, 2004), 159–180; R.L. Kremer, “John of Murs, Wenzel Faber and the Computation of True Syzygy in the Fourteenth and Fifteenth Centuries”, in J.W. Dauben et. al. (eds), Mathematics Celestial and Terrestrial: Festschrift für Menso Folkerts zum 65. Geburtstag (Halle [Saale], 2008), 147–160.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_011

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Together with John of Lignères and John of Saxony, he recast these tables into what are now called the Parisian Alfonsine Tables. Although in 1317 John of Murs was a convinced defender of the Toledan Tables, as attested in his earliest known work, beginning Auctores kalendarii …, a few years later he adhered to Alfonsine astronomy and remained faithful to it thereafter. In particular, in his Expositio intentionis regis Alfonsii circa tabulas eius (probably composed in 1321) he explained parameters and models already found in these tables and described many of the features of what he called “the tables of Alfonso”, but remained silent on the way he got access to them. In any case, the Alfonsine material was available to him (and others) in Paris by 1321.2 Reconstructing the transmission of astronomical ideas is a complex task that is especially difficult for a period when scholars rarely mentioned the names of their contemporaries or near contemporaries on whose work they depended, as is the case in the late Middle Ages. In his Expositio John of Murs mentions only one of his predecessors in Paris, William of Saint-Cloud (end of the thirteenth century), and he even reproduces parts of Saint-Cloud’s Almanach Planetarum of 1292 almost word for word (without marking any of these passages as quotations).3 Yet some parameters of Alfonsine origin that John of Murs incorporated into his own tables (such as the maximum value of the solar equation, 2;10°) had already reached Paris, for we find them in the tables of John Vimond for 1320.4 It seems unlikely that John of Murs did not know about John Vimond since both astronomers came from the same region, Normandy, and both worked on planetary tables in Paris at the same time. John Vimond composed his tables “for the use of students at the University of Paris” and was therefore a known participant in the Parisian astronomical community. The texts under consideration in this paper have not previously been studied in detail,5 but we claim that they are of great importance for understanding the transmission of Alfonsine astronomy from Toledo to Paris. In fact, there were at least five texts produced in Paris in 1320 and shortly thereafter that bear on this transmission: (1) John Vimond’s tables of 1320, (2) John of Murs’s

2 Chabás and Goldstein, Alfonsine Tables of Toledo (ref. 1), 277–281. 3 See E. Poulle, “Jean de Murs et les tables alphonsines”, Archives d’histoire doctrinale et littéraire du moyen âge, xlvii (1980), 241–271, especially pp. 261–265; Chabás and Goldstein, Alfonsine Tables of Toledo (ref. 1), 263–264, 279–280. 4 See B.R. Goldstein and J. Chabás, “The Maximum Solar Equation in the Alfonsine Tables”, Journal for the History of Astronomy, xxxii (2001), 345–348; J. Chabás and B.R. Goldstein, “Early Alfonsine Astronomy in Paris: The Tables of John Vimond (1320)”, Suhayl, iv (2004), 207–294. 5 For a brief account, see North, “Alfonsine Tables in England” (ref. 1), 284–285.

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Expositio, (3) John of Murs’s tables of 1321, (4) John of Murs’s Patefit, and (5) John of Lignères’s canons and tables of 1322. We have previously discussed the evidence in items (1), (2), and (5), and in this paper we focus on items (3) and (4).6 As we noted in an article published in 1994 (with José Luis Mancha), one lengthy chapter on planetary velocity in the Latin canons to the tables of the Parisian astronomer, John of Lignères (fl. 1320–1335), is almost identical with a chapter in the Castilian canons to the Alfonsine Tables of Toledo, composed some 50 years earlier.7 As we demonstrate in this paper, John of Murs’s tables of 1321 and the tables for syzygies in his Patefit are based on the same models and parameters that underlie the Parisian Alfonsine Tables (although the formats for these tables are entirely different), indicating that, for matters other than presentation, John of Murs’s contribution to Alfonsine astronomy was made all at once. Some of these parameters are not known from any text or table prior to those of John Vimond and John of Murs, although others can be discerned in Castilian material of the late 13th century.8 Hence, what we learn from items (3) and (4) confirms our results, based on items (1), (2), and (5). In other words, despite the fact that the Alfonsine Tables of Castile are not extant, the evidence we present strongly supports the claim that the Parisian material produced in the 1320s relied on a Castilian tradition associated with King Alfonso x, as John of Murs himself asserts.9 John of Murs compiled several sets of tables. They have not yet been thoroughly examined, except the set called Tabulae permanentes, which is restricted to the computation of the time from mean to true syzygy.10 The tables of 1321 are his first and most extensive set, and they are entirely devoted to the planets and the two luminaries, the Sun and the Moon, that is, matters related

6

7 8 9

10

See, e.g., Chabás and Goldstein, Alfonsine Tables of Toledo (ref. 1), 266–284; Chabás and Goldstein, “John Vimond and the Alfonsine Trepidation Model”, Journal for the History of Astronomy, xxxiv (2003), 163–170; Chabás and Goldstein, “John Vimond” (ref. 4). B.R. Goldstein, J. Chabás, and J.L. Mancha, “Planetary and Lunar Velocities in the Castilian Alfonsine Tables”, Proceedings of the American Philosophical Society, cxxxviii (1994), 61–95. See J. Chabás, “Were the Alfonsine Tables of Toledo First Used by Their Authors?”, Centaurus, xlv (2003), 142–150. For the evidence in John of Murs’s Expositio, see Goldstein and Chabás, “Maximum Solar Equation” (ref. 4), 347 n. 2. For contrary views, see E. Poulle, “The Alfonsine Tables and Alfonso x of Castille”, Journal for the History of Astronomy, xix (1988), 97–113; and idem, “Les astronomes parisiens au xive siècle et l’ astronomie alphonsine”, Histoire littéraire de la France, xliii (2005), 1–54. B. Porres and J. Chabás, “John of Murs’s Tabulae permanentes for finding true syzygies”, Journal for the History of Astronomy, xxxii (2001), 63–72.

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to the daily rotation and trigonometry are not mentioned.11 That John of Murs is the author of these tables is attested by a note accompanying the text that refers to them as “istas tabulas magistri Jo. de muris” (Oxford, Bodleian Library, ms Can. Misc. 501, f. 106v). A close examination of this material throws light on the transmission of Alfonsine parameters, although the author does not address this issue at all. As far as we know, John of Murs’s Tables of 1321 are extant in only two manuscripts: Lisbon, ms Ajuda 52-xii-35 (henceforth ms L); and Oxford, Bodleian Library, ms Can. Misc. 501 (henceforth ms O). They are accompanied by a text (L 65r–66v, O 103r–105v) consisting of about three pages and beginning Si vera loca planetarum per presentes tabulas volueris invenire a tempore incarnationis domini dato perfecto deme 1320 … After roughly one page we read Expliciunt canones super revolutiones planetarum. These very short “canons” give some indication of the way to use the tables, but the text is too condensed to be meaningful for anyone without previous familiarity with them, that is John of Murs was either writing for himself or for a very select audience. This is consistent with a sentence where he refers to his tables: in hac arte nulli scientifico ignotum est. Several comments—or notes—are appended concerning tables for conjunctions with the Sun, mostly on the periods of anomaly of the Moon and the planets. At the end of these comments there is another explicit: Explicit compositio tabularum de certis revolutionibus planetarum. In this text John of Murs avoided the type of canons which explain at length how to use a set of tables, such as those written by John of Saxony in 1327 for the Parisian Alfonsine Tables, filling 30 pages in the editio princeps of the Alfonsine Tables,12 or the Castilian canons of the Alfonsine Tables of Toledo, completed no later than 1272 which explain the use of these tables in 54 chapters.13 Unfortunately, as mentioned above, the original tables that corresponded to these canons are not extant. A description of the tables follows, based on both manuscripts. We note that “physical” signs of 60° are generally, although not systematically, used in both manuscripts (here incorporated into sexagesimal notation such that a circle contains 6,0° = 6 physical signs).

11 12 13

For a brief account of their contents, see Poulle, “Les astronomes parisiens” (ref. 9), 24–26, where these tables are dated no earlier than 1325. Tabule astronomice illustrissimi Alfontij regis castelle, edited by E. Ratdolt (Venice, 1483), ff. a2r–b8v. For a transcription of these canons, see Chabás and Goldstein, Alfonsine Tables of Toledo (ref. 1), 19–94.

312 table 1

chapter 9 Mean conjunctions of the Sun and Saturn (excerpt)

Year

Day

Month

Time (h)

Motus (º)

Centrum (º)

Verus locus (º)

1320 1 2 … 59

100 113 126

Apr 10 Apr 23 May 6

15;17 17;29 19;41

0,27;14 0,39;54 0,52;34

2,15;51 2,28;31 2,41;10

0,22;59 0,36;43 0,50;35

102

Apr 11

20;53

0,29; 8

2,17;15

0,24;59

add

1

5;36

1;54

1;24

1

Tables in L and O

L 26r–v and O 55r–v display a table for the mean conjunctions of the Sun and Saturn (Table 1). It is called tabula principalis in its heading and lists the dates of 58 successive conjunctions of the Sun and Saturn from 1320 to 1359 (where no such conjunction took place in years 21 and 50 after the radix), as well as the corresponding motus (mean motion), centrum (mean argument of center), and verus locus (true longitude) of the planet. The title of this table is: Tabula medie coniunctionis principalis solis et saturni secundum radices per Alfonsum regem Castele (L adds “ultimo”) verificatas super Toletum (L: Tollectum) distans a Parisius in occidente per 48 m. hore (L: 48 hore). It is highly significant that the title in both manuscripts indicates that the radices, i.e., the values corresponding to the initial time, were verified (verificatas) for Toledo by Alfonso x, king of Castile. The title also specifies that Toledo is located 48 minutes of an hour west of Paris (L erroneously reads 48 hours). The same reference to Toledo and to King Alfonso is also found in all the tables discussed below with radices for the rest of the celestial bodies, a clear sign that John of Murs took the initial values from Alfonsine astronomy as computed in Toledo and then converted these data to the meridian of Paris. And this is indeed the case, as we shall see. The first column in Table 1 displays the years from 1320 to 1359; the second column displays the number of days in that year (counted from the last day of the previous year) that have elapsed; the third column replaces the number of days displayed in the second column by the date. Below the table are the amounts to be added to the first entry in a given column to arrive at the final entry in that column.

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As derived from the entries, the time between two successive mean conjunctions of the Sun and Saturn, is 1 year, 13 days, and 2;12h (L 65v and O 103v give a more precise value: 378 days 2;12,13,12h). Thus, in a little over 59 years, 57 mean conjunctions of the Sun and Saturn occur. According to our calculations, using a spreadsheet to compute astronomical positions with the standard Alfonsine Tables,14 a mean conjunction of the Sun and Saturn took place in Toledo on 10 April 1321 at 15;17h at a mean longitude of 27;14,25° for both celestial bodies, when the true longitude of Saturn was 22;57,59° and its mean argument of center was 135;51,34°, in good agreement with the values corresponding to the radix. It follows that the epoch “1320” has to be understood as 1320 completed, that is, 1321 “current”, which is our usual reckoning.15 It also indicates that these entries were computed using a solar model with a maximum value of the solar equation of 2;10° and a mean motion of 0;59,8,19,37,19,13,56°/d, as well as a model for the motion of Saturn with a maximum value of the equation of center of 6;31°, a maximum value of the equation of anomaly of 6;13° and a mean motion of 0;2,0,35,17,40,21°/d, corresponding to the basic parameters of the standard Alfonsine Tables used in the aforementioned programme for the Sun and Saturn.

14

15

For recomputations according to the editio princeps of the Alfonsine Tables (1483), we have used a spreadsheet provided to us by Richard L. Kremer (Dartmouth College, usa), which was prepared by Lars Gislén (Lund University, Sweden). By the standard Alfonsine Tables we mean the collection of tables found in the editio princeps, edited by E. Ratdolt (ref. 12) that, by and large, goes back to a compilation made in Paris in about 1327 with canons by John of Saxony. Among many others, it includes: tables for the differences between the eras, tables to transform dates of various eras, a set of radices for various eras, a table for the movement of the 8th sphere (with a maximum of 9;0°), tables of the mean motions (presented as 60 consecutives multiples of the daily mean motions), equations of the luminaries (with maximum values of 2;10° and 4;56° for the Sun and the Moon), equations of the planets, etc. There is no modern edition of these tables which would have to be based on the vast number of extant manuscripts, but we have examined many manuscripts containing them, and none has exactly the same collection as the first edition. However, for example, Madrid, Biblioteca Nacional, ms 10002, and Vienna, Nationalbibliothek, ms 2288, share most of the characteristics of the editio princeps although, in contrast to it, they do not have a star table. There were two conventions for dates counted from the Incarnation in the Middle Ages: (1) the number of years that have been completed, and (2) the current year which has not been completed. So for a date as we reckon it in current years, e.g., noon, 15 Feb. 1321, one might say equivalently that 1320 years have been completed plus 1 month (January) plus 14 days.

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table 2

Correction for Saturn when not in conjunction with the Sun (excerpt)

w d Days Hours 0,0° hd 0,12° … 2,12° hd 2,24° … 5,36° hd 5,48° hd (º) (′) (º) (º) (′) (º) (º) (′) (º) (′) 1 3 … 27 … 52 54

5 4

12 14;48 25 4;57

0 189

1; 6

1;29 34 2;56 35

0;21 … 2;52 28 1;46 … 1;16 29

1;55 … 0;17 …

3;51 34 5;19 34

2;44 37 4;11 37

5;35 43

4;10 … 1;13 38

2;29 …

8;30 42

7; 5 45

1 365 11;43 9;53 33 8;46 … 7;23 33 8;29 … 12;17 35 11; 8 37 0 378 2;12 11;24 33 10;18 … 9; 7 33 10;13 … 13;44 35 12;34 35

In order to find the true longitude of Saturn when the planet is not in conjunction with the Sun, another table, called contratabula (Table 2), is needed. It appears on O 56r–58v and on three folios in L, labelled successively 27r–v, 26r–v, and 27r–v, for the manuscript has two folios numbered 26 and two numbered 27. In the heading of Table 2, “w” stands for week, “d” for day of the week, and “hd” for half the difference (in minutes of arc) between two successive values of the mean argument of center of the planet. In this double argument table of 34 columns and 30 rows, the first four columns are for the time after a mean conjunction, that is, the “age of Saturn” (given as a number of days, expressed in weeks and days within a week, and hours). The rows are evenly separated, and the last one corresponds to 54 weeks (378 days or 1 year and 13 days) and 2;12h. As indicated above, this is roughly the time between two successive mean conjunctions of Saturn and the Sun, and consequently it is equal to the time between the radix and the first mean conjunction listed in the table called principalis (see Table 1, above). The headings for the rest of the columns represent values of the mean argument of center of Saturn, from 0,0° to 5,48° at intervals of 12°. The entries give, in degrees and minutes, the correction to be added, or subtracted, to the mean motion of the planet to obtain its true longitude. In Figure 9.1 we represent three rows of this double argument table, showing the entries (in minutes of arc on the y-axis) for three selected times (12 days, 189 days, and 378 days) after a mean conjunction and for the mean argument of center from 0° to 360° on the x-axis. As seen at a glance, for a given value of the age of the planet, the entries are distributed along a sinusoidal curve with a period of 360°.

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figure 9.1 Correction for Saturn as a function of its mean argument of center (the curves, from lowest to highest, correspond to 12 days, 189 days, and 378 days, respectively)

To illustrate how the contratabula works, let us derive the true longitude of Saturn exactly one mean conjunction after the radix (text: 36;43°; see Table 1, second row), from the mean values of the preceding conjunction. One begins by entering Table 2 with the mean argument of center (2,15;51°) already given in the Table 1. After interpolation between the values 2,12° and 2,24° for the mean argument of center corresponding to a time of 54 weeks and 2;12h, one finds that 9;29° is the positive correction to be applied. By adding this amount to the mean motion of the previous conjunction, one finds 36;43° (= 27;14° + 9;29°), which is exactly the entry found in Table 1. The same procedure applies to any other time after a mean conjunction. Direct recomputation with the standard Alfonsine Tables confirms this result. L 28r–50r and O 59r–80r display tables for the conjunctions with the Sun (tabula principalis) and corrections (contratabula) for each of the other planets. In all cases, the titles indicate that the tables were computed for Toledo and use the radices given by King Alfonso. The times between successive mean conjunctions with the Sun can be derived from the tables, but the text on L 65r–66v and O 103r–105r gives more precise data: Saturn Jupiter Mars Venus Mercury

378 days 2;12,13,12h 398 days 21;12, 8,24h 779 days 22;22,34h 583 days 22;14, 3h 115 days 21; 5, 2, 7h

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table 3

Mean conjunctions of the Sun and the Moon (excerpt)

Year Days

Time (h)

motus (º)

argum. lune (º)

1 2 … 75 76

18 7

2; 3 5, 5;38 10;51 4,54;55

4,26;53 3,36;41

10 28

13;49 4,58;15 11;22 5,16;39

2, 9;12 1, 4;49 *

* L: 1,44;49.

These values for the periods of anomaly of the planets are not difficult to recompute. For the three superior planets, they result from dividing 360° by the difference between the daily mean motions of the Sun and each planet, whereas for the inferior planets they are obtained by dividing 360° by the daily mean motions in anomaly of each planet. All these periods are consistent with the standard mean motions of the planets in Alfonsine astronomy, as they appear, for example, in the editio princeps of 1483. As was the case for the radices of Saturn, the entries corresponding to those of the other planets can be recomputed from the standard Alfonsine Tables. Therefore, all the values for the mean motions of the planets that were later used in the Parisian Alfonsine Tables are already embedded in the Tables of 1321 by John of Murs. Next we find tables for the conjunctions of the Sun and the Moon (L 50v–57v, O 81r–87r). They share the same format as those for the five planets, and have the same references to King Alfonso and Toledo in the title. The first table (Table 3), also called tabula principalis, lists the dates of the first mean conjunction of the Sun and the Moon for each year in 76 consecutive years, as well as the corresponding motus (mean position) and argumentum lune (mean lunar anomaly). Although no date is specified in Table 3, the first row corresponds to the first conjunction of 1322. Indeed, recomputation with the standard Alfonsine Tables indicates that a mean conjunction between the Sun and the Moon took place at Toledo on 18 January 1322 at 2;3h, when the mean longitude of the Sun and the Moon, here called motus, was 305;38°, in perfect agreement with the tabulated data. The rest of the rows correspond to the first conjunction of the luminaries (occuring in January) of each year after 1322. This table thus covers 76 years, i.e., the least common multiple of a solar cycle of 4 years and a lunar

317

john of murs’s tables of 1321

cycle of 19 years. Contrary to the tables for the planets, this table begins in 1322, and it is possible that the row for the radix (1321, that is, “1320 completed”) is missing in both manuscripts. In that case, one should expect to find 28 January at 17;14h (time), 5,16;21° (mean longitude), and 5,17;5° (mean lunar argument of anomaly), according to our computation for Toledo using the standard Alfonsine Tables. This is confirmed by the table for mean conjunctions given in the Patefit (see Table 14, below). L 51r and O 81r display three short tables related to the Moon and they contain the following data for one lunation: Time Mean motion Mean lunar anomaly Mean argument of lunar latitude Daily mean motion of the Moon

29 days 12;44, 3, 3h * 29; 6,24,12° 25;49, 6,30° 30;40,13,48° 0;13,10,35, 0°/d

* O: 29 days 12;44, [blank], 3h. Once again, the title refers to King Alfonso and indicates that his radices are not included in the mean motion: Tabula mediarum coniunctionum solis et lune infrascriptarum absque radice per Alfonsum. All these data later became characteristic of the Parisian Alfonsine Tables. As was the case for the planets, in order to find the true longitude of the Moon when not in conjunction with the Sun, another table, again called contratabula, is needed. It is displayed on L 51v–57r and O 81v–87r (Table 4). In this double argument table the entries are presented in 60 columns and 30 rows. The vertical argument is the time after a mean conjunction, that is, the “age” of the Moon (given here in days) and the horizontal argument is the mean lunar argument of anomaly from 0,6° to 6,0° at intervals of 6°. Again, “hd” stands for half the differences (in minutes of arc) between two successive values of the lunar argument for a given time. For example, 0;14° is half the difference, rounded to minutes, between 0,11;24° and 0,10;57°, corresponding to 0,6° and 0,12°, respectively, for day 1; hence the entry 14′ in the first column labelled “hd” in that row. The entries, in degrees and minutes, give the correction to be added to the mean motion of the Moon to obtain its true longitude, as can be seen in the following example. Consider the mean conjunction for 18 January 1322, which occurred at 2;3h at Toledo; the mean motion and the mean lunar argument of anomaly at that time were 5,5;38° and 4,26;53°, respectively (see Table 3). If we wish to know the position of the Moon one or more days after that time, we enter in Table 4, with the mean lunar argument of

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table 4

Days

1 2 … 15 … 20 … 29 30

Correction for the Moon when not in conjunction with the Sun (excerpt)

0,6° (º)

hd (′)

0,12° (º)

…

3,0° (º)

…

5,54° (º)

hd (′)

0,0° (º)

hd (′)

0,11;24 0,23;14

14 12

0,10;57 0,22;50

… …

0,14;45 0,29;30

… …

0,12;19 0,24; 7

14 13

0,11;51 0,23;40

13 13

3,19;45

15

3,20;14

…

3,16;18

…

3,18;41

16

3,19;14

16

4,30;10

8

4,29;56

…

4,16;35

…

4,30;28

3

4,30;21

5

0,20;20 * 13 ** 0,19;53 0,32;26 12 0,32; 3

… …

0,23;47 0,38; 6

… …

0,21;15 0,33;14

15 13

0,20;46 0,32;52

13 12

* L: 0,20;6. ** L: 11.

anomaly. By interpolation between the values 4,26° and 4,30° we find, for exactly, say, one day after mean conjunction, 17;56° as the correction to be applied. Adding this value to the mean motion at conjunction, we find 5,23;34°; recomputation based on the standard Alfonsine Tables yields 5,23;34,41°, in very good agreement with it. Figure 9.2 shows the behavior of the correction for the Moon (in degrees) as a function of the mean lunar argument of anomaly for a given time of the synodic month or “age” (taken here as 20 days after a mean conjunction), whereas Figure 9.3 displays the correction of the Moon as a function of the “age” of the Moon for a given value of the mean lunar argument of anomaly (taken here as 4,0°). As was the case for the planets, the values for the mean motion of the Moon as well as the maximum lunar equation (4;56°) and the maximum solar equation (2;10°), all parameters later used in the Parisian Alfonsine Tables, are already embedded in John of Murs’s tables. In his Patefit, John of Murs also compiled other tables for true syzygies, and they will be examined in section 4, below. It is very interesting to contrast John of Murs tables for the mean conjunctions of the luminaries and the position of the Moon (Tables 3 and 4) with those for the same purpose by John Vimond.16 Vimond’s tables are only extant 16

Chabás and Goldstein, “John Vimond” (ref. 4), 213 and 229, Tables 1 and 8.

john of murs’s tables of 1321

319

figure 9.2 Correction for the Moon for a given “age” (20 days)

figure 9.3 Correction for the Moon for a given value of mean lunar argument of anomaly (4,0°)

in one manuscript, Paris, Bibliothèque nationale de France, ms lat. 7286c, and the same manuscript also contains the canons and tables for 1322 by John of Lignères. In Vimond’s tables the epoch of the first mean conjunction is 10 March 1320 (computed for Paris), whereas in John of Murs’s table it is 18 January 1322 (computed for Toledo), indicating that John Vimond’s tables preceded those by John of Murs. Both astronomers use values of the mean synodic month that are almost identical: the value in John Vimond’s tables is not explicit, but it can be derived from them, namely, 29 days 12;44,3,6h

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(= 29;31,50,7,44,35d), whereas that in John of Murs’s table is given as 29 days 12;44,3,3h. The difference amounts to 3 sixtieths of a second of an hour. It also happens that both use double argument tables to find the true position of the Moon between syzygies, and the tables look very similar—although the vertical and horizontal arguments are switched—and the entries are given to the minutes in both cases. However, John of Murs’s table is larger, for the argument of anomaly is taken at 6°-intervals in contrast to 12°-intervals in the tables of John Vimond, and covers 30 days whereas that of Vimond covers only 14 days; hence, it has more than four times as many entries (60 × 30 for John of Murs and 30 × 14 for John Vimond). We note that in John of Murs’s tables the entries for half the differences between the corrections corresponding to two consecutive values of the argument of anomaly are given to one place (minute of arc), whereas in John Vimond’s tables the entries for interpolation represent another quantity, the difference between two consecutive corrections for the same value of the argument of anomaly, and are displayed to two places (second of arc). That is, we are offered two interpolation schemes that complement each other: that of John de Murs applies for a fixed day, whereas that of John Vimond is for a fixed argument of anomaly. The 420 entries for the corrections common to both tables agree, except for copyist’s errors (see Table 5 for a comparison between two selected columns in both tables). This does not happen by chance. We are thus faced with two possibilities: the most likely is that John of Murs used the principle established by John Vimond and expanded the table, but it could also be that they both depended on a table in an otherwise unknown prior text. The approach for the motion of the lunar nodes (L 59v, O 87v) parallels that for the Sun. In this table we are given the true position of the node for “1320” (4,57;14°) and then at four year intervals to 1392, adding a final row for 1393, thus using the standard period of 93 years used by astronomers in the Alfonsine tradition.17 At the bottom of this page is a short table for the increments in years 1, 2, and 3 within the four-year calendaric cycle (for year 1 the entry is 5,4;40° in both manuscripts, whereas they should read 5,40;40°). Another sub-table lists the true motion of the node for the days of the year (given by the month, beginning in January, and the day within the month), at intervals of 6 days. Note that the entry corresponding to 31 December is correctly given as 5,40;40°.

17

E.g. Abraham Zacut: see J. Chabás and B.R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the Transition from Manuscript to Print (Philadelphia, 2000), 60 and 117.

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john of murs’s tables of 1321 table 5

Comparison of extracts ( for mean arguments of anomaly 12° and 180°) of the tables of John Vimond and John de Murs for the correction of the Moon

J. of Murs J. Vimond Days 0,12° 0s12° (º) (s) (º) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 …

0,10;57 0,22;50 0,34;45 0,46;48 0,59; 3 1,11;46 1,24;47 1,38; 5 1,51;53 2; 6;12 2,20;57 2,35;55 2,50;51 3, 5;37 …

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

10;57 22;50 4;45 16;48 29; 3 11;46 24;47 8; 5 21;53 6;12 20;57 5;55 20;51 5;37

…

J. of Murs J. Vimond 3,0° 6s 0° (º) (s) (º)

… …

… … … … … … … … … … … … … … …

0,14;45 0,29;30 0,44; 5 * 0,58;38 1,12;44 1,26;20 1,39;36 1,52;36 2, 5;11 2,17;22 2,29;14 2,40;56 2,52;39 3, 4;27 …

… … … … … … … … … … … … … … …

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

14;45 29;30 14;11 28;38 12;44 26;20 9;36 22;36 5;11 17;22 29;14 10;56 22;39 4;27

* L mg.: al. 11 (i.e., other manuscripts read 0,44;11); O: 0,44;11.

The daily motion of the node resulting from the table is –0;3,10,38,11,34°/d, in good agreement with the same parameter in the Parisian Alfonsine Tables (–0;3,10,38,7,14°/d), and quite far from other parameters historically used for the motion of the nodes. As was the case with the previous tables, the title indicates that the radix was derived from the tables compiled by Alfonso x, King of Castile, for Toledo. And it is indeed so: our recomputation, using the standard Alfonsine Tables for noon of the first day of year 1321, indicates that the true longitude in Toledo was 4,57;12,12°, in close agreement with the entry in the text. Then come tables for the planets: Saturn (L 60r–v, O 88r–v), Jupiter (L 60r and 61r, O 88r and 89r),18 Mars (L 61v–62r, O 89v–90r), Venus (L 62v–63r, O 90r–v), and

18

The columns for the arguments, the equation of center, and the stations of Saturn and

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table 6

Days 12 25 38 … 189 … 353 365 378

Argument of anomaly and argument of center of Saturn (excerpt)

table 7

Anom. (º)

Center (º)

0,12 0,24 0,36

0;25 * 0;51 1;16

0, 6 0,12 0,18

3, 0

6;20

5,36 5,48 0, 0

11;49 12;15 12;40

Equation of center and stations of Saturn (excerpt)

Mean center (º) (º)

Eq. center (º)

Station (º)

5,54 5,48 5,42

0;40 1;17 1;55

22;45 22;47 22;49

1,30

4,30

6;31

24;11

2,48 2,54 3, 0

3,12 3, 6 3, 0

1;25 0;43 0; 0

25;27 25;28 25;30

* L: 0;15.

Mercury (L 63v–64r, O 91v–92r). In addition to the tables for the conjunctions of the planets with the Sun described above, for each of the five planets we are given three tables: one for the argument of anomaly and the argument of center, a second for the equation of center and the stations, and a third for latitude. Excerpts of the three tables for Saturn follow (Tables 6, 7, and 8). In Table 6 the first column is for the argument, expressed in days within a period of anomaly (378 days 2;12,13,12h), as indicated above. The second column represents the mean motion of the argument of anomaly, in degrees, and the third column displays the mean motion of the argument of center, in degrees and minutes. In Table 7 the first two columns represent the mean argument of center, in degrees. The third column is for the equation of center of Saturn, in degrees and minutes. Its maximum, 6;31°, agrees with that in the Almagest, the zij of al-Battānī, and the Toledan Tables, as well as the Parisian Alfonsine Tables. The fourth column represents the first station, in degrees and minutes, but we note that the zodiacal signs (required for the entries to be meaningful) are not given, a feature shared by both manuscripts that will be addressed below.

Jupiter are displayed on a single page, whereas the latitudes of these two planets are presented separately.

john of murs’s tables of 1321

323

The tables for the planets other than Saturn present the same characteristics. The maximum values for the equation of center of the other planets are 5;57° (Jupiter), 11;24° (Mars), 2;10° (Venus), and 3;2° (Mercury). This indicates that John of Murs’s Tables of 1321 adhere to the tradition represented by the Toledan Tables for all planets except for Jupiter and Venus, where the latter have 5;15° and 1;59°, respectively. However, all 5 parameters used by John of Murs were already in Paris at the time, for they are found, once again, in the tables of John Vimond, including the “new” ones for Jupiter and Venus.19 For the stations of Saturn and Jupiter there is no column for the signs, in contrast to each of the other three planets. This is most peculiar in the case of Saturn, for which we are given entries ranging from 22;45° (for 0,6° = 6°) to 25;30° (for 3,0° = 180°), as shown in Table 7. Now, in the Toledan Tables, which use zodiacal signs of 30°, the entries range from 3s 22;44° (for 0s 0°) to 3s 25;30° (for 6s 0° = 180°), with 3s 22;45° for 0s 6°. It would therefore seem that John of Murs, who used physical signs of 60° in this very same table, took the entries for Saturn from a table, such as that in the Toledan Tables, using zodiacal signs of 30°, and omitted the signs. In sexagesimal notation the values displayed would range from 1,52;45° to 1,55;30°. We mentioned the Toledan Tables as an example only, for other tables with the same values, using zodiacal signs, were available in Paris at the time: see, e.g., those of John Vimond which display the same entries, although shifted differently depending on the planet.20 The tables for planetary latitudes are of great interest because they are also presented as double argument tables, a feature for which we know of no precedent. As is often the case, superior and inferior planets are treated differently. Table 8 reproduces an excerpt of the table for Saturn. For the superior planets, the entries, in degrees and minutes, are presented in 7 columns and 31 rows (L 60v–61r and 62r; O 88v–89r and 90r). The vertical argument is the argument of center of the planet, shifted +50° in the case of Saturn, –20° for Jupiter, and with no shift for Mars, in accordance with the instructions given in Almagest xiii.6. In the heading, “hd” stands for half the difference (in minutes of arc) between the entries of two successive columns for a fixed argument of center. The extremal values appear in the column corresponding to 180°: Saturn +3; 2° (North) –3; 5° (South) Jupiter +2; 5° (North) –2; 8° (South) Mars +4;21° (North) –6;30° (South) 19 20

Chabás and Goldstein, “John Vimond” (ref. 4), 244. Chabás and Goldstein, “John Vimond” (ref. 4), 242.

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table 8

Center (º) 310 316 … 34 40 46 … 124 130

Latitude of Saturn (excerpt)

0°/360° hd 30°/330° (º) (′) (º)

…

150°/210° hd 180°/180° (º) (′) (º)

310 304

2; 3 2; 1

3 3

2; 8 2; 7

… …

2;57 2;55

3 3

3; 2 3; 0

226 220 214

0;13 0; 0 0;13

0 0 0

0;14 0; 0 0;13

… … …

0;19 0; 0 0;19

0 0 0

0;19 0; 0 0;20

136 130

2; 0 2; 1

2 2

2; 3 2; 5

… …

2;55 2;57

4 4

3; 3 3; 5

We note that the northern (positive) and southern (negative) limits for Saturn agree with the tradition (Almagest, al-Battānī, Toledan Tables, and also Vimond) but differ from the values that were to become part of the Parisian Alfonsine Tables (+3;3° and –3;5°, respectively). As for Jupiter, the northern limit agrees with that in the Toledan Tables and the tables of Vimond, and departs slightly from that in the Almagest and the zij of al-Battānī (+2;4°) and in the Parisian Alfonsine Tables (+2;8°), whereas the southern limit agrees with all the sets of tables mentioned so far. This is also the case for the northern limit of Mars, but not for its southern limit (Almagest and al-Battānī: –7;7°; Toledan Tables, Vimond’s tables, and Parisian Alfonsine Tables: –6;30°). For each of the inferior planets, we are given two double-argument tables for latitude (L 63r and 64r¸ O 91r and 92r): one for the inclination (here called declinatio) and one for the slant (reflexio). The entries, in degrees and minutes, are displayed in 7 columns and 16 rows. Tables 9 and 10 display excerpts for the latitude of Venus. In all four tables (two for each of the inferior planets), the first four columns are for the argument, which is shifted +60° in the table for the inclination of Venus and +90° in the table for the inclination of Mercury. They are small double argument tables, with 7 columns and 16 rows, where the maximum value for Venus is 7;22° (cf. Almagest and the zij of al-Battānī: 6;22°; Toledan Tables: 7;24°; and Vimond’s tables and the editio princeps of the Alfonsine Tables: 7;12°) and that for Mercury, 4;5° (in agreement with the values in the Almagest, the zij of al-Battānī, the Toledan Tables, and Vimond’s tables, as well as in the editio princeps of the Alfonsine Tables). Most notable is the

325

john of murs’s tables of 1321 table 9

Latitude of Venus, inclination (excerpt)

Center (º) 300 306 … 342 … 24 30

0°/360° hd 30°/330° … 150°/210° (º) (′) (º) (º)

hd (′)

180°/180° (º)

300 120 120 244 114 126

1; 3 1; 3

3 3

0;57 0;57

… …

3; 3 3; 2

129 127

7;22 7;16

258

78 162

0;47

3

0;42

…

2;15

95

5;25

216 210

36 204 30 210

0; 7 0; 0

0 0

0; 6 0; 0

… …

0;18 0; 0

15 0

0;47 0; 0

presentation of a column for the 3rd component of latitude, or deviation (called 3ª latitudo in L), in the tables for the slants of both inferior planets. This is indeed a very unusual feature in medieval tables, for there are not many which list values for the deviation. These values were mentioned, but not tabulated, in Almagest xiii.6, with a limit of 0;10° (north) for Venus and a limit of 0;45° (south) for Mercury. It is significant that the canons to the Castilian Alfonsine Tables explicitly addressed this particular problem, giving instructions to take into account the third component of latitude appearing in the tables they describe.21 To this we can add that John Vimond included the deviation of Venus and Mercury in his own tables and used the values mentioned in the Almagest.22 As can be seen in Table 10, John of Murs gave 0;10° (septentrionalis) for the maximum deviation of Venus, but for that of Mercury he used 0;23° (meridionalis), a value about half that of Ptolemy and all others who worked in the Ptolemaic tradition. There are no tables for the third component of latitude in the Parisian Alfonsine Tables. The last table in both manuscripts is for the days in a year (L 64v, O 102v), where each day of the year is assigned an ordinal number from 1 to 365. In addition to the contents of John of Murs’s Tables of 1321 that appear in both manuscripts, there are a few other tables closely related to them that are only found in one of the two manuscripts.

21 22

Chabás and Goldstein, Alfonsine Tables of Toledo (ref. 1), 42–43. Chabás and Goldstein, “John Vimond” (ref. 4), 257–258.

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table 10

Latitude of Venus, slant (excerpt)

Center (º) 0 6 … 42 … 84 90

2

0°/360° hd 30°/330° … 150°/210° hd 180°/180° 3rd lat. (º) (′) (º) (º) (′) (º) (′)

360 180 180 354 174 186

0; 0 0; 0

20 20

0;41 0;41

… …

2;22 2;20

70 70

0; 0 0; 0

10 10

318 138 222

0; 0

15

0;30

…

1;46

53

0; 0

8

276 270

0; 0 0; 0

2 0

0; 4 0; 0

… …

0;15 0; 0

7 0

0; 0 0; 0

1 0

96 264 90 270

Tables in L but not in O

On L 57v–58v there is a table for the true positions of the Sun for each day in a year: see Table 11. The table is also called principalis, and in the title we are again told that the radices were computed by King Alfonso of Castile for the city of Toledo. The entries are given in physical signs, degrees, minutes, and seconds. The date for which the table is valid is not indicated, but it corresponds to 1321, as shown from the recomputations displayed in Table 11, where the true positions of a sample of entries are compared with recomputations for 1321. Note that the values for text and computation do not differ by more than 7 seconds. Reinforcing our claim that Vimond’s tables were composed prior to those by John of Murs is the fact that Vimond displays a one-year calendar with syzygies valid for 1320. It would seem odd for either astronomer to compile a one-year table for the (recent) past. It is far more likely that such a table was produced at (or near) the beginning of the year in question. The Sun has also its contratabula (L 59r), of which an excerpt is given in Table 12, showing the extremal values of the various columns. Note that this contratabula differs from those reviewed above, among other things for the fact that it is not a double argument table, although its purpose is substantially the same. The argument in Table 12 is the day of the year given at intervals of 6 days, beginning in January. The second column lists the correction to be subtracted from the true position of the Sun for 1321, found in the preceding table, in order to determine its true position for dates corresponding to one, two, or three years after 1321. For example, let us consider the true longitude of the Sun for

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john of murs’s tables of 1321 table 11

True solar positions for 1321 (excerpt)

Text (t) Jan. 1 Feb. 1 Mar. 1 Apr. 1 May 1 June 1 July 1 Aug. 1 Sep. 1 Oct. 1 Nov. 1 Dec. 1

4,49;47, 5° 5,21;17,33° 5,48;19,14° * 0,19;46,14° 0,48;43, 1° 1,18;16,45° 1,46;47,35° 2,16;24,35° 2,46;20,24° ** 3,16; 2, 2° 3,47;12,50° 4,17;47,50°

t–c Computation (c) (in seconds) 4,49;47, 6° 5,21;17;33° 5,49;19,13° 0,19;46,18° 0,48;43, 1° 1,18;16,45° 1,46;47,36° 2,16;24,35° 2,46;26,24° 3,16; 2, 2° 3,47;12,50° 4,17;47,57°

–1 0 +1 –4 0 0 –1 0 0 0 0 –7

* Instead of 5,49;19,14°. ** Instead of 2,46;26,24°.

6 January 1322. The procedure is to note the true solar longitude exactly one year before, 6 January 1321, which is given in Table 11 as 294;53,39°. Then we have to subtract from it the difference for one year displayed in the column labelled “correction” for that date in Table 12, and it is 0;14;52°; the result is 294;38,47°, in good agreement with our recomputation based on the standard Alfonsine Tables (294;38,27°). Note that the column for the correction exhibits a minimum value of 0;13,48° (31 May–30 June) and a maximum value of 0;14;56° (6–18 December). The approach to finding the position of the Sun in these two tables is strongly reminiscent of an almanac, in the sense that we are given the true position of the Sun at noon for all days in a year (tabula principalis) and a simple procedure to find its true position at noon for each day in the next three years (contratabula), with no indication of the underlying parameters and no rule for computing true positions beyond this 4-year period. The third column gives the hourly velocity of the Sun in minutes and seconds per hour, from 0;2,23°/h to 0;2,34°/h, for purposes of interpolation between the entries in the previous column. These are the same limits as in the Parisian Alfonsine Tables.23 23

See, e.g., Tabule astronomice illustrissimi Alfontij (ref. 12), ff. g6r–g7r.

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table 12

Contratabula for the Sun (excerpt)

Month Day Correction Velocity Eq. of time (′) (′/h) (min) Jan … Jan … May … Jun … Jul … Oct … Dec …

6

14;52

2;33

5; 2

31

14;43

2;32

0; 0

6

13;55

2;24

20;58

18

13;48

2;23

15;30

24

13;55

2;24

12; 0

24

14;43

2;32

32;38

12

14;56

2;34

16;53

In the fourth column we find the equation of time. There are not many medieval astronomical tables where the argument is expressed in days of the year and the entries are given in time, as is the case for this table. In fact, we only know of two of them, one in Abraham Zacut’s Ḥibbur and another in the Tabule Verificate for Salamanca.24 In Table 12, the extremal values of the entries are the following: Min: 0; 0h max: 0;20,58h min: 0;12, 0h Max: 0;32,38h

24 Jan–12 Feb 6 May 18–24 July 24 Oct

When converted into time-degrees, these extremal values correspond to 0°, 5;14,30°, 3;0°, and 8;9,30°. As far as we know, these values for the equation of time are unprecedented in the astronomical literature. In particular, this table differs both in format and in content from the table of Peter of St. Omer for

24

See Chabás and Goldstein, Abraham Zacut (ref. 17), 108–109.

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john of murs’s tables of 1321

1292–1293, another astronomer working in Paris shortly before John of Murs, and reproduced by John of Lignères, with the following extremal values:25 Min: 0; 0° max: 5;21° min: 2;49° Max: 7;57°

Aqr 18–25° Tau 25–27° Leo 5° Sco 8–9°

However, all other parameters for the Sun in these tables are those that are found in the Parisian Alfonsine Tables.

3

Tables in O but not in L

On O 92v–98r there are five tables for the conjunctions of the planets with the Sun that are very similar to those described in section 1. However, the entries are different, for they correspond, as indicated in the titles, to 1452 “perfecto” (i.e., 1453 in the usual reckoning) and were computed for Paris, not Toledo. The titles add that the radices are those of Alfonso x, king of Castile. For each planet we are only given the tabula principalis, on the assumption that the corresponding contratabula to be used is the one already found among John of Murs’s Tables of 1321. Note also that the column for the true position has been eliminated and the accuracy of the entries has been improved. Table 13 displays an excerpt of the table for the mean conjunctions of the Sun and Saturn, beginning in 1453 (1452 completed). table 13

Year

1452 1 2 … 59

25

Mean conjunctions of the Sun and Saturn for 1453 (excerpt)

Day Month of the year and day

Time (h)

med. motus centrum med. (º) (º)

283 296 309

Oct 10 Oct 23 Nov 6

10; 9,14 12;21,27 14;33,40

3,28;20,24 3,41; 0,17 3,53;40,10

5,15;37,31 5,28;16,49 5,40;56, 7

285

Oct 11

15;45,42

3,30;14, 9

5,16;58, 0

See Chabás and Goldstein, Alfonsine Tables of Toledo (ref. 1), 186.

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It is clear that these tables do not belong to John of Murs’s Tables of 1321 and are simply an extension of them made more than a century later, indicating that his table was taken as a model by at least one subsequent user. The same can be said about the tables on O 98r–99v concerning the conjunctions of the Sun and the Moon beginning in 1453 (1452 completed), which follow the pattern of Table 3, above.

4

Syzygies in the Patefit

In addition to the Tables of 1321, John of Murs composed another work containing tables, associated with the canon beginning Patefit ex Ptolomei disciplines in libro suo …, and traditionally dated 1321.26 The Patefit survives in very few copies. Three manuscripts have previously been noted: London, British Library, Royal ms 12.c.xvii; Erfurt, Biblioteca Amploniana, mss 4° 360; and Erfurt, Biblioteca Amploniana, 4° 371.27 We can now add to this short list Vatican, Biblioteca Apostolica, ms lat. 3116. We have also found other manuscripts containing extracts of this work, as is the case with the Lisbon manuscript itself (L 1r–22v), described above in relation to the Tables of 1321. Among the tables associated with the Patefit there are five directly concerning syzygies, two of which are also double argument tables. For the description of the tables, below, we have followed the London manuscript (henceforth called B), where zodiacal signs are used (except in one case), in contrast to the Tables of 1321 and the Lisbon manuscript where physical signs are used. The first is a table for the mean conjunctions and oppositions for a period of 76 years from 1321 to 1396 (expressed in current years), covering 1880 (= 940 × 2) successive syzygies. The table has no title and, under the headings “conjunctions” and “oppositions”, four quantities are given in each case: time in days, hours, and minutes; mean motion of the Sun in zodiacal signs, degrees, and minutes; mean lunar anomaly in zodiacal signs, degrees, and minutes; mean argument of lunar latitude in zodiacal signs, degrees, and minutes (see Table 14, an excerpt of B 155v–168r). The entries in Table 3, above (limited to mean conjunctions), agree with the corresponding entries in Table 14 of the Patefit, except for the minutes in a few cases. We also note that in this table, and in general in the Patefit, years

26 27

According to Poulle, this work was probably composed in the late 1320s: cf. Poulle, “Les astronomes parisiens”, (ref. 9), 26–27. See, e.g., Kremer, “John of Murs, Wenzel Faber” (ref. 1), 148.

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john of murs’s tables of 1321 table 14

Jan Feb Mar … Oct Nov Dec

Mean conjunctions and oppositions for 1321–1396 (excerpt)

1321 Conjunctions Day Time motus arg. lune arg. lat. (h) (s) (º) (s) (º) (s) (º)

1321 Oppositions Day Time motus arg. lune arg. lat. (h) (s) (º) (s) (º) (s) (º)

28 17;14 10 16;21 10 17; 5 27 5;58 11 15;28 11 12;54 28 18;42 0 14;34 0 8;43

13 22;52 10 1;48 4 4; 9 6 12 11;36 11 0;54 4 29;59 7 14 0;20 0 0; 1 5 25;48 8

21 11;51 20 0;35 19 13;19

7 8;19 8 7;25 9 6;32

0 20;40 1 21;20 2 22; 1

6 9;26 9 26;42 7 5;15 10 27;22 8 1; 4 11 28; 2

1322 Conjunctions Day Time motus arg. lune arg. lat. (h) (s) (º) (s) (º) (s) (º) Jan 18 2; 3 10 5;38 8 26;53 0 28;42 Feb 16 14;47 11 4;44 * 9 22;42 * 1 29;22 * Mar 18 3;31 0 3;51 10 18;31 3 0; 3 …

5;20 6; 1 6;41

6 17;28 6 23;46 11 26;31 3 11;22 5 6;13 7 22;52 0 22;20 4 12; 2 4 18;57 8 21;58 1 18; 9 5 12;42 1322 Oppositions Day Time motus arg. lune arg. lat. (h) (s) (º) (s) (º) (s) (º) 3 7;41 9 21; 5 2 13;58 6 13;22 1 20;25 10 20;11 3 9;47 7 14; 3 3 9; 9 11 19;18 4 5;36 8 14;43

* In L these entries are given as 5,34;45° (= 11s 4;45°), 4,52;43° (= 9s 22;43°), and 1,0;22° (= 2s 0;22°), respectively. … 1396 Conjunctions Day Time motus arg. lune arg. lat. (h) (s) (º) (s) (º) (s) (º) Jan 10 13;49 9 28;16 … Dec 0 9;53 8 18;27 Dec 29 22;48 * 9 17;33

4 9;15

0 12;15

1 23;14 11 19;34 2 19; 5 0 20;– **

1396 Oppositions Day Time motus arg. lune arg. lat. (h) (s) (º) (s) (º) (s) (º) 25

8;11 10 12;49 10 22;10 6 27;33

15 15;32 8 3;53 7 10;20 5 15 4;16 9 3; 0 8 6; 9 6

4;15 4;55

* In L this entry is given as 22;38h. ** Vatican, ms lat. 3116: 0s 20;14°. The column for the zodiacal signs corresponding to the lunar anomaly for oppositions is missing in this manuscript in most cases.

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refer to the current year. Thus, Table 14 was also computed for the meridian of Toledo, following the radices set up by King Alfonso, a feature not specified in the title. The same table is found in Vatican, lat. 3116, ff. 11r–23v, and all the entries displayed above agree with those in B.28 In L the information contained in this table is split into four different tables: the times of 940 successive mean conjunctions for the 76-year period, 1321–1396; the mean motion of the Moon for 235 successive conjunctions (that is, a lunar cycle of 19 years); the mean lunar anomaly for 251 successive conjunctions (that is, the cycle according to which 251 synodic months = 269 returns in lunar anomaly); and the mean argument of lunar latitude for 223 successive conjunctions (that is, the Saros cycle of about 18 years): for these cycles see, e.g., Almagest iv.2.29 The second table displays the times of true conjunctions and oppositions for a period of 76 years (1321–1396), given in days, hours, and minutes (see Table 15, an excerpt of B 168r–172r). The title, Tabula continens veras coniunctiones et oppositiones ad Tholetum per 48 m. hore distante a Parisius, indicates that the entries were computed for Toledo, which is distant by 48 minutes of an hour from Paris. This table is not in Vat. lat. 3116. In the heading of Table 15, “m” stands for month. As expected, the entries in this table were computed with Alfonsine models and parameters for the Sun and the Moon. The third table deals with true conjunctions although in the three manuscripts we have examined (London, Vatican, and Lisbon) the title refers both to conjunctions and to oppositions: Tabula vere coniunctionis et opposicionis solis et lune. Five quantities are given: true longitude of the Moon at mean conjunction, in physical signs, degrees, and minutes; lunar time correction, in hours and minutes; hourly lunar velocity, in minutes and seconds; true longitude of the Sun at mean conjunction, in physical signs, degrees, and minutes; solar time correction, in hours and minutes (see Table 16, an excerpt of B 172v–174r). The columns dealing with the Moon cover 251 conjunctions, whereas those for the Sun only cover 235 conjunctions. It is noteworthy that in this table physical signs of 60° are used, contrary to the other tables in the Patefit. Also in L 9r–12v physical signs are used. The same table is found in Vatican, lat. 3116, ff. 24r–25v. Note that the time correction for the Moon and the time correction for the Sun are the two terms in which the time from mean to true syzygy, Δt, is divided. 28

29

On L 8r–v there is another table containing much the same information with the title, Tabula medie coniunctionis solis et lune in annis ad meridiem Tholeti secundum radices Alfonsii regis castelle. We are only given entries for the last conjunction of each of the 76 years, but with a higher accuracy, both for time (to seconds of an hour) and for the three other quantities (to seconds of an arc). G.J. Toomer, Ptolemy’s Almagest (New York and Berlin, 1984), 174–176.

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john of murs’s tables of 1321 table 15

Time of true conjunctions and oppositions for 1321–1396 (excerpt)

1321 Conj. Time (h) (d)

Opp. (h)

m.

Time (d)

28 27 29

13;58 7;29 0;36

14 12 14

9;16 20;16 5; 8

1 2 3

21 19 19

7;12 17;29 5; 6

6 5 5

12;54 5;17 1;32

10 11 12

m.

Time (d)

1 2 3 … 10 11 12

1322 Conj. Time (h) (d)

Opp. (h)

…

1396

18 16 18

18;45 9;54 1;36

3 2 3

18;47 9;35 21;26

… … …

10 9 9

23;24 9;41 0; 2

25 24 23

7;22 2; 7 21;17

… … …

To illustrate this, consider the mean conjunction for 28 January 1321, occurring at 17;14h after noon (see Table 14). The true conjunction occurs at 13;58h (see Table 15). Thus, true conjunction precedes mean conjunction, and Δt = –3;16h. As readily seem in Table 16, the sum of the lunar correction (–6;40h) and the solar correction (+3;24h) is –3;16h. Computing the time from mean to true syzygy is an issue that interested many medieval astronomers and they offered a variety of methods to give a proper answer.30 In a recent paper Kremer analysed this table and argued convincingly (1) that cols. 2 and 5 are based on the parameters for the lunar and solar corrections that later appeared in the Parisian Alfonsine Tables; (2) that col. 4 is based on the lunar velocity table in the Toledan Tables (which is identical with the corresponding table in the zij of al-Battānī); and (3) that cols. 3 and 6 are based on a computational scheme similar to the one that underlies the table by Nicholaus de Heybech (fl. 1400) for finding the time from mean to true syzygy, separating the solar and lunar components.31 The fourth table has the heading Tabula veri motus lune ad dimidiam lunationem. Argumentum lune ad coniunctionem mediam inventum (see Table 17, an excerpt of B 177r–182v). It is a double argument table where the horizontal scale is for the argument of lunar anomaly, ranging from 0s 3° to 12s 0° at 3° intervals, and the vertical scale is the number of days after mean conjunction or oppo-

30 31

See J. Chabás and B.R. Goldstein, “Computational Astronomy: Five Centuries of Finding True Syzygy”, Journal for the History of Astronomy, xxviii (1997), 93–105. For additional details, see Kremer, “John of Murs, Wenzel Faber” (ref. 1), 148–155.

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table 16

True conjunctions: positions and velocities (excerpt)*

True lunar Time correction Hourly lunar True solar Time correction position for the Moon veloc. position for the Moon (º) (h) (′/h) (º) (h) 1 2 3 … 235 236 … 250 251

5,19;30 5,46;45 0,13;49

–6;40 –2;50 1;29

30;52 30;24 30;20

5,17;58 5,47;35 0,16; 8

3;24 4;31 4;25

4,51;42 5,21;22

–8;17 –9;44

34;20 32;59

4,48; 2 0, 0; 0

1;20 0; 0

0, 8;51 0,37;29

–9;44 –9; 5

32;56 31;46

0, 0; 0 0, 0; 0

0; 0 0; 0

* We have added a minus sign (–) to the entries under deme (subtract) in the table.

sition, up to 16 days. The same table is found in Vatican, lat. 3116, ff. 29r–34v. An extract is also found in Brussels, Bibliothèque Royale, ms 1086–1115, f. 29v. This table is essentially the same as Table 4 but has twice as many columns and about half as many rows. Another difference is the use of zodiacal signs rather than physical signs. And yet the most striking difference is that the entries common to both tables differ slightly in a systematic way. In Table 18 we compare excerpts of these two tables of John de Murs, one belonging to the Tables of 1321 (Table 4) and other to the Patefit (Table 17). table 17

True lunar positions for each day between successive syzygies (excerpt)

Days

0s 3° (s) (º)

diff. 0s 6° (′) (s) (º)

…

3s 0° (s) (º)

… 11s 27° diff. 12s 0° (s) (º) (′) (s) (º)

1 2 3 … 15 16

0 11;36 0 23;24 1 5;12

14 13 11

0 11;22 … 0 8;10 … 0 12; 4 14 0 23;11 … 0 21;38 … 0 23;51 13 1 5; 1 … 1 5;26 … 1 5;36 12

0 11;50 14 0 23;38 14 1 5;24 12

6 19;30 * 16 7 4; 7 * 13

6 19;44 … 6 22;19 … 6 18;57 17 7 4;20 … 7 4;54 … 7 3;39 14

6 19;14 16 7 3;53 16

* Vatican, ms lat. 3116 has mistakenly 19;16 and 4;13, respectively.

diff. (′)

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john of murs’s tables of 1321 table 18

Comparison of extracts ( for mean arguments of anomaly 12° and 180°) of the two tables of John of Murs for the correction of the Moon

t. of 1321 Patefit Days 0,12° 0s12° (º) (s) (º) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0,10;57 0,22;50 0,34;45 0,46;48 0,59; 3 1,11;46 1,24;47 1,38; 5 1,51;53 2; 6;12 2,20;57 2,35;55 2,50;51 3, 5;37 3,20;14 3,34;38

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

10;55 22;47 4;41 16;43 28;58 11;41 24;42 8; 0 21;48 6; 8 20;54 5;53 20;51 5;38 20;15 4;44

…

t. of 1321 Patefit 3,0° 6s 0° (º) (s) (º)

… …

… … … … … … … … … … … … … … … …

0,14;45 0,29;30 0,44; 5 * 0,58;38 1,12;44 1,26;20 1,39;36 1,52;36 2, 5;11 2,17;22 2,29;14 2,40;56 2,52;39 3, 4;27 3,16;18 3,28;14

… … … … … … … … … … … … … … … …

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

14;47 29;33 14;15 28;43 12;49 26;25 9;41 22;41 5;16 17;26 29;17 10;58 22;40 4;27 16;18 28;11

* L mg.: al. 11; O: 0,44;11.

As is readily seen, the corresponding entries in these tables differ systematically in the minutes, and the differences range from –0;5° to +0,5°. We note, however, that when computing the true position of the Moon exactly one day after the mean conjunction of 18 January 1322 at Toledo we found it to be 5,23;34°, using the correction deduced from Table 4 (17;56°) and in good agreement with recomputation, 323;34,41°. Had we used Table 17, the correction to be applied would have been 18;1°, and thus the true position of the Moon 5,23;39°, in worse agreement with recomputation. There is a fifth table whose heading is Tabula invencionis veri loci lune incipiendo a coniunctione eius a sole (B 183r–188v). Again, it is a double argument table, for 30 days and the argument of lunar anomaly, and very similar to Tables 17 and 4, the latter of which is also for 30 days. This table is not in Vat. lat. 3116.

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Conclusions

John of Murs’s Tables of 1321 are exclusively concerned with the luminaries and the planets. Their most significant feature is the organizational principle: the mean motions of the planets are presented in tables for the mean conjunctions of each planet with the Sun, and their equations are given in double argument tables. This approach meant that astronomers could avoid a lot of cumbersome computations to determine true planetary positions, compared with using tables previously available in Latin. In setting up his tables that way, John of Murs took advantage for the planets of the pattern used for the Moon in its conjunctions with the Sun. We know of no other example of this presentation for the planets in the Alfonsine corpus. On the other hand, as far as we know, double argument tables were a novelty in Europe. North noted that they had been used by Ibn Yūnus (tenth century, Cairo) and al-Baghdādī (thirteenth century, Baghdad), but we are not aware of any double argument tables produced in the Iberian Peninsula prior to 1320.32 In the tables devoted to syzygies in the Patefit, John of Murs split the time from mean to true syzygy, Δt, into two separate terms, one for the Sun and one for the Moon, and was probably the first to introduce this approach to a complicated problem. The tables he compiled, among them double argument tables, helped practitioners of astronomy in completing their tasks by reducing the number of required computations to reach a certain result. John of Murs compiled his tables with material already in place in Paris and, in particular, there is evidence that he shared parameters and approaches with his contemporary, John Vimond. Indeed, we have shown that there is a stronger relationship between John Vimond’s work and John of Murs’s than previously thought. As repeatedly indicated in the titles of the tables, John of Murs used radices for Toledo which he attributed to Alfonso, king of Castile. No wonder, for John of Murs had a thorough knowledge of what he called “the tables of Alfonso” that he had described in his Expositio intentionis regis Alfonsii circa tabulas eius. Indeed, in his Tables of 1321 John of Murs used parameters that occur in that text. As Lejbowicz put it regarding John of Murs, “l’appropriation des héritages fournit aux novateurs l’ appui nécessaire à leur travail”.33 Moreover, it is also clear that all basic parameters for the planets and the two luminaries (for mean motions and equations) that some years

32 33

North, “Alfonsine Tables in England” (ref. 1), 279 and 293; Chabás and Goldstein, “Finding True Syzygy” (ref. 30), 93 and 104 n. 3. Lejbowicz, “Présentation de Jean de Murs” (ref. 1), 175.

john of murs’s tables of 1321

337

later formed the core of the Parisian Alfonsine Tables are already embedded in his Tables of 1321. Hence credit should properly be given to John of Murs for these innovations that played a decisive role in the transmission of Alfonsine astronomy.

Acknowledgements We thank Richard L. Kremer (Dartmouth College) for his helpful comments on a draft of this paper.

chapter 10

Isaac Ibn al-Ḥadib and Flavius Mithridates: The Diffusion of an Iberian Astronomical Tradition in the Late Middle Ages* Isaac Ibn al-Ḥadib (or al-Aḥdab) first appears in the literature, when he was in Castile in the 1370s, as a student of Judah ben Asher ii (then resident in Burgos), the great-grandson of Asher ben Yeḥiel of Cologne (d. c. 1328) who became chief rabbi of Toledo in 1305. Judah ben Asher ii (d. 1391) composed a set of astronomical tables that are poorly preserved in a unique copy;1 he was killed during the anti-Jewish riots that took place all over Spain beginning in 1391 and, as a result of these riots, many Jews left Spain around that time. Ibn al-Ḥadib was a member of a prominent Jewish family in Castile and arrived in Sicily no later than 13962 when the island was ruled by Joan i (d. 1396), King of Aragon and eldest son of Pere iii of the house of Barcelona. Both Pere and Joan were keen on astronomy and had Jewish scholars at the royal court.3 Ibn al-Ḥadib’s main astronomical work was a set of tables in Hebrew for conjunctions and oppositions of the Sun and the Moon, called Oraḥ selulah (“the paved way”: cf. Prov. 15:19). This text is mentioned by several later astronomers, notably by Abraham Zacut in chapter 5 of his Great composition (ha-Ḥibbur ha-gadol), composed in 1478.4 Ibn al-Ḥadib died in Sicily around 1426. Flavius Mithridates was a name assumed by William Raymond of Moncada, a shadowy figure who was active in Italy in the late 15th century.5 Mithridates * Journal for the History of Astronomy, 37 (2006), 147–172. 1 B.R. Goldstein, “Abraham Zacut and the Medieval Hebrew Astronomical Tradition”, Journal for the History of Astronomy, xxix (1998), 177–186, p. 179. 2 M. Steinschneider, Mathematik bei den Juden, 2nd edn (Hildesheim, 1964), 168; B.R. Goldstein, “Descriptions of Astronomical Instruments in Hebrew”, in Essays in Honor of E.S. Kennedy, ed. by D.A. King and G. Saliba, Annals of the New York Academy of Science, d (1987), 105–141, p. 128. 3 A. Rubió i Lluch, Documents per l’història de la cultura mig-eval (Barcelona, 1908–1921); J.M. Millás, Las Tablas Astronómicas del Rey Don Pedro el Ceremonioso (Madrid–Barcelona, 1962); J. Chabás, with the collaboration of A. Roca and X. Rodríguez, L’astronomia de Jacob ben David Bonjorn (Barcelona, 1992). 4 See, e.g., Oxford, Bodleian Library, ms Opp. Add. 8° 42, f. 17b. Cf. F. Cantera Burgos, “El judío salmantino Abraham Zacut”, Revista de la Academia de Ciencias Exactas, Físico-Químicas y Naturales de Madrid, xxvii (1931), 63–398, espec. pp. 113, 171. 5 See K. Lippincott and D. Pingree, “Ibn al-Ḥātim on the Talismans of the Lunar Mansions”,

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_012

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reports that he was the son of a Sicilian rabbi, Nissim Abū l-Faraj (“Rabi nissim abu ilfaragh”: Vatican, ms Urb. lat. 1384, f. 3a).6 It has been established that he converted to Christianity in the 1460s and that before his conversion his name was Samuel ben Nissim.7 His new name was presumably chosen in honour of his first patron, William Raymond of Moncada, Count of Adernò (that is, Adrano, in Sicily). A document dated 1473 indicates that Guillelmo Raymundo de Moncata was a student at the University of Naples, and some evidence shows that by 1477 Mithridates had reached Rome where he attracted the attention of Federico of Montefeltro, Duke of Urbino (1422–1482), among others.8 Federico played a major role in political and military affairs in Italy and he was also a leading patron of the arts during the Renaissance. Mithridates is best known as an adviser to the great humanist, Giovanni Pico della Mirandola (d. 1494), for whom he translated many kabbalistic texts from Hebrew into Latin around the year 1486.9 We will focus on Mithridates’s interest in astronomy, rather than kabbalah, and on his astronomical tables, uniquely preserved in Vatican, ms Urb. lat. 1384. Samuel’s father was a student of Isaac Ibn al-Ḥadib10 and, as we shall see, there is a strong connection between the astronomical tables of Ibn al-Ḥadib and those of Mithridates. The main purpose of Ibn al-Ḥadib’s tables is to compute the time and position of true syzygy, and the circumstances for solar and lunar eclipses.11 Some of

6 7

8 9

10 11

Journal of the Warburg and Courtauld Institutes, l (1987), 57–81, pp. 58–59. Although William Raymond of Moncada did not take Flavius Mithridates as his nom de plume until the 1480s, we generally refer to him by the name by which he is best known. Cf. R. Starrabba, “Ricerche storiche su Guglielmo Raimondo Moncada, ebreo convertito siciliano del xv”, Archivio Storico Siciliano, iii (1878), 15–91, p. 87. A. Scandaliato, “Le radici familiari culturali di Guglielmo Raimondo Moncada, ebreo convertito del rinascimento, nella Sicilia del. sec. xv”, in Una Manna Buona per Mantova. Man Tov le-Man Tovah. Studi in onore di Vittore Colorni, ed. by M. Perani (Florence, 2004), 203–240; E. Engel, “A Palaeographical Analysis of Mithridates’ Hebrew Autographs”, in Guglielmo Raimondo Moncada alias Flavio Mitradate: Un ebreo converso siciliano. Atti del Convegno Internazionale, Caltabellotta (Agrigento), 23–24 ottobre 2004, ed. by M. Perani (Palermo, 2008), 201–223. Starrabba “Guglielmo Raimondo Moncada” (ref. 6), 39, 41, and 47–48; M. Steinschneider, Die hebraeischen Übersetzungen des Mittelalters (Berlin, 1893), 986–987. G. Busi, et al., The Great Parchment: Flavius Mithridates’ Latin Translation, the Hebrew Text, and an English Version (Turin, 2004) pp. 16–17; C. Wirszubski, Pico della Mirandola’s Encounter with Jewish Mysticism (Cambridge (Mass.) and London, 1989). Munich, Staatsbibliothek, ms Heb. 246, f. 83a; see M. Steinschneider, Die hebraeischen Handschriften der K. Hof- und Staatsbibliothek in Muenchen (Munich, 1895), 120. For other approaches to this problem, see J. Chabás and B.R. Goldstein, “Computational

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his tables are based on previous identifiable astronomical material, originally compiled in the Iberian peninsula and used there in the fourteenth century. Indeed, there was a rich astronomical tradition in Hebrew in the fourteenth and fifteenth centuries both in the Iberian peninsula and in southern France, and a great many sets of tables in Hebrew were composed at that time, each with its own special characteristics, by Levi ben Gerson, Jacob ben David Bonjorn of Perpignan (known as ha-Poel), Immanuel Bonfils of Tarascon, Abraham Zacut, and Judah ben Verga, among others.12 It is quite unusual to find astronomical tables in Latin compiled in the fifteenth century that do not depend on either the Toledan Tables or the Parisian Alfonsine Tables. In fact, most sets of tables in Latin that are independent of these two “families” derive from the astronomical tradition in Hebrew; this applies to Mithridates’s set, for it depends primarily on the zij of Ibn al-Ḥadib. To be sure, the various tables composed in this period all depend on the Ptolemaic tradition as it was elaborated in al-Andalus (i.e., Muslim Spain) and then diffused to the Jewish and Christian communities in the Iberian peninsula, later spreading to other parts of Europe. Ibn al-Ḥadib’s tables are preserved in about 20 Hebrew manuscripts of which we have inspected the following: Vatican, ms Heb. 379; Paris, Bibliothèque nationale de France, ms Heb. 1086; Munich, Staatsbibliothek, ms Heb. 343. We know of no Latin version of these tables, but they were preserved in Greek: Venice, ms Marc. gr. 326 (ff. 135–139); and Mount Athos, Vatopedi, ms 188 (ff. 113–116v). According to Tihon and Mercier,13 Matthew Camariotes (d. 1490/1)

12

13

Astronomy: Five centuries of Finding True Syzygy”, Journal for the History of Astronomy, xxviii (1997), 93–105. On Ibn al-Kammād (early 12th century), see J. Chabás and B.R. Goldstein, “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for History of Exact Sciences, xxxxviii (1994), 1–41; and on Ibn al-Raqqām (d. 1315), see, E.S. Kennedy, “The Astronomical Tables of Ibn al-Raqqām a Scientist of Granada”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, xi (1997), 35–72. See, e.g., Immanuel ben Jacob Bonfils, Sefer shesh kenafayim [Hebrew edition of The Six Wings, bound with Isaac ben Solomon, Sefer or ha-levanah] (Zhitomir, 1872); Millás, Las Tablas Astronómicas (ref. 3); J. Chabás, “The Astronomical Tables of Jacob ben David Bonjorn”, Archive for History of Exact Sciences, xlii (1991), 279–314; Chabás, L’astronomia (ref. 3); B.R. Goldstein, Levi ben Gerson’s Astronomical Tables (New Haven, 1974); idem, “Astronomy in the Medieval Spanish Jewish Community”, in Between Demonstration and Imagination: Essays in the History of Science and Philosophy Presented to John D. North, ed. by L. Nauta and A. Vanderjagt (Leiden, 1999), 225–241; J. Chabás and B.R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the Transition from Manuscript to Print (Philadelphia, 2000); B.R. Goldstein, “The Astronomical Tables of Judah ben Verga”, Suhayl, iv (2001), 227–289. A. Tihon and R. Mercier, Georges Gémiste Pléthon: Manuel d’astronomie (Louvain-laNeuve, 1998), 12.

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composed an adaptation of the canons, extant in Leiden, ms bpg 74e, which does not include the tables. Vatican, Biblioteca Apostolica, ms Urb. lat. 1384, is a manuscript of 88 folios, probably copied in 1480–1481, containing three items: (1) De imaginibus coelestibus, a translation into Latin of a work in Arabic attributed to Ibn al-Ḥātim; (2) canons and tables on eclipses; and (3) a translation from Arabic into Latin of the Quran.14 In item 2, the canons appear on ff. 30r–43v, and its title includes a short dedication to Federico of Montefeltro, Duke of Urbino, together with the name of the author, given here as Guillelmus Raymundus de Moncata, i.e., William Raymond of Moncada. There are similar dedications for the two other texts that begin on ff. 1r and 65r. On f. 31r the epoch of the tables is given explicitly as Sunday, January 8, 1475. We do not think this date has any special significance other than the fact that it is the date of a mean conjunction of the Sun and the Moon, and that the civil day January 7 (having 12 hours in common with the astronomical day January 8 that begins at noon on January 7 in the civil calendar) is Saint Raymond’s day in the Christian calendar. Several Greek and Muslim authorities are mentioned in the canons: Aristotle, Ptolemy, and Ibn al-Ḥātim on f. 39r; Ibn al-Ḥātim on f. 42v; and Ibn Sina and Ptolemy on f. 43v. On f. 42v we are told that Ibn al-Ḥātim observed a solar eclipse in al-Andalus on July 19, 939.15 Of particular interest is that three other astronomers are cited on f. 32v: al-Battānī, Ibn al-Raqqām, and Ibn al-Kammād: (…) apud Il Bactani et Ibn il raccam nec non et Ibn il chimadi. These are the same three astronomers mentioned by Ibn al-Ḥadib in the introduction to his Oraḥ selulah.16 We are not aware of any other Latin text of the fourteenth or fifteenth centuries in which Ibn al-Raqqām’s name appears. Throughout the canons there are a few references to tables, and there is even a complete table for the possibility of an eclipse (f. 35v), where the limits for the distances from a lunar node are ±15° for lunar eclipses and ± 10;16° for solar eclipses (cf. Vatican, ms Heb. 379, f. 4a). According to Ptolemy the limit for the possibility of a lunar eclipse corresponds to distances from a lunar node of ± 12;12°; whereas for solar eclipses, the limits are 17;41° to the north and 8;22°

14

15 16

For a detailed table of contents and discussion of items 1 and 3, see Lippincott and Pingree, “Ibn al-Ḥātim” (ref. 5); K. Lippincott, “More on Ibn al-Ḥātim”, Journal of the Warburg and Courtauld Institutes, li (1988), 188–190. Cf. Lippincott and Pingree, “Ibn al-Ḥātim” (ref. 5), 58. Vatican, ms Heb. 379, f. 2a.

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to the south.17 As Pedersen explains, an eclipse may be possible even if at true syzygy the true Sun is 3° from the place of the mean Sun at mean syzygy.18 This means that a lunar eclipse may be possible when the mean Sun at syzygy is about ±15° from a lunar node. In the tables on ff. 47v and 52v, below, the lunar eclipse limits are ±12° from the lunar node. On f. 31v, lines 6–9, the text refers to the movement on the 8th sphere, which is called “motus proprie et per se”, whereas the movement on the 9th sphere is called “motus naturalis”. This is the same terminology as that used in previous astronomical texts in Castilian and in Arabic.19 On f. 35r, the difference between the 8th and 9th spheres, that is the difference between sidereal longitudes and tropical longitudes, is said to be 12;30° without an explanation of the way this value was determined (see also Tabula gradus solaris, f. 56v, below). A commentary on Ibn al-Ḥadib’s tables by Abraham Gascon (Cairo, midsixteenth century) contains a worked example for the solar eclipse of August 11, 1542 (New York, Jewish Theological Seminary of America [jtsa], ms 2571, ff. 1a–8b), based on Ibn al-Ḥadib’s zij. In this manuscript the author’s name is consistently given as Ibn al-Ḥadib and not Ibn al-Aḥdab as in some other manuscripts.20 In Gascon’s commentary (f. 3a) the difference between the 8th and 9th spheres, which he calls “the motion in access” (tenucat ha-haqbalah), is taken to be 12°, without any explanation of the way this value was determined. Ibn al-Ḥadib does not address this issue in the canons to his tables except to say that his tables are arranged for 8th sphere, i.e., his coordinates are sidereal (Vatican, ms Heb. 379, f. 2b). However, in a commentary on these tables by Ibn al-Ḥadib’s son, Jacob, “the motion in access” is said to be 12° (New York, jtsa, 2571, f. 16a), and this may be the source for Gascon’s remark.21 The tables of Mithridates are on ff. 44r–61v, following the canons. We offer a brief description of them and their relationship to those of Ibn al-Ḥadib, with special attention to the parameters embedded in them, in order to identify lines of transmission of the astronomical material. The first four tables, together with two other tables on f. 61r–v, list the solar and lunar equations at syzygy, and it is worth noting that these tables come before the tables for mean motions (see

17 18 19 20 21

Almagest vi.5; G.J. Toomer, Ptolemy’s Almagest (New York, 1984), 286–287. O. Pedersen, A Survey of the Almagest (Odense, 1974), 229–230. See J. Chabás and B.R. Goldstein, The Alfonsine Tables of Toledo (Dordrecht and Boston, 2003), 217–218. On Gascon, see B.R. Goldstein, “The Hebrew Astronomical Tradition: New Sources”, Isis, lxxii (1981), 237–251. Another copy of Jacob’s commentary is extant in London, British, Library, Or. 2806, ff. 20b– 39b.

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ff. 54r–55r and 56v–57r), whereas the standard order in medieval sets of tables is the opposite.

f. 44r. Prime adequationes temporis in .xij. signis This table displays the correction for the solar position (in hours, minutes, and seconds) at syzygy as a function of the solar anomaly given for each degree. The maximum is 3;54,22h at anomaly 3s 1° (see Table a). The entries are the same as those in the column for the “correction for the time” in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, ff. 7a–8a (see Table aa), where the same maximum value is found. The letter “s” at the top left-hand margin of f. 44r means that the entries are to be subtracted between arguments 0s 1° and 5s 30°, and the letter “a” at the bottom left-hand margin indicates that the entries are to be added between 6s 0° and 11s 29°. It is worth noting that the maximum occurs at 91°; we are not aware of any other table for the solar equation with a maximum value at that argument. After inspection of the surrounding entries, it seems that this particular entry is an isolated error and that the intended maximum was 3;54,20h at an argument of 3s 2°. To determine the underlying maximum solar equation, we have compared Table a with various tables for the solar equation: Ibn al-Kammād’s table where the maximum is 1;52,44°;22 al-Battānī’s where the maximum is 1;59,10°;23 and the Parisian Alfonsine Tables where the maximum is 2;10,0°.24 Dividing an entry in each of these tables by the corresponding entry in Table a leads to the result that the best fit is with the entries in al-Battānī’s table. table a

s 1 2 3 4 22 23 24

The solar equation in time at syzygy (Mithridates, f. 44r)

0[s]

1[s]

2[s]

3[s]

4[s]

5[s]

0; 4, 1h 0; 7,56 0;11,52 0;15,48

1;57,10h 2; 0,16 2; 4, 0 2; 7,22

3;21,11h 3;23,15 3;25,17 3;27,15

3;54,22h 3;54,20 3;54,18 3;54, 8

3;23, 0h 3;21,55 3;20, 8 3;18, 1

1;56,55h 1;53,13 1;49,25 1;45,35

Chabás and Goldstein, “Ibn al-Kammād” (ref. 11), 6 ff. C.A. Nallino, Al-Battānī sive Albatenii Opus Astronomicum (2 vols, Milan, 1903–1907), ii, 78 ff. Tabule astronomice illustrissimi Alfontij regis castelle ed. by E. Ratdolt (Venice, 1483), e2v– e4r.

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table a

The solar equation in time at syzygy (Mithridates, f. 44r) (cont.)

s

0[s]

1[s]

2[s]

3[s]

4[s]

5[s]

5 … 10 … 20 … 30

0;19,41

2;10,43

3;29, 9

3;53,51

3;15,51

1;41,45

0;39,20

2;26,41

3;37,28

3;51,56

3; 3,44

1;22,21

1;16,35

2;55,32

3;49,21

3;41,15

2;33, 0

0;42, 5

1;53,52

3;19, 6

3;54, 2

3;24,40

2; 0,41

0; 0, 0

a

11[s]

10[s]

9[s]

8[s]

7[s]

6[s]

0s 20. Read: 1;17,35. table aa

Table for the solar anomaly (Ibn al-Ḥadib, Vatican, ms Heb. 379, f. 7a), the first of 6 similar sub-tables

0[s] Subtract Solar Anomaly * 1 2 3 4 5 … 10 … 20 … 30

Corr. for the Lunar Anomaly

Corr. for the Time

Corr. for the Position

0; 2,47° 0; 5,32 0; 8,11 0;11,12 0;13, 3

0; 4, 1h 0; 7,56 0;11,52 0;15,48 0;19,41

0; 2,12° 0; 4,20 0; 6,29 0; 8,40 0;10,49

0;27,15

0;39,20

0;21,36

0;55,25

1;16,35

0;42,37

1;19,32

1;53,42

1; 2,24

11[s] Add

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isaac ibn al-ḥadib and flavius mithridates * In some of the sub-tables in Vatican, ms Heb. 379, the manuscript reads “Lunar Anomaly”, but in Paris, ms Heb. 1086, 7a–8a, the heading is consistently “Solar Anomaly”.

f. 44v. Secunde adequationes temporis in totidem signis This table displays the correction for the lunar position (in hours, minutes, and seconds) at syzygy as a function of the lunar anomaly given for each degree. The letters “a” and “s” in the margins have the same meaning as in the previous table, but their placement is interchanged. The maximum is 9;42,6h at anomaly 3s 6° (see Table b); comparison with the surrounding entries indicates that the entries around the maximum value are mistaken. The entries in this table are the same as those in the column for the “correction for the time” in the corresponding table in Vatican, ms Heb. 379, ff. 8b–9b (see Table bb), where the same maximum value is found. Again, when we divide an entry in each of the tables of other authors for the lunar equation by the corresponding entry in Table b we find that the best fit is with the entries in al-Battānī’s table, where the maximum lunar equation is 5;1,0°. But an underlying value of 4;56° cannot be definitely excluded. table b

The lunar equation in time at syzygy (Mithridates, f. 44v)

a

0[s]

1[s]

2[s]

3[s]

4[s]

5[s]

1 2 3 4 5 6 … 10 … 20 … 30

0; 9,20h 0;18,40 0;28, 2 0;37,20 0;46,40 0;56, 0

4;38,25h 4;46,52 4;55, 0 5; 3, 6 5;11,12 5;19,19

8; 9, 5h 8;14,25 8;18,36 8;22,46 8;27,50 8;32, 2

9;40, 1h 9;41,35 9;41,54 9;42, 4 9;42, 5 9;42, 6

8;39,46h 8;36,15 8;31,18 8;26, 9 8;20,49 8;15,14

5; 5,30h 4;55,47 4;46,20 4;36,37 4;26,53 4;17, 3

1;33, 0

5;50,32

8;49,41

9;40,11

7;51,29

3;36,47

3; 3,49

7; 1,20

9;22,55

9;21,53

6;40,13

1;50,30

4;30,15

8; 1,44

9;40,11

8;45,42

5;14,26

0; 0, 0

s

11[s]

10[s]

9[s]

8[s]

7[s]

6[s]

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3s 1. Probably a mistake for 9;41,0h; the same mistake is found in Vatican, ms Heb. 379. In Hebrew, using alphabetical notation for numbers, 41 is ma (with no space between the letters) and 40,1 is m a (with a space between the letters). table bb

Table for the lunar anomaly (Ibn al-Ḥadib, Vatican, ms Heb. 379, f. 8b), the first of 6 similar sub-tables

0[s] Add Lunar Anomaly 1 2 3 4 5 … 10 … 20 … 30

Corr. for the Lunar Anomaly

Corr. for the Time

Corr. for the Position

0; 5, 5° 0;10,10 0;15,15 0;20,20 0;25,25

0; 9,20h 0;18,40 0;28, 2 0;37,20 0;46,40

0; 0; 0; 0; 0;

0;50,48

1;33, 0

0; 3,49

1;40, 4

3; 3,49

0; 7,32

2;28, 7

4;30,15

0;11,14

0,23° 0,46 1, 9 1,32 1,55

11[s] Subtract

Thus, the two preceding tables (a and b) were intended for computing a first approximation of the time between mean and true syzygy and, presumably, were based on tables for the solar and lunar equations with maxima of 1;59,10° and 5;1,0°, respectively, both of which are well represented in the astronomical literature in the Iberian peninsula. On the other hand, the next two tables (c and d) are intended for computing a first approximation of the difference in longitude between mean and true syzygy. These tables treat the effect of each luminary separately and differ substantially from those compiled for the same purpose at about the same time.25 25

Cf. Chabás and Goldstein, “True Syzygy” (ref. 11).

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f. 45r. Prime adequationes motus draconis et loci solis in .xij. signis This table displays the correction for the solar position (in degrees, minutes, and seconds) as a function of the solar anomaly given for each degree. The letters “s” and “a” in the margins have the same meaning as in the previous tables. The maximum is 2;8,51° at anomaly 3s 1° (see Table c). The entries are the same as those in the column for the “correction for the position” in the corresponding table in Vatican, ms Heb. 379, ff. 7a–8a (see Table aa), where the same maximum value is found. table c

The solar equation at syzygy (Mithridates, f. 45r)

s

0[s]

1[s]

2[s]

1 2 3 4 5 … 10 … 20 … 30

0; 2,12° 0; 4,20 0; 6,29 0; 8,40 0;10,49

1; 4,22° 1; 6, 5 1; 8, 9 1;10, 0 1;11,50

1;50,38° 1;51,23 1;52,49 1;53,53 1;54,56

a

11[s]

3[s] 2; 2; 2; 2; 2;

8,51° 8,47 8,45 8,40 8,39

4[s]

5[s]

1;51,34° 1;49,58 1;49,39 1;48,49 1;47,38

1; 4,18° 1; 2,11 1; 0, 7 0;58, 9 0;55,55

0;21,36 1;20,36 1;59,31 2; 7,28 1;41,30 0;45,14 0;42,37 1;35,59 2; 6, 1 2; 1,35 1;26, 8 0;23, 7 1; 2,29 1;49,26 2; 8,44 1;52,22 1; 6,20 0; 0, 0 10[s]

9[s]

8[s]

7[s]

6[s]

The entries in this table, here given in degrees, are equivalent to those in Table a, given in hours. If ai and ci are the corresponding entries in Tables a and c, then the expression ci = 0;32,56 · ai, where 0;32,56°/h, the mean lunar velocity, yields fairly good results. This seems to mean that an entry in Table c is the distance the Moon travels at its mean velocity corresponding to the time computed in Table a. We note that 0;32,56°/h is an approximation of the lunar mean motion in longitude, 13;10,35°/d.

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f. 45v. Secunde adequationes motus draconis et loci solis in totidem signis This table displays the correction for the lunar position (in degrees, minutes, and seconds) as a function of the lunar anomaly given for each degree. There is no indication in the table or the margins concerning the addition or subtraction of the entries; we have introduced a and s in accordance with the presentation in Ibn al-Ḥadib’s corresponding table. The maximum is 0;23,55° at anomaly 3s 4° (Table d). The entries are the same as those in the column for the “correction for the position” in the corresponding table in Vatican, ms Heb. 379, ff. 8b–9b (see Table bb), where the same maximum value is found. table d

[a] 1 2 3 4 5 … 10 … 20 … 30 [s]

The lunar equation at syzygy (Mithridates, f. 45v)

0[s] 0; 0; 0; 0; 0;

0,23° 0,46 1, 9 1,32 1,55

1[s]

2[s]

3[s]

4[s]

5[s]

0;11,25° 0;11,46 0;12, 7 0;12,26 0;12,46

0;20, 6° 0;20,21 0;20,30 0;20,40 0;20,52

0;23,52° 0;23,53 0;23,54 0;23,55 0;23,54

0;21,21° 0;21,13 0;21, 0 0;20,50 0;20,35

0;12,32° 0;12, 9 0;11,44 0;11,21 0;10,58

0; 3,49 0;14,23 0;21,46 0;23,50 0;19,27 0; 8,56 0; 7,32 0;17,18 0;23, 8 0;23,20 0;16,27 0; 4,31 0;11, 4 0;19,47 0;23,50 0;21,36 0;12,50 0; 0, 0 11[s]

10[s]

9[s]

8[s]

7[s]

6[s]

The entries in this table, here given in degrees, are equivalent to those in Table b, given in hours. If bi and di are the corresponding entries in Tables b and d, then the expression di = 0;2,28 · bi, where 0;2,28°/h, the mean solar velocity, yields good results. This seems to mean that an entry in Table d is the distance the Sun travels at its mean velocity

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corresponding to the time computed in Table b. We note that 0;2,28°/h is an approximation of the mean motion of solar longitude, 0;59,8°/d. Two tables on f. 61r–v, with the same format as Tables a, b, c, and d, are described below.

f. 46r. Tabula multiplicationis numerorum In this multiplication table we find the numbers 1 to 10 across the top and the numbers from 1 to 10 and then multiples of 5 from 15 to 60 at the beginning of each row. However, Vatican, ms Heb. 379, f. 15r, displays a multiplication table with the numbers 1 to 10 across the top and the numbers 1 to 30 (at intervals of 1) at the beginning of each row. On f. 15v the table continues with 1 to 10 across the top and 31 to 60 (at intervals of 1) at the beginning of each row. Mithridates kept the heading for the columns, but reduced the number of rows.

f. 46v–47r. Tabula horarum et minutorum meridierum in his latitudinibus The entries in this table correspond to the time of half-daylight (in hours and minutes) for various latitudes (0;0°, 8;28°, 16;50°, 24;0°, 30;20°, 36;0°, 40;30°, 45;0°, 48;30°, 51;30°, 59;45°, 63;0°, 64;45°, and 66;25°) as a function of the solar longitude. The argument is given at 10°-intervals, and the columns for the last 4 latitudes are not completely filled in. This table is similar to that of Ibn al-Ḥadib (Vatican, ms Heb. 379, f. 10a), but the argument there is given at 5°-intervals and for 9 latitudes that are slightly different from those in Mithridates’s table. The table for lunar eclipses on f. 47v and the two tables for lunar eclipses on f. 48r that follow agree with the corresponding tables in “Wing 4” of the Six Wings by Immanuel Bonfils of Tarascon (Provence, Southern France), composed in Hebrew c. 1365.

f. 47v. Quot puncta globi lunaris deficient This table displays the digits of a lunar eclipse (in digits and minutes of a digit), where 1 digit is a twelfth of the lunar diameter, as a function of the argument of latitude (from 0° to 12°, for each integer degree) and the lunar

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anomaly (from 0s to 11s, for each integer sign). The maximum value is 21;36 digits, corresponding to an argument of latitude of 0s 0°. The entries are the same as those in the columns for the “digits of the diameter” in Ibn al-Ḥadib’s tables for lunar eclipses: see Vatican, ms Heb. 379, f. 11a. The maximum in this table agrees with the maximum in Wing 4 in the tables of Immanuel Bonfils.26 This maximum is also found in al-Battānī’s tables (Nallino 1903–1907, 2:90). In the Almagest the maximum is called “entire” following an entry of 21 digits; however, Neugebauer shows that the parameters in the Almagest lead to a maximum of 21;36 digits.27

f. 47v. Tabula colorum The heading for this table refers to the colour of eclipses. Only the frame and the arguments are given; no other entries are displayed. The argument runs from 1 to 12 digits, for each integer digit. A table for colours of solar eclipses with these arguments appears in Vatican, ms Heb. 379, f. 13b.28 The text preceding the tables mentions a table for the colours of eclipses (f. 38v, line 2).

f. 47v. Alia secundum il chimadi The heading for this table also refers to the colour of eclipses. As in the previous table, the frame and the arguments are given without any other entries. The argument runs from 1 to 24 digits, for each integer digit, and there are two successive numbers in each of the 12 rows. The heading mentions Ibn al-Kammād.29 There are two tables for the same purpose on f. 53v. These arguments probably refer to digits of a lunar eclipse.

26

27 28 29

See Bonfils 1872, pp. 38–40; cf. P. Solon, The Hexapterygon of Michael Chrysococces. (Ph.D. dissertation, Brown University, 1968); and idem, “The Six Wings of Immanuel Bonfils and Michael Chrysokokkes”, Centaurus, xv (1970), 1–20, p. 7. Toomer, Almagest (ref. 17), 307; O. Neugebauer, A History of Ancient Mathematical Astronomy (Berlin and New York, 1975), 136. B.R. Goldstein, “Colors of Eclipses in Medieval Hebrew Astronomical Tables”, Aleph, v (2005), 11–34. For colours of eclipses according to Ibn al-Kammād, see Chabás and Goldstein, “Ibn al-Kammād” (ref. 11), 18–19.

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f. 48r. Quot horis durabit medius defectus This table displays the time of the half-duration of lunar eclipses (in hours and minutes) as a function of the argument of latitude (from 0° to 12°, for each integer degree) and the lunar anomaly (from 0s to 11s, for each integer sign). The maximum value is 2;2h, corresponding to an argument of latitude of 0s 0° and a lunar anomaly of 0s. The entries are the same as those in the columns for the “half-duration” in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, f. 11a. We note that the term defectus is used for “eclipse”; on f. 42v the term ekleipsis is written in Greek characters. This maximum for the half-duration of a lunar eclipse is also found in the tables of Immanuel Bonfils where the time from first contact to the beginning of totality is given as 1;8h and the half-duration of totality is given as 0;54h, for 1;8h + 0;54h = 2;2h.30

f. 48r. Numeri medie obscuritatis This table displays the time of the half-duration of totality of lunar eclipses (in hours and minutes) as a function of the argument of latitude (from 0° to 12°, for each integer degree) and the lunar anomaly (from 0s to 11s, for each integer sign). The maximum value is 0;54h, corresponding to an argument of latitude of 0s 0° and a lunar anomaly of 0s. The entries are the same as those in the columns for the “half-duration of totality” in Ibn al-Ḥadib’s tables: see Vatican, ms Heb. 379, f. 11a. They also agree with the values in Wing 4 of the Six Wings by Immanuel Bonfils.31

f. 48r. Numeri defectus s[eptentrionalis] aut m[eridionalis] fuerint In this table only the frame is given with headings for the columns, but there are no other entries. It has the same pattern as the previous tables for the digits of the diameter, the half-duration, and the half-duration of totality.

30 31

Bonfils 1872, pp. 38–40; cf. Solon, “The Six Wings” (ref. 26), 7. Ibid.

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f. 48v. Diversus aspectus in primo climate ad latitudinem xvi The entries in this table for the longitudinal and latitudinal components of the adjusted parallax are almost identical with those in Ptolemy’s Handy Tables for latitude 16;27° and in al-Battānī’s zij for latitude 16;32°, according to the heading.32

f. 49r. Diversus aspectus in secundo climate ad latitudinem xxiiii The entries in this table for the longitudinal and latitudinal components of the adjusted parallax are almost identical with those in Ptolemy’s Handy Tables for latitude 23;51° and in al-Battānī’s zij for latitude 24;0°, according to the heading.

f. 49v. Diversus aspectus in tertio climate ad latitudinem xxx The entries in this table for the longitudinal and latitudinal components of the adjusted parallax are almost identical with those in Ptolemy’s Handy Tables for latitude 30;22° and in al-Battānī’s zij for latitude 30;40°, according to the heading.33 This table appears in Vatican, Heb. 379, f. 12a, with the heading: Table for parallax in longitude and latitude for [geographical] latitude 30°.

f. 50r. Diversus aspectus in quarto climate ad latitudinem xxxvi The entries in this table for the longitudinal and latitudinal components of the adjusted parallax are almost identical with those in Ptolemy’s Handy Tables for latitude 36;0° and in al-Battānī’s zij for latitude 36;22°, according to the heading.34

32

33 34

See W.D. Stahlman, The Astronomical Tables of Codex Vaticanus Graecus 1291 (Ph.D. dissertation, Brown University, 1959; University Microfilms, No. 62–5761), 268–269; and Nallino, Al-Battānī (ref. 23), ii, 95. See Stahlman, Astronomical Tables (ref. 32), 272–273; and Nallino, Al-Battānī (ref. 23), ii, 97. See Stahlman, Astronomical Tables (ref. 32), 274–275; and Nallino, Al-Battānī (ref. 23), ii, 98.

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f. 50v. Diversus aspectus in quinto climate ad latitudinem xxxx The entries in this table for the longitudinal and latitudinal components of the adjusted parallax are almost identical with those in Ptolemy’s Handy Tables for latitude 40;56° and in al-Battānī’s zij for latitude 41;15°, according to the heading.35 This table appears in Vatican, Heb. 379, f. 12b, with the heading: Table for parallax in longitude and latitude for [geographical] latitude 40°.

f. 51r. Diversus aspectus in sexto climate ad latitudinem xxxxv The entries in this table for the longitudinal and latitudinal components of the adjusted parallax are almost identical with those in Ptolemy’s Handy Tables for latitude 45;1° and in al-Battānī’s zij for latitude 45;22°, according to the heading.36

f. 51v This table has no title and very few entries, but it was certainly intended for parallax for a latitude where the longest daylight (from sunrise to sunset) is 16;30h.

f. 52r This table has no title and the entries are given in minutes (see Table e). The argument reads “gradus motus differenctia”, here meaning the lunar anomaly, and the entries represent minutes of proportion. Its purpose is to correct the parallax when the Moon is not at its mean distance at syzygy. Parallax varies inversely with the lunar distance from the Earth (and lunar distance is a function of lunar anomaly) but the parallax tables assume that the Moon is at its mean distance at syzygy. In Ptolemy’s lunar model for syzygy, the radius of

35 36

See Stahlman, Astronomical Tables (ref. 32), 276–277; and Nallino, Al-Battānī (ref. 23), ii, 99. See Stahlman, Astronomical Tables (ref. 32), 278–279; and Nallino, Al-Battānī (ref. 23), ii, 100.

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table e

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Table for correcting parallax

0[s]

1[s]

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 4 4 3 3 3

2[s] 3[s] 4[s] 5[s] 3 3 3 2 2 2 2 2 1 1 1 1 1 1 1

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

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

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

28 26 24 22 20 18 16 14 12 10 8 6 4 2 0

11[s] 10[s] 9[s] 8[s] 7[s] 6[s]

the epicycle is 5;15.37 Hence, the maximum lunar distance is 65;15 at apogee (0° anomaly), the minimum distance is 54;45 at perigee (180° anomaly), where the mean distance is 60 (near 90° anomaly). Therefore, the correction for parallax is negative near apogee and positive near perigee and, in general, p′(α) = p + e(α) · p, where p is the value in one of the parallax tables, e(α) is the entry in Table e for an anomaly α, and p′(α) is the corrected value for parallax as a function of α. This table also appears in Vatican, ms Heb. 379, f. 13b, where the heading is: Table for correcting parallax. Above the columns for 0s, 1s, and 2s, is the word for subtract, and above the columns for 3s, 4s, and 5s, is the word for add. The argument is labelled: degrees of [lunar] anomaly [ḥoq]. Similar tables are found

37

In addition to the value 5;15 for this parameter, there are variants 5;13 and 5;14: see Almagest iv.6; trans. Toomer, Almagest (ref. 17), 197, 202, and 209.

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in the astronomical literature in the Iberian peninsula: in an expanded version (to two sexagesimal places) in the Tabule Verificate for Salamanca (c. 1461), and in shorter versions in Abraham Zacut’s ha-Ḥibbur ha-gadol as well as in its Latin version, the Almanach Perpetuum.38

f. 52r–v This table has no title, and its entries represent the time between noon and nonagesimal, here called medium celum (midheaven), given in hours and minutes, for each zodiacal sign. The argument is the geographical latitude (from 30° to 51°, for each integer degree). The entries are the same as those in the corresponding table in Vatican, ms Heb. 379, f. 11b, although the horizontal and vertical arguments are interchanged. In Ibn al-Ḥadib’s table the geographical latitude varies from 30° to 45°, at intervals of one degree.

f. 52v This table has no title, and the entries (in degrees, minutes, and seconds) represent the lunar latitude as a function of the argument of lunar latitude from 0° to 12° in steps of 1°. The maximum value, 1;2,16°, is reached at argument 12°. The entries are the same as those in the first two columns of a table in Vatican, ms Heb. 379, f. 11a which has the heading: Table for eclipses of the Moon and its latitude at the time of an eclipse. The argument of 12° represents the limit for the Moon’s distance from the lunar node at the time of a lunar eclipse and, with a maximum lunar latitude of 5°, the value for 12° of argument would be 1;2,22°, which is close to the value here (cf. the table on f. 47v).

f. 53r Again, this table has no title. The entries represent the velocity of the Moon relative to that of the Sun, in minutes and seconds of arc per hour, from 0;27,30°/h to 0;32,30°/h, as a function of lunar anomaly, given in degrees for every other integer degree. The relative velocity increases monotonically by 0;0,4°/h for arguments between 0s 0° and 6s 0°, and then decreases monotonically by

38

Chabás and Goldstein, Zacut (ref. 12), 33–34, 62, 118–122.

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0;0,4°/h for arguments between 6s 0° and 12s 0°. The entries are the same as those in the corresponding table in Vatican, ms Heb. 379, f. 11b.

f. 53r As was the case in several of the previous tables, this one is untitled. The entries give the fraction of the solar diameter that is darkened in a solar eclipse (in digits and minutes), where 1 digit is a twelfth of the solar diameter, as a function of the true lunar latitude (in minutes and seconds of arc): see Table f. It is noteworthy that the entries for the latitude are not rounded whereas the digits of the diameter are such that the successive differences between them are 0;20 digits or 0;15 digits. The entries are the same as those in the column for “digits of the diameter” in the corresponding table in Vatican, ms Heb. 379, f. 13a. We note, however, that in Mithridates’s table the argument (true latitude) increases downward, but in Ibn al-Ḥadib’s table it decreases. table f

Digits of a solar eclipse

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

0 1 12 44 31 48 8 4 56 54 51 46 45 35 30 21 20

Digits 11 11 11 10 10 10 9 9 9 8 8 8 7 7 7 6 6

40 20 0 40 20 0 40 20 0 40 20 0 40 20 0 40 20

True latitude

Digits

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

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

19 15 12 8 4 1 57 53 50 47 44 39 35 31 28 23 20 17 13 0

0 40 20 0 40 20 0 40 20 0 40 20 0 40 20 0 45 30 15 0

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f. 53v This is a double argument table for the half-duration of a solar eclipse and it is mentioned in the text (f. 42r). The vertical argument is the corrected lunar latitude, in degrees, at 2°-intervals, and the horizontal argument is the velocity of the Moon relative to that of the Sun, in minutes and seconds of arc per hour, from 0;27,30°/h to 0;32,30°/h. The entries give the half-duration of a solar eclipse, and they are the same as those in the corresponding table in Vatican, ms Heb. 379, f. 13a. As was the case with the table on f. 53r, the vertical argument (corrected lunar latitude) increases downward, whereas in Ibn al-Ḥadib’s table it decreases. This table is identical (but for minor variants) to those in Ibn al-Kammād’s zij (Madrid, Biblioteca Nacional, ms 10023, f. 54r) and in the Tables of Barcelona (Table 49).39

f. 53v There are three frames for small tables, also untitled, but two of them have the word Colores in the heading, as was the case with the two tables on f. 47v whose entries are missing. The following tables yield mean positions and times for conjunctions. In the canons to Ibn al-Ḥadib’s tables, chapter 2 (Vatican, ms Heb. 379, f. 2b), we are told that these motions are computed for the western extremity (not, e.g., for Toledo) and each day begins at noon of the day preceding it, e.g., Sunday begins at noon on Saturday and ends at noon on Sunday. In other respects the tables are arranged according to the Jewish calendar with its 19-year cycle, beginning with Molad Tohu, the conjunction of Tishri, year 1 Anno mundi, i.e., the era of creation. According to the Jewish tradition, the conjunction of creation took place on Monday (day 2) at 5h 204 ḥelaqim (where 1h = 1080 ḥelaqim) counting from sunset,40 at a location whose distance from the western extremity is 75;43,45° in the view of Ibn al-Ḥadib. The epoch of the Jewish calendar, Tishri

39

40

See Chabás and Goldstein, “Ibn al-Kammād” (ref. 11), 23; Millás, Las Tablas Astronómicas (ref. 3), p. 237; and J. Chabás, “Astronomía andalusí en Cataluña: Las Tablas de Barcelona”, in From Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet, edited by J. Casulleras and J. Samsó (Barcelona, 1996), 477–525, pp. 511–512. See, e.g., Maimonides: Sanctification of the New Moon, trans. by S. Gandz, introduction by J. Obermann, astronomical commentary by O. Neugebauer (New Haven, 1956), 116.

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1, 1 Anno mundi, is equivalent to Oct. 7, 3761bc (jdn 347998). The radices for the conjunction of creation are given as the entries for the first month, Tishri. As we will see, the parameters for the mean motion tables are the same for Mithridates as they are for Ibn al-Ḥadib, but the presentation is different and the initial values are different. In the case of Mithridates there are places where radices were to be given but they are filled with names which do not correspond to any numerical values. As we learn from the worked example in Gascon’s commentary, Ibn al-Ḥadib’s mean motion tables are set up such that the entries in them refer to the current 19-year cycle, the current year within the cycle, and the conjunction at the beginning of each month.

f. 54r. Tabula argumenti solis This table is arranged for conjunctions and it consists of 3 sub-tables, giving the solar anomaly (in signs, degrees, minutes, and seconds) for 13 consecutive months, 19 consecutive years, and groups of 19-year cycles (for cycles 1, 2, 3, …, 10, 20, 30, …, 100, …, 700), respectively. This table also displays a numerical value, 0s 14;33,10°, together with the word FRAEDERICO, referring to Federico, Duke of Urbino. The same value is found for the solar anomaly in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, f. 6b, and it represents the motion of the solar anomaly in half a mean synodic month, i.e., the time between successive syzygies. Indeed, it is so labelled in Vat. 379, f. 6b, and is intended for making the tables usable for oppositions. The entries for two successive 19-year cycles show a constant difference of 359;44,50°, and this is also the entry for 1 cycle. This line-by-line difference is the same as in the column for the “solar anomaly” in Ibn al-Ḥadib’s table in Vatican, ms Heb. 379, f. 6a, but that table is presented differently: the information is only given for cycles 273 to 280 (11s 20;54,8° to 11s 19;7,58°), where cycle 273 refers to the beginning of cycle 273, i.e., 272 cycles have been completed or 272 · 19 = 5168 years have passed since the creation according to the Jewish tradition. For an explanation of the way the entries in this column were computed, see the comments on Tabula gradus solaris (f. 56v), below. The constant difference of 359;44,50° means that in one 19-year cycle the Sun progresses 6839;44,50°. Now, the first entry in the table for the time on f. 55r, below, associates 2d 16;33,3,20h with a 19-year cycle. (This is also identical to the value underlying the column for “time” in Ibn al-Ḥadib’s table in Vatican, ms Heb. 379, f. 6a.) This time corresponds to the excess over an integer number of weeks of the time between two consecutive 19-year cycles, which is taken here as 991 weeks + 2d 16;33,3,20h. Thus, the time between two consecutive 19-year cycles is 6939;41,22,38,20d.

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Dividing the amount progressed by the Sun by this time we obtain a value for the solar mean motion of 0;59,8,9,16°/d and it is a sidereal value. The closest value of which we are aware is that of Ibn al-Kammād: 0;59,8,9,21,15°/d. In the sub-table for single years the entry for year 1 is 11s 20;41,15°, whereas in the tables of Ibn al-Ḥadib the entry for year 1 is 0;0°. In the sub-table for months the entry for month 1 is 9s 7;14,49,42°, whereas for Ibn al-Ḥadib it is 3s 15;20,10° which, according to Ibn al-Ḥadib’s canons, is the radix for the conjunction of creation. We have no explanation for these differences but is seems likely that Mithridates has changed the epoch.

f. 54v. Tabula motus diuersi This table consists of 3 sub-tables giving the lunar anomaly (in signs, degrees, minutes, and seconds) for 13 consecutive months, 19 consecutive years, and groups of 19-year cycles (for cycles 1, 2, 3, …, 10, 20, 30, …, 100, …, 700), respectively, as in the previous table. This table also displays a numerical value, 6s 12;54,30°, together with the words DUCI URBINI, referring to Federico, Duke of Urbino. The same value for the lunar anomaly is found in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, f. 6b, and it represents the lunar motion in anomaly in half a mean synodic month. The entries for two successive 19-year cycles show a constant difference of 306;54,52°, as in the column for the “lunar anomaly” in Ibn al-Ḥadib’s table in Vatican, ms Heb. 379, f. 6a, where this information is presented differently. The entry for cycle 1 is 10s 6;54,52°. In Ibn al-Ḥadib’s tables the entry for 273 refers to the beginning of the cycle (as we learn from Gascon’s commentary). Since the radices for this epoch are given by Ibn al-Ḥadib as entries for the month Tishri, the table for cycles should not have radices embedded in them. Ibn al-Ḥadib’s entry for cycle 273 is 320;42,12°, and it corresponds to the motion in anomaly in 272 completed cycles of 19 years. To derive this value from the constant difference of 306;54,52°, we first compute: 272 · 306;54,52° = 83480;43,44° = 320;43,44°. Now let us take 83480;42,12° and divide it by 272; the result is 306;54,51,40°, which rounds to 306;54,52°. This indicates that Ibn al-Ḥadib’s table was constructed from a basic parameter of 306;54,51,40° rather than 306;54,52°. The constant difference of 306;54,52° means that in one cycle the Sun progresses (251 · 360) + 306;54,52 = 90666;54,52°. Dividing the Moon’s progress by the time between two consecutive 19-year cycles, 6939;41,22,38,20d, we obtain a

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value for the mean motion in lunar anomaly of 13;3,53,55,56,18°/d. This is a classical parameter, to be compared, for example, with those in Ptolemy’s Almagest (13;3,53,56,17,52°/d) and the Toledan Tables (13;3,53,56,17,57 …º/d). In the sub-table for single years the entry for year 1 is 11s 15;38,0°, whereas in the tables of Ibn al-Ḥadib the entry for year 1 is 0;0°. In the sub-table for months the entry for month 1 is 2s 12;53,26,31°, whereas for Ibn al-Ḥadib it is 8s 21;51,25° which, according to Ibn al-Ḥadib’s canons, is the radix for the conjunction of creation. We have no explanation for these differences but it seems likely that Mithridates has changed the epoch.

f. 55r. Tabula temporum This table consists of 3 sub-tables giving the excess over an integer number of weeks (in days, hours, minutes, seconds, and thirds) for 13 consecutive months, 19 consecutive years, and groups of 19-year cycles (for cycles 1, 2, 3, …, 10, 20, 30, …, 100, …, 700), respectively, as in the two previous tables. This table also displays a numerical value, 7d 18;22,1,50h, together with the word GUILLELMUS, referring to William Raymond of Moncada. Almost the same value is found for the time in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, f. 6b, where we have 0d 18;22,2h. It represents the duration of half a mean synodic month in days (mod 7) and hours. We note that Mithridates’s value has a higher precision than that in Vatican, ms Heb. 379. We also note that 18;22,1,50h = 0;45,55,4,35d and twice this amount is 1;31,50,9,10d, i.e., the excess of a mean synodic month over 28 days (= 4 weeks). The entries for two successive 19-year cycles show a constant difference of 2d 16;33,3,20h (= 2;41,22,38,20d), which is also the value for the first entry. This is the same value that can be derived from the column for “time” in Ibn al-Ḥadib’s table in Vatican, ms Heb. 379, f. 6a, where this information is presented differently and with a lower precision (the entries are rounded to seconds of time). The entry for cycle 273 is 3d 13;51,7h (= 3;34,37,47,30d) which implies that the entry for cycle 0 would be 4d 7;26,57h = 4;18,37,22,30d (for 4;18,37,22,30d + 273 · 2;41,22,38,20d = 4;18,37,22,30d + 6;16,0,25d (mod 7) = 3;34,37,47,30d (mod 7) = 3d 13;51,7h). We have no explanation for Ibn al-Ḥadib’s initial value. The value for the constant different in a 19-year cycle is equivalent to 2d 16h and 595 ḥelaqim and this parameter is exactly that given, for example, in Maimonides’s Sanctification of the New Moon.41

41

Neugebauer, Maimonides (ref. 40), 115.

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In the sub-table for single years the entry for year 1 is 1d 21;30,10,0h, whereas in the tables of Ibn al-Ḥadib the entry for year 1 is 0d 0;0h. In the sub-table for months the entry for month 1 is 2d 1;43,6,51h, whereas for Ibn al-Ḥadib it is 2d 6;8,25h which, according to Ibn al-Ḥadib’s canons, is the radix for the conjunction of creation. We have no explanation for these differences but it seems likely that Mithridates has changed the epoch.

f. 55v. In quoto die mensium coniungium vel dissidium futurum sit This table displays the day of the (Christian) month in a 19-year cycle when a conjunction takes place (entries range from 1 to 31). The letter “p” is written opposite years 2, 5, 8, 10, 13, 16, and 18 in the 19-year cycle, indicating a “leap year”, i.e., a year of 13 lunar months.

f. 56r There are 4 small tables concerning weekday numbers. In one of them the words Federico, Duke, Count, William, Raymond, Moncata, and Urbino are assigned to the first day of the month, and in another, numbers between 1 to 7 are assigned to these 7 words. In the two other tables, a number between 0 and 6 is ascribed to each month in a year beginning in January, whether a common year or a leap year.

f. 56v Tabula gradus solaris This table consists of 3 sub-tables giving the mean longitude of the two luminaries at mean conjunction (in signs, degrees, minutes, and seconds) for 13 consecutive months, 19 consecutive years, and groups of 19-year cycles (for cycles 1, 2, 3, …, 10, 20, 30, …, 100, …, 700), respectively, as in previous tables. This table also displays three numerical values, 6s 0;0,0°, 0s 12;30,0°, and 0s 14;33,9,59°. The first indicates the amount to the added to the lunar longitude at opposition; the second is a parameter already mentioned in the text (f. 35r, lines 7–10), to be used to change from sidereal to tropical coordinates; and the third represents the motion of the Sun in longitude in half a mean synodic month. These values are presented together with the word RAYMUNDUS, referring to William Raymond of Moncada. Almost the same value for the solar motion for the position occurs in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, f. 6b where, for the

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third value, we find 0s 14;33,10º. We note that, as was the case in the analogous table on f. 55r, Mithridates’s value has a higher precision than that in Vatican, ms Heb. 379, although elsewhere Mithridates gives it as 0s 14;33,10° (f. 54r: see above). The entries for two successive 19-year cycles show a constant difference of 359;48,50°, as in the column for the “position” in Ibn al-Ḥadib’s table in Vatican, ms Heb. 379, f. 6a, where this information is presented differently. This means that in one cycle the Moon progresses (253 · 360) + 359;48,50° = 91439;48,50°. Dividing the Moon’s progress by the time between two consecutive 19-year cycles, 6939;41,22,38,20d, we obtain a value for the lunar mean motion in longitude of 13;10,34,52,40,30°/d. This parameter may be compared with the value in Ptolemy’s Almagest (13;10,34,58,33,30,30°/d) and those in the zij of al-Khwārizmī, the zij of Ibn al-Kammād, and the Toledan Tables (13;10,34,52,46°/d). In the sub-table for single years the entry for year 1 is 11s 20;43,15°, whereas in the tables of Ibn al-Ḥadib the entry for year 1 is 0;0°. In the sub-table for months the entry for month 1 is 5s 7;3,48,40°, whereas for Ibn al-Ḥadib it is 5s 16;41,31° which, according to Ibn al-Ḥadib’s canons, is the radix for the conjunction of creation. We have no explanation for these differences but it seems likely that Mithridates has changed the epoch. We can also deduce from the text the solar mean motion in longitude: (18 · 360 + 359;48,50)/6939;41,22,38,20d = 0;59,8,11,20°/d. The difference between the solar mean motion in longitude and the motion of the solar anomaly is the proper motion of the solar apogee: 0;59,8,11,20°/d – 0;59,8,9,16°/d = 0;0,0,2,4°/d (or 1° in about 286 Julian years). Standard values for the proper motion of the solar apogee are close to this amount, e.g., for Ibn al-Kammād it is 1° in about 290 Julian years (Chabás and Goldstein 1994, p. 28). If we subtract the mean solar anomaly at epoch (the conjunction of creation) according to Ibn al-Ḥadib from his value for the mean solar position at epoch, the result is the solar apogee at epoch: 166;41,31° – 105;20,10° = 61;21,21°. To verify that a sidereal solar position derived from Ibn al-Ḥadib’s tables is approximately correct, we compare the value based on his tables for the solar eclipse that took place on Aug. 11, 1542 with the sidereal solar position according to the Toledan Tables: in both cases the result is about 135° (see below). Hence, the sum of the entries for the solar position in Ibn al-Ḥadib’s tables is acceptable (by the standards of his day). But there is a problem: we expect the radix to be shown in the table for months opposite Tishri, as explained in Ibn al-Ḥadib’s canons, corresponding to the conjunction of creation. But there is another “radix” embedded in the table for the mean motions. First, let us compute the mean solar position for Aug. 11, 1542, the mean

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conjunction of Elul [month 12], 5302am, which took place on Ab [month 11] 29, in year 1 of cycle 280 (note that the entry for year 1 in Ibn al-Ḥadib’s table is 0°). cycle 280: 0s 7;43,58° Elul: 4s 6;51,11° sum:

4s 14;35, 9°

The entry for Elul can be computed by adding 11 times the increase in longitude in a mean synodic month (= 29;6,20°: Vatican, ms 379, f. 6b) to the radix (the entry for Tishri): 11 · 29;6,20° + 5s 16;41,31° = 320;9,40° + 166;41,31° = 126;51,11°. On analogy with the computation of the mean motion in lunar anomaly (see Tabula motus diuersi, f. 54v), one expects the entry for the beginning of cycle 280 to be 279 times the line-by-line differences in the column for solar position in the table for cycles, i.e., 279 · 359;48,50° = 308;4,30°. But the entry for cycle 280 is 7;43,58°, i.e., it exceeds the expected value by 59;39,28°. Now the sum we computed for Elul, cycle 280, can be considered to have 4 components: 308;4,30° – 59;39,28° + 320;9,40° + 166;41,31° = 126;51,11°. The components 308;4,30° and 320;9,40° are just the result of multiplying the mean motion parameter, and so they are fixed. But it is unclear why Ibn al-Ḥadib did not consider as radix 107;2,3° (= 166;41,31° – 59;39,28°), combining the two other components. The only reason that comes to mind is that the creation, according to the Jewish tradition, took place at the beginning of the month Tishri which is associated with the autumnal equinox. Moreover, the purpose of the 19-year cycle is to keep each month in the same season. A radix of 107;2,3° entered in the month Tishri would mean that the creation took place closer to summer solstice than to the autumnal equinox. The same procedure also works for the mean motion in solar anomaly, that is, we find that the shift in the entries for cycles is the same amount, 59;39,28°. This is appropriate since the difference between the solar position and the solar

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anomaly is the longitude of the solar apogee; hence, a shift in one requires a shift in the other.

f. 57r. Tabula motus draconis This table consists of 3 sub-tables giving the argument of latitude (in signs, degrees, minutes, and seconds) for 13 consecutive months, 19 consecutive years, and groups of 19-year cycles (for cycles 1, 2, 3, …, 10, 20, 30, …, 100, …, 700), respectively, as in previous tables. It also displays a numerical value, 0s 15;20,7,1°, together with the words DE MONCATHA, referring to William Raymond of Moncada. Almost the same value for the argument of lunar latitude occurs in Ibn al-Ḥadib’s tables in Vatican, ms Heb. 379, f. 6b, where we find 0s 15;20,7º. It represents the motion of the argument of lunar latitude in half a mean synodic month. As was the case in previous tables, Mithridates’s table exhibits a higher precision than Vatican, ms Heb. 379. The entries for two successive 19-year cycles show a constant difference of 7;35,20°, as in the column for the “argument of latitude” in Ibn al-Ḥadib’s table in Vatican, ms Heb. 379, f. 6a, where this information is presented differently. This means that in one cycle the Moon progresses (255 · 360) + 7;35,20° = 91807;35,20°. When we divide this amount by the time between two consecutive 19-year cycles, 6939;41,22,38,20d, we obtain a value for the mean motion of the argument of latitude of 13;13,45,39,47,10°/d Subtracting the mean motion in argument of latitude from the lunar mean motion, we find –0;3,10,47,6,40°/d. Again, this value for the mean motion of the node is close to those of alKhwārizmī and Ibn al-Kammād, whereas the value for the mean motion of the argument of latitude is close to Ptolemy’s 13;13,45,39,48,56,37°/d. In the sub-table for single years the entry for year 1 is 5s 14;26,9°, whereas in the tables of Ibn al-Ḥadib the entry for year 1 is 0;0°. In the sub-table for months the entry for month 1 is 4s 29;29,28,46°, whereas for Ibn al-Ḥadib it is 6s 3;40,27° which, according to Ibn al-Ḥadib’s canons, is the radix for the conjunction of creation. We have no explanation for these differences but it seems likely that Mithridates has changed the epoch.

ff. 57v–60r. Quarundam urbium longitudines & latitudines This is a list of the geographical coordinates (longitude and latitude) of 435 localities, of which many are in Sicily, ordered from west to east. The coordinates are mostly given in degrees and minutes, e. g., for the city called Agrigen-

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tum we are given 38;30° (longitude) and 36;24° (latitude). However, those for Toledo are 10° (longitude) and 42° (latitude). The prime meridian for geographical longitude is the one used by Ptolemy (and many other medieval tables such as the Toledan Tables), passing through the Canary Islands, which he referred to as the “Fortunate Islands”. It differs from the so-called “meridian of water”, usually considered to be 17;30° to the west of Ptolemy’s prime meridian, and used in some Andalusian zijes.42

f. 60v This table is for transforming time-degrees (for each integer degree from 1° to 180°) into hours (from 0;4h to 12;0h). The following two tables have the same format as those on ff. 44r–45v.

f. 61r. Prime adequationes motus diversi in .xij. signis This table displays the correction for the lunar anomaly (in degrees, minutes, and seconds) as a function of the solar anomaly, given for each degree. The letters “s” and “a” in the margins have the same meaning as in Tables a, b, and c. The maximum is 2;43,35° at anomaly 3s 2° (see Table g). The entries are the same as those in the column labelled “correction for the lunar anomaly” in the corresponding table in Vatican, ms Heb. 379, ff. 7a–8a (see Table aa), where the same maximum value is found. We have not succeeded in accounting for the entries in this table despite the fact that its entries differ from those in Tables a and c by factors of proportion, and despite our expectation that the explanation for Table g should be analogous to that for Table h (see below).

f. 61v. Secunde adequationes motus diversi in totidem signis This table displays the correction for the lunar anomaly (in degrees, minutes, and seconds) as a function of the lunar anomaly, given for each degree. There is no indication in the table or the margins concerning the addition or sub-

42

M. Comes, “The ‘Meridian of Water’ in the Tables of Geographical Coordinates of alAndalus and North Africa”, Journal for the History of Arabic Science, x (1992–1994), 41–51.

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table g

First correction for the lunar anomaly

s

0[s]

1[s]

2[s]

3[s]

4[s]

5[s]

1 2 3 4 5 … 10 … 20 … 30

0; 2,47° 0; 5,32 0; 8,18 0;11,12 0;13,50

1;21,47° 1;24, 4 1;26,42 1;29, 8 1;31,33

2;22,28° 2;24,45 2;25,36 2;26,30 2;27,42

2;43,31° 2;43,35 2;43,34 2;43,28 2;43,19

2;24,30° 2;23,30 2;21,53 2;20,29 2;18,40

1;22,10° 1;20, 1 1;17,40 1;13,10 1;12,30

a

11[s]

0;27,15 1;43,15 2;33, 0 2;41,27 2; 9,36 0;59,10 0;55,25 2; 2,57 2;39,36 2;34,20 1;49,10 0;29,36 1;19,32 2;21,50 2;43,26 2;24,30 1;25,11 0; 0, 0 10[s]

9[s]

8[s]

7[s]

6[s]

traction of the entries, as was the case for Table d. The maximum is 5;16,55° at anomaly 3s 4° (see Table h). The entries are the same as those in the column labelled “correction for the lunar anomaly” in the corresponding table in Vatican, ms Heb. 379, ff. 8b–9b (see Table bb), where the same maximum value is found. As was the case with Table d, the entries in this table, here given in degrees, are equivalent to those in Table b, given in hours. If bi and hi are the corresponding entries in Tables b and h, then the expression hi = 0;32,40 · bi, where 0;32,40°/h, the velocity in lunar anomaly, yields good results. We note that this value is an approximation of the daily mean motion in lunar anomaly, 13;3,54°/d. Thus, Tables b, d, and h are identical, but for factors of proportion and for scribal errors, and underlying them is the lunar equation. In order to explain how the tables for the equations (Tables a, b, h, and g) work, and to compare the results derived from them with those based on other methods that depend on other tables for the same purpose, let us determine the time from mean to true syzygy for a given conjunction. Consider the mean conjunction of July 20, 1327 for which a comparison of the methods is

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isaac ibn al-ḥadib and flavius mithridates table h

1 2 3 4 5 … 10 … 20 … 30

Second correction for the lunar anomaly

0[s]

1[s]

2[s]

3[s]

4[s]

5[s]

0; 5, 5° 0;10,10 0;15,15 0;20,20 0;25,25

2;31,33° 2;36, 9 2;40,35 2;45, 7 2;49,25

4;26,12° 4;29, 9 4;31,26 4;33,42 4;36,26

5;16,17° 5;16,37 5;16,47 5;16,55 5;16,52

4;42,57° 4:41, 2 4;38,21 4;35,32 4;32,21

2;46,18° 2;41, 1 2;35,52 2;30,35 2;25,17

0;50,48 3; 7,39 4;48,20 5;15,50 4;10,40 1;57,58 1;40, 4 3;49,21 5; 6,26 5; 5,43 3;37,51 1; 0,12 2;28, 7 4;22,15 5;15,50 4;46,11 2;51,10 0; 0, 0 11[s]

10[s]

9[s]

8[s]

7[s]

6[s]

already available for the following astronomers: Ptolemy, Ibn al-Kammād, Yosef Ibn Waqār, John of Saxony, Levi ben Gerson, Immanuel Bonfils, Nicholaus de Heybech, and John of Gmunden.43 The basic magnitudes for that conjunction that occurred at 3;58,10h after noon in Toledo, are 35;25,4° for the mean solar anomaly and 222;26,7° for the mean lunar anomaly. In the case of the Sun, Ibn al-Ḥadib assumes that the solar anomaly does not change significantly from mean to true to conjunction (i.e., the correction for the solar anomaly is the same at mean and true conjunction). Thus, we enter with 35;25,4° in Table a and, after interpolation, we obtain –2;12,5h as the contribution of the Sun to the time between mean and true conjunction. The mean lunar anomaly has two corrections; the first depends on Table g and the second on Table h. With the corrected lunar anomaly, one enters Table b to find the contribution of the Moon to the time between mean and true conjunction. Following Ibn al-Ḥadib’s instructions in his canons (chapter 5), with the solar anomaly, 35;25,4°, we obtain in Table g a correction for the lunar anomaly of –1;32,34°. Thus, α1, the first corrected lunar anomaly, is 222;26,7° – 1;32,34° = 220;53,33°. With α1 as argument we enter Table h and obtain –3;41,36°; hence α2, the second corrected lunar anomaly, is 217;11,57°. Then with α2 as argument

43

Chabás and Goldstein, “True Syzygy” (ref. 11), 102.

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we enter Table b and obtain –6;17,38h as the contribution of lunar anomaly to the time from mean to true conjunction. Finally, we add the solar and lunar contributions and the result is very nearly –8;30h (= –2;12,5h – 6;17,38h) which is very close to the results using other tables.

Conclusion Our most important result is that Flavius Mithridates’s astronomical tables in Latin were almost entirely derived from Isaac Ibn al-Ḥadib’s Oraḥ selulah in Hebrew. Mithridates does not mention his principal source and perhaps this ‘oversight’ was intentional—in order to improve his standing with his patron, Federico of Montefeltro, Duke of Urbino, to whom these tables are dedicated. It is most likely that Mithridates, or Samuel ben Nissim as he was called before his conversion to Christianity, acquired a copy of Oraḥ selulah from his father, Nissim Abū l-Faraj, who was a student of Ibn al-Ḥadib in Sicily. Mithridates’s tables address problems concerning the times and circumstances of solar and lunar eclipses which in turn depend on finding the time and position of a true conjunction or opposition of the Sun and the Moon. Ibn al-Ḥadib’s goal, which he shared with many contemporary astronomers, was to compose a set of “user-friendly” tables that would ease the burden on the user. In order to achieve this, Ibn al-Ḥadib tried to separate the effects of the Sun and the Moon when computing the time from mean to true syzygy, following a system that has partially resisted explanation. As is the case with some other sets of tables in Hebrew compiled in the late Middle Ages (e.g., The Six Wings by Immanuel Bonfils, and the tables by Jacob ben David Bonjorn, both of which are cited at the beginning of Ibn al-Ḥadib’s canons: Vatican, ms Heb. 379, f. 1b), Oraḥ selulah and thus Mithridates’s tables concentrate on the motions of the luminaries and contain nothing concerning planetary motion or astrology. The astronomical tradition on which these tables are based ultimately derives (for the most part) from Ptolemy’s Almagest, and many attempts were made to build on it. In this respect, Muslim astronomers in al-Andalus played a major role in its transmission and elaboration. In particular, Azarquiel (d. c. 1100) and Ibn al-Kammād (fl. 1115) had a significant impact on the work of their successors in Spain and elsewhere. One new feature of this Andalusian tradition was to assign a proper motion to the solar apogee and Ibn al-Ḥadib (and hence Mithridates and Gascon) accepted it. In contrast, the Toledan Tables and the Parisian Alfonsine Tables have nothing about a proper motion of the solar apogee. In sum, given the wealth and variety of astronomical tables in the late Middle Ages that we have explored in recent years, we are

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increasingly convinced that the analysis and comparison of such tables is a powerful tool for understanding the diffusion and elaboration of ideas and methods in mathematical astronomy.

Acknowledgements We thank Y.T. Langermann for sending us a copy of the manuscript containing Flavius Mithridates’s tables, and B. Porres for her help in reading some passages in the Latin text. Biographical data about Mithridates appears both in Hebrew and Latin on the last page of Munich, ms Heb. 246, but in the film available in Jerusalem that has been used by most recent investigators the Latin in very hard to read. We are therefore grateful to P. Kunitzsch who inspected the manuscript and reported that it has been seriously affected by water damage, that is, the manuscript itself has deteriorated.

part 4 Other Tables

∵

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Ibn al-Kammād’s Star List* The study of medieval astronomical texts has provided many examples of the interaction of scholars writing in Arabic, Latin, and Hebrew in a tradition that largely depended on Ptolemy’s Almagest, written in Greek in the 2nd century, ad. The influence of the Almagest was sometimes direct, for there were translations of it into all three of these languages, but sometimes it was indirect through adaptations and abridgements. Star lists have been shown to exemplify the variety of transmission and adaptation of the Almagest, largely due to the efforts of Paul Kunitzsch who has examined, edited, and classified a great number of such texts in Latin and Arabic (Kunitzsch 1959, 1966, 1986, 1990; on star lists in Hebrew, see Goldstein 1985b and Fischer et al. 1988). In addition to translations of Ptolemy’s catalogue of over 1000 stars, there were also a great number of short lists, corrected for precession from the epoch of Ptolemy’s catalogue (137ad) to some later epoch. In this article we will be concerned with a list of 30 stars that clearly depends on Ptolemy’s star catalogue, and it is represented in texts in Arabic, Latin, and Hebrew. This star list by Abū Jacfar Aḥmad b. Yūsuf Ibn al-Kammād, an Andalusian astronomer active in Córdoba in the 12th century, was surprisingly successful, if we are to judge by its persistence over many centuries. Although working in Córdoba, Ibn al-Kammād was probably from Sevilla; the date 510 Hijra (1116/17) in relation to Ibn al-Kammād is mentioned in the zij by Ibn Isḥāq of Tunis (fl. early 13th century) preserved uniquely in ms Hyderabad, Andra Pradesh State Lib., 298. Ibn al-Kammād is supposed to be the author of three zijes, two of which (al-Kawr ʿalā al-dawr and al-Amad ʿalā al-abad) are not extant, while the other (al-Muqtabis) is only preserved in a Latin translation, completed by John of Dumpno in 1260 in Palermo.1 Al-Zīj al-Muqtabis includes quite a number of astronomical tables, as well as 30 canons. It is only extant in a 14th century Latin manuscript now at the Biblioteca Nacional de Madrid,

* Centaurus 38 (1996), 317–334. 1 We continue to use the term al-Muqtabis that generally appears in the secondary literature on Ibn al-Kammād, but we are informed by Prof. Dr. Kunitzsch that the correct vocalization is al-Muqtabas (in the passive, rather than the active, voice). He also informed us that, according to the catalogue of Naqshabandi (1982), there is an Arabic text of Ibn al-Kammād in Baghdad: Iraq Museum, ms 296 [782].

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_013

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ms 10023, and has been described extensively in a recent study (Chabás and Goldstein 1994). In general, the canons offer an explanation of the tables, but these explanations are restricted to the way the tables are to be used with no hint of the method by which they were constructed. Among the many tables in al-Muqtabis there is one that we noted briefly in our previous paper. In ms Madrid 10023, f. 47r, we find a list of stars and, for each of them, the following information is given: magnitude, name, ecliptic coordinates (longitude and latitude), and the planets associated with it for astrological purposes. As is often the case in al-Muqtabis, the canons give no valuable information on the star table. This list was very popular among astronomers in medieval Spain, and it was still being transmitted as late as the very end of the 15th century, independently of the rest of al-Muqtabis. We have located this star list in the following Latin, Arabic, and Hebrew manuscripts. Note that several Hebrew manuscripts give the star names in Arabic written in Hebrew characters. Latin: ms Madrid 10023, f. 47r ms Vienna 5311, f. 129v

(A) (W)

Arabic: ms Chester Beatty 4087, f. 41a ms Escorial Ar. 909, f. 33b ms Museo Naval de Madrid (unnumbered), f. 31b ms Cairo dm 1081

(C) (E) (N)

Hebrew: ms Vatican heb. 356, f. 65b ms Vatican heb. 379, f. 195b ms Sassoon 823, p. 193 ms Vatican heb. 390, f. 163b ms London, Jews College, heb. 135, f. 149a ms Munich heb. 230, f. 59a

(B) (V) (S) (T) (J) (M)

Some of these manuscripts add other information on the stars, but all retain what is specific to Ibn al-Kammād’s list. Moreover, there are related lists in other Arabic, Latin, and Hebrew manuscripts that share an important characteristic

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375

with this list, namely the stellar longitudes are 6;38° greater than those found in the Almagest. Our interest in this star list is to establish the character of a medieval astronomical tradition, rather than to reconstruct its original form. Accordingly, we present transcriptions of selected manuscripts with a full set of variants from some others. In our view, adding more variants would serve no useful purpose. ms A, the closest witness we have to the work of Ibn al-Kammād, serves as the base manuscript for the present edition. In this manuscript the names of the stars are given in Latin (11 items) or are transliterated from Arabic (19 items). In Table 1, we list the star names in Latin, Arabic, and Hebrew in mss A, V, and B, respectively.2 In Table 2 we display the longitudes, latitudes, magnitudes, and associated planets in ms A, as well as the corresponding modern designations of the stars. In Table 3 we show the longitudes that appear in various manuscripts and compare them with the corresponding longitudes in the Almagest, adding 6;38° to each of them. In Table 4 we present the latitudes that appear in the same manuscripts and compare them with the corresponding latitudes in the Arabic versions of the Almagest. Kunitzsch (1966, pp. 99–102 [Type xv]) transcribed this star list as it appears in ms W, a 15th-century Latin manuscript, along with variants from ms A. The entries in ms W agree very well with the values derived from the Almagest, but we do not cite any variants from Kunitzsch’s edition of that list, for it is independent of ms A, as we argue below. However, we do give the variants from ms W for the associated planets, for they do not appear in Kunitzsch’s edition (see Notes to Table 2). In Ptolemy’s Tetrabiblos (i, 9; ed. Robbins 1964, pp. 46ff.), the powers of specific fixed stars are associated with planets; there are many lists in medieval manuscripts that depend on this passage. Indeed, the planets associated with the fixed stars in our list are in good agreement with those described by Ptolemy. Despite the corruption of the text in all the surviving manuscripts that affects both the star names and the coordinates, the stars are generally not difficult to identify (see Table 2). The difference in longitude of 6;38° for each star as compared with its longitude in the Almagest is a plausible amount of precession for 485 years (taking the Hijra as the epoch for the longitudes in this list, and 137ad as the epoch of Ptolemy’s catalogue: 622 – 137 = 485 years), for a precession of 6;38° in 485 years yields a rate of 1° in about 73 years (for

2 For the Arabic terms we have followed the spellings that appear in the Hebrew manuscript cited. Note that in the Middle Ages Jews often used Hebrew script to write Arabic. For the standard Arabic forms, see the various works of Kunitzsch (e.g., Kunitzsch 1959, 1966).

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table 1

Star names in Ibn al-Kammād’s list

Star Latin: no. ms A

Arabic: ms V

Hebrew: ms B

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.

akhir al-nahr ra’s al-ghūl al-dabaran rijl al-jawzā al-cayyūq mankib al-jawzā suhail al-cabūr al-tawm al-shamālī al-tawm al-janūbī al-ghūmayṣa faqar al-shujāb qalb al-asad mijraf al-safiṭ matn al-asad dhanab al-asad al-aczal al-rāmiḥ munīr al-fāṣa al-kiffa al-janūbiyya al-kiffa al-shamāliyya jabhat al-caqrab rukbat al-rāmī

aḥarit ha-nahar rosh ha-saṭan ceyn ha-shor regel teʾomim ha-nilḥam ketef teʾomim kesil ha-kelev ha-gadol

raselgul achar emahar adebran pes geminorum algaihuc brachium geminorum suheil alhabur teum septentrionalis teum meridionalis algumeisa (…)el sugia cor leonis migdeb elsaffima corpus leonis cauda leonis alagzel arremach munir elfike alfike septentrionalis alfike septentrionalis hayt elhacrab cor scorpionis rucbit errami annesir elgueca annesir ettair cauda piscis cauda gallina brachium (…) alkef elchadib

rukba al-dajāja mankib al-farīr al-kaff al-khaṣīb

satum ha-cayin ḥuliyot ha-gibbor lev ha-aryeh ṣawar ha-aryeh zenav ha-aryeh zenav ha-aryeh al-aczal bacal ha-romaḥ meʾir ha-catarah ha-kaf deromit ha-kaf ha-ṣefonit meṣaḥ ha-caqrav lev ha-caqrav arkuvat semurah(?) be-qeshet nesher nofel nesher mecofef pi ha-dag zenav ha-tarnegolet shekem ha-sus ha-yad ha-ṣevucah

377

ibn al-kammād’s star list table 2

Star no. 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.

Longitudes, latitudes, magnitudes, and associated planets of the stars in Ibn al-Kammād’s list: ms A

Longitude Ari Tau Tau Tau Gem Gem Gem Gem Gem Cnc Cnc Leo Leo Leo Leo Vir Lib Lib Lib Lib Lib Sco Sco Sgr Sgr Cap Aqr Aqr Psc Psc

6;10° 6;18° 19;28° 26;38° 1;38° 8;38° 23;49° 24;39° 24;59° 3;18° 5;49° 6;38° 9; 8° 15; 8° 24;44° 1; 8° 3;38° 3;38° 21;18° 24;38° 28;49° 12;18° 19;18° 23;49° 23;58° 10;28° 13;38° 15;48° 8;49° 24;10°

Latitude *

Magn. #

(–)13;30° 23; 0° –5;10° –31;30° 22;30° –17; 0° –75; 0° –39;10° 9;40° 6;15° –16;10° –20;30° 0;10° –24;40° 13;40° 11; 7° –2; 0° 31;30° 44;30° 0;40° 8;30° –1;20° –3; 0° –18; 0° 62; 0° 29;10° –23; 0° 40; 0° 31; 0° 29; 0°

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

Assoc. planets Jup Mar? Mar ? Ven/Mer Mar? Sat Jup?/Mar Mer/Ven Mar/Ven Mer/Mar Sat/Ven Mar/Jup Sat/Jup Sat/Mar Sat/Ven Ven/Mer Mar/Jup Ven/Mer Sat/Mer Sat/Mar Sat/Mer Mar/Jup Jup/Mer Ven/Mer Mar? Mar/Mer Ven?/Mer Ven/Jup Ven

Modern designation θ β α β α α α α α β α α α χ δ β α α α α β β α α α α α α β α

Eri Per Tau Ori Aur Ori Car CMa Gem Gem CMi Hya Leo Car Leo Leo Vir Boo CrB Lib Lib Sco Sco Sgr Lyr Aql PsA Cyg Peg And

378 table 3

Star no. 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.

chapter 11 Longitudes of the stars in various manuscripts

Longitude Alm.+ 6;38° Ari Tau Tau Tau Gem Gem Gem Gem Gem Cnc Cnc Leo Leo Leo Leo Vir Lib Lib Lib Lib Lib Sco Sco Sgr Sgr Cap Aqr Aqr Psc Psc

6;48° 6;18° 19;18° 26;28° 1;38° 8;38° 23;48° 24;18° 29;58° 3;18° 5;48° 6;38° 9; 8° 15; 8° 20;48° 1; 8° 3;18° 3;38° 21;18° 24;38° 28;48° 12;58° 19;18° 23;38° 23;58° 10;28° 13;38° 15;48° 8;48° 24;28°

A

M

B

V

S

J

6;10 = 19;28 26;38 = = + 24,39 24;59 = + = = = 24;44 = 3;38 = = = + 12;18 = 23;49 = = = = + 24;10

+ = = = = = + = + = + = = = = = = = = = + = = 23;49 = 10;18 = 10;48 + 24;10

+ 6;38 = = = = + = + = + = = = 20;44 Lib = = = = = + = = 23;49 = = 13;58 = + 24;10

+ 6;38 = = = = + = 27;59 = + = = 9; 8 20;44 Lib = = 3;35 = = + = = 23;49 = = 13;58 = + 24;10

+ 7;38 = = = = + = 24;59 = + = = = 20;42 Lib = = = = = + 12;29 = 23;49 = = 13;58 = + 24; 4

= = = = = = + = = = = = = = 20;44 1; 5 = = = = + 12;18 = 23;49 = = = = + 24; 8

379

ibn al-kammād’s star list table 4

Star no. 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.

Latitudes of the stars in various manuscripts

Latitude Ar. Alm.

A

–53;40° (–)13;30 23; 0° = –5;10° = –31;30° = 22;30° = –17; 0° = –75; 0° = –39;10° = 9;40° = 6;15° = –16;10° = –20;30° = 0;10° = –69;40° –24;40 13;40° = 11;50° 11; 7 –2; 0° = 31;30° = 44;30° = 0;40° = 8;50° 8;30 1;20° –1;20 –4; 0° –3; 0 –18; 0° = 62; 0° = 29;10° = –23; 0° = 60; 0° 40; 0 31; 0° = 26; 0° 29; 0

M

B

V

S

J

–13;30 = = = = = = = = = = –22;30 = = = = = = = = 8;30 –1;20 = = = = = = = 29; 0

–13;30 = = = = = = = = = = = 0; 6 –49;40 = 11; 7 = = = 0;20 8; 0 –1; 0 –3; 0 –18;10 42; 0 29; 0 = = = 29; 0

–13;30 = –5; 0 = = = = –89,10 = = = = = –49;40 = 11; 7 = = = 0;20 8; 0 –1; 0 –3; 0 –18;10 42; 0 29; 0 = = 37; 0 29; 0

–13;30 = = 31;30 = = –55; 0 = = = = = = –49;40 = 11; 7 = = = 0;24 8; 0 1; 0 –3; 0 –18;10 42; 0 29; 0 = = 38; 0 29; 0

–13;30 = = = 24;30 = = = = = = –23;30 = –49;40 = 11; 7 = 24;30 = 0; 6 8;20 –1;40 –3; 0 = = 29; 0 = = 41; 0 =

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Table 1. Variants in mss M, S for the entries in Arabic: (1) M: akhir al-nahār. (9) S: ?. (12) M: qafār al-shujāc; S: qafār al-sūjay. (14) M: mijdāb al­safina; S: miqdaf al-safiṭ. (19) M, S: munīr al-fakka. (23) M, S: qalb al-caqrab. (24) M: rukbat al-dalw; S: danbat al-rāmī. (25) M, S: al-nasr al-wāqic (26) M, S: al-nasr al-ṭāyir. (27) M, S: fum al-ḥūt. (28) M, S: danab al-dajāja. (29) M: mankib al-faras; S: mankib al­faraṣ. (30) M: al-kaff al-khaḍīb; S: al-kaff al-naṣīb. Some of the entries in ms V are corrupt or missing, and we offer the following suggestions: (12) faqār al-shujāc (with ms E). (14) mijdāf al-safina (the rudder of the ship: cf. Kunitzsch 1966, p. 100). (19) munir al-fakka (with mss E, M). (23) qalb al-caqrab (with mss E, M). (24) rukbat al-rāmī (with ms E, and ms V, No. 23). (25) al-nasr al-wāqic (with mss E, M, S). (26) al-nasr al-ṭāyir (with mss E, M, S). (27) fum al-ḥūt (with mss E, M, S). (28) dhanab al-dajāja (with mss E, M, S)—note that ‘d’ and ‘dh’ in Arabic script differ by a dot; hence the readings in M and S, cited above, do not disagree with ms E. (29) mankib al-faras (with mss E, M). (30) al-kaff al-khaḍīb (with mss E, M). In ms B the entries for (14) and (15) are corrupt. Notes to Table 2: (*) Positive and negative values for latitude represent “north” (septentrionalis) and “south” (meridionalis), respectively, and the symbol (–) indicates that meridionalis in the ms is faint. The number of degrees for the latitude of star (3) is difficult to read: the “5” is probably a correction in a later hand, written over a “1”. (#) The magnitudes in mss M, B, V, and S agree with those in ms A in all cases. The magnitudes in ms J agree with those in ms A in all but one case: see note for star 17. Where variants are given below, “m” stands for Magnitude, and “a” stands for Associated Planets. 2(1) a(W): Jup/Ven. (2) a(W): Mar. (4) a(W): Sat/Jup. (6) a(W): Mar/Mer. (8) a(W): Jup/Mar. (10) a(W): Mar. (17) m(J): 2. (22) In the Almagest the magnitude is 3. (23) In the Almagest the magnitude is 2. a(W): Jup/Mar. (26) a(W): Mar/Jup. (28) In the Almagest the magnitude is 2. a(W): Ven/Mer. (30) Also called δ Peg. Note to Table 3: The symbol = means that the value in that ms is the same as in the Greek version of the Almagest (as reported in Toomer 1984), plus 6;38°; the symbol + means that the value in that ms is the same as in the Greek version of the Almagest, plus 6;39°. Since both values are represented in all mss, this variation is not due to copyists’ errors.

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Note to Table 4: Positive and negative values for latitude represent “north” and “south”, respectively, and the symbol (–) indicates that meridionalis in the ms is faint. The symbol = means that the value in that ms is the same as in the Arabic versions of the Almagest (see Kunitzsch 1986). In two cases the Arabic versions of the Almagest differ from the edited Greek text which has the following values: (9) 9;30° [one Greek ms reads 9;40°], (27) –20;20° [one Greek ms reads –23;0°]. For (22), one Arabic version (al-Ḥajjāj) has 1;20° in agreement with the Greek text, while the other (Isḥāq) has 1;30°. similar values, see Goldstein 1994). Since the coordinates in the planetary tables of al-Muqtabis are sidereally fixed for the beginning of the Hijra, it is natural to expect the star list to be arranged for that date as well. But the epoch for the star list has been the subject of some controversy. Based on a passage in the Treatise on the motion of the fixed stars by Azarquiel (11th century), Comes (1991, p. 86) has claimed that the values for the longitudes of the stars were calculated for the beginning of the motion in precession according to Azarquiel’s first model, a date which she computes as 25 January 581 (about 41 years prior to the Hijra). The passage reads as follows (cf. Millás 1943, p. 316): The ratios and the amounts of the motions approximate [the following amounts?] with respect to the greatest distance on the circle [i.e., apogee] according to the first hypothesis: if the first point of Aries at [the time of] the observation of Hipparchus was behind [i.e., in regress] the vernal point to the west by 9 degrees exactly [lit.: complete], and the distance of the point from the greatest distance was 72 4/5 degrees, and also the regress of the head of Aries from the observation of Ptolemy was 6;38 degrees, and the distance of the point from the greatest distance was 45 3/4 degrees, and at the [time of] the observation of al-Battānī the head of Aries had progressed 4;18 degrees and the distance of the point was in the opposite direction from the greatest distance 27 1/4 degrees, and the head of Aries had also progressed at [the time of] our observation 6 3/4 degrees, and the distance of the point from the greatest distance is 46;5 degrees. The key remark in this passage is that at the time of Ptolemy, the longitudes of stars were 6;38° less than they were at the epoch of Azarquiel’s first model for precession. According to Samsó (1994, viii, pp. 26–27), the parameters

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as preserved do not yield consistent results. But, in the same treatise by Azarquiel (cf. Millás 1943, p. 338; Samsó 1994, viii, p. 22), concerning the third model, we read: Indeed, when the head of Aries is on the equator, i.e., when the head of Aries is coincident with the vernal point, as happened about 40 years before the Hijra, that is, at the moment of the Prophet’s birth, because the positions of the Sun, Moon, and Stars calculated with the Mumtaḥan system were, at that time, close to those derived with the Indian and Iranian systems, which are similar. This epoch, 581ad, may not have been Ibn al-Kammād’s date for his star list, for he never mentions it, and it was apparently unknown to his successors as well. Moreover, according to the standard Muslim tradition, Muḥammad was born in the year of the Elephant (570/71ad), about forty years before receiving his first prophetic message (see, e.g., Watt 1953, p. 33; von Grunebaum 1970, p. 28; Pingree 1968, p. 34 n). Indeed, Ibn al-Kammād’s readers probably considered the Hijra to be the epoch of this star list, because that is the epoch of the other tables in his zij. One of his successors, Abū l-Ḥasan cAli al-Marrākushī (13th century), explicitly gives this date as the epoch, but it is not found in any other text we have seen. Although one can recognize some of the essential characteristics of our list in al-Marrākushī’s table, the latter is not a variant of our list because it has 240 stars rather than Ibn al-Kammād’s 30 stars. On the other hand, the list by Ibn al-Bannāʾ (1256–1321) is so close to that of Ibn al-Kammād that we take it to be a variant of it. Al-Marrākushī’s table includes all the stars listed by Ibn al-Kammād, and all the stellar longitudes in it are also 6;38° greater than those in the Almagest. The table showing the ecliptic coordinates bears a date in the heading: the beginning of the Hijra (cf. Sédillot and Sédillot 1834, p. 140), and this date is corroborated in several passages of the accompanying text. Three Arabic manuscripts (mss C, E, N) that contain our list are copies of the zij of Ibn al-Bannāʾ (see Vernet 1952). We have not included variants from these manuscripts although we have used one of them for establishing the Arabic names of some of the stars (see Notes to Table 1). The zij of Ibn al-Bannāʾ, entitled Minhāj al-ṭālib li-tacdīl al-kawākib, is preserved in ms Escorial Ar. 909 (ms E), ms Chester Beatty 4087 (ms C: very faint in the microfilm copy available to us), ms Museo Naval de Madrid (ms N), and ms Alger 1.454 (cf. Samsó 1994, Essay x, p. 2: this ms was not available to us). In ms E, f. 33b, there is a list of 28 stars (cf. Vernet 1952, p. 107); ms N has the same list (with some variants), and it has been reproduced photographically in Comes (1991), p. 93.

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All stars in Ibn al-Kammād’s table are included in Ibn al-Bannā’s list, except for Nos. 9, 10, and 20. One additional star is also listed: ʿunq al-ḥayya for which Sco 0;58° and 25;20° are given as its longitude and latitude, respectively. The list of Ibn al-Bannāʾ does not include the associated planets. In the heading for this list Ibn al-Bannāʾ has: “The positions of the fixed stars from the proper [or: very] beginning (min al-mabdaʾ al-dhātī)”. If this refers to a date, it seems unlikely that it would be year 581. However, a more likely interpretation of this difficult passage is that the expression “the proper beginning” refers to a point on the ecliptic other than the vernal equinox, rather than to the epoch of the list.3 Ibn al-Kammād’s list is found in various manuscripts in addition to ms A and the zij of Ibn al-Bannāʾ. We begin with the Tables of Barcelona (Millás 1962), and we will pay particular attention to mss B, V, and S (all in Hebrew characters). This set of tables was compiled in Barcelona under the patronage of King Pere el Ceremoniós (1319–1387) for epoch 1321, and no star list has previously been associated with it. Recently, it has been shown that these tables, specifically those concerning eclipse theory, depend very strongly on the work of Ibn al-Kammād, whose star list has now to be considered as the star list of the Tables of Barcelona (Chabás 1996). Another copy of this list is found in ms T, a 15th-century Hebrew manuscript, whose variants are not reported here. Next, we note that Joseph Ibn Waqār compiled a set of astronomical tables for Toledo in 1359/60 that is uniquely extant in ms M (cf. Goldstein 1985a, p. 237), and it includes the same list that we find in Ibn al-Kammād’s zij with the star names in Arabic written in Hebrew characters. Of some interest for us is that the heading for the table is almost identical with the heading in ms E: curiously, the words “from the beginning” (min al-mabdaʾ) have been omitted while retaining the word for “proper” or “very” (al-dhātī). Finally, ms J contains an extensive set of astronomical tables for Burgos compiled by Juan Gil de Castiello (ca. 1350: cf. Rubió 1908, pp. 152, 155–156, 164; Beaujouan 1969, p. 11; Goldstein 1985a, p. 237). On f. 149a there is a list of stars that includes all those in Ibn al-Kammād’s list, adding two more: mankib al-asad and danab al-jady. Their longitudes are given as Leo 8;48° and Aqr 3;0°, and their latitudes 8;20° N and 2;10° S, respectively. They can be identified with γ Leo and γ Cap (or δ Cap), respectively. Besides the name,

3 We are grateful to Prof. Dr. Kunitzsch for this suggestion, based on a parallel passage in the Arabic text of al-Marrākushī’s Comprehensive Collection of Principies and Objectives in the Science of Timekeeping (Sezgin 1984, i, 42; cf. Sédillot and Sédillot 1834, pp. 126–131).

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magnitude, ecliptic coordinates, and associated planets for each star, Juan Gil’s table also gives declinations and co-culminations. Although it has additional information, Juan Gil’s table is clearly dependent upon that of Ibn al-Kammād, whose name he cited repeatedly and whose work he used extensively. Abraham Zacut (d. after 1515, known in Latin as Abraham Zacutus) was a Jewish astronomer in Spain at the time of the expulsion in 1492, and spent the rest of his life in Portugal, North Africa, and Palestine (Cantera 1931; Cantera 1935; Goldstein 1981). His main astronomical work, ha-Ḥibbur ha-Gadol, was written in Hebrew and subsequently translated into Latin and several other languages. A version of his work was published in 1496 in Leiria (Portugal) under the title Tabulae tabularum coelestium motuum sive Almanach Perpetuum. According to the colophon, José Vizinho was responsible for this edition which contains a summary of the canons and most of its tables. In fact, two editions came out of the printing house that year: one with the canons in Castilian, the other in Latin; the tables are identical in the two editions but for their order. The Leiria edition has a list of coordinates of 56 stars but, oddly, the names of the stars are omitted, making this list of dubious value to the reader. The Hebrew original of this list appears in ms Lyon heb. 14, ff. 114v–114r (folio numbers were assigned to this ms in the Latin order rather than in the Hebrew order: the text begins on f. 215v). The Hebrew version has 61 entries arranged as follows: magnitude, star name, longitude, latitude, and associated planets; the Leiria edition preserves the first 56 entries of that list with very few variants. A similar star list is found in a Latin manuscript of Zacut’s Almanach Perpetuum (ms Madrid 3385, f. 101v) with 83 entries. Here the longitudes are 14° greater than those for the same stars that also appear in the Leiria edition. The correction of 14° is for the precession from the epoch of Ibn al-Kammād’s star list (which we take to be 622ad) to some later unspecified epoch, presumably near the end of the 15th century. This is the only example we have found where this type of star list has been corrected for precession. From Tables 3 and 4 it is evident that the lists in mss B, V, and S are closely related; this is not surprising, for they all belong to the Tables of Barcelona. It is also clear that the entries in M are closest to those derived from the Almagest, after adding 6;38° or 6;39° in the case of the longitudes. Regarding the latitude of star 16, note that “7” and “50” in Arabic alphabetic nmerals are easily confused. Regarding the latitude of star 30, the entries in all mss displayed in Table 4 except J read 29;0°. The Almagest, as well as mss J and W, reads 26;0°, and in ms W the “26” is corrected (above) to “29”. The same value, 29;0°, is also found in mss C and E, as well as in the earlier star lists of al-Qaṭṭān and Maslama al-Majrīṭī (see below). For star 22 all mss in Table 4 (except S) read “south” while for Ptolemy it is a northern star. ms W also reads “south” but, comparing tables 3

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and 4 with ms W, we conclude that this manuscript lies in a different line of transmission. First, in ms W only 6;38° occurs as the longitudinal difference with respect to the Almagest, whereas our mss have used both 6;38° and 6;39° (see Note to Table 3). Moreover, the longitude of star 24 in ms W agrees with the entry in the Almagest, Sgr 23;38°, whereas all our manuscripts have Sgr 23;49°. In the case of the latitudes, for star 1 ms W has –53;30° in agreement with the Almagest, whereas the entry in all the mss displayed in Table 4 is –13;30° (as is also the entry in ms E). As we have already indicated, Ibn al-Kammād’s list has a characteristic value for precession of 6;38° and a specific set of 30 stars. So far in this paper we have concentrated on lists with the same value for precession. Now we turn our attention to texts of early Islamic astronomers that contain lists with different values for precession but a similar set of stars. There follow some of the results of our search. Ibn al-Kammād’s list does not seem to bear any close relation to the star catalogue (epoch 880ad) in the widely influential zij by al-Battānī (d. 929: cf. Nallino 1903–1907, 2:144ff.). Moreover, Ibn al-Kammād’s list differs from the list (epoch: 1104ad) by Abraham bar Ḥiyya of Barcelona of 28 stars of magnitudes 1 and 2 that depends on al-Battānī’s catalogue (cf. Goldstein 1985b; Millás 1959). The first documented star list in al-Andalus seems to be the list of 16 stars in the Kitāb al-hayʾa by Qāsim ibn Muṭarrif al­Qaṭṭān (mid-10th century). It was calculated for the city of Córdoba for ah300 (ad912/13), and it is extant in ms Istanbul Carullah 1279 (Comes 1994, p. 95). All the stars in this list are also found in our table. The two extant lists associated with Maslama al-Majrīṭī (d. 1007) were calculated for the end of year ah367 (ad978: cf. Vernet and Català 1965, p. 45; and Kunitzsch 1966, pp. 16–17, who calls them “a” and “a”). They contain 21 stars: 18 of them are also found in Ibn al-Kammād’s list (assuming that the star al-kaff al-khaḍīb listed by Kunitzsch (p. 17) is not β Cas but α And: No. 30 in Ibn alKammād’s list). Surprisingly enough, the latitude of α Sco in Maslama’s list “a” is –3;0°, and the same erroneous value is also found in our list for that star (see Table 4). This suggests a common ancestor. All stars in Ibn al-Kammād’s list, except for six (Nos. 1, 6, 14, 15, 19, 22), are also found in the Toledan Tables (cf. Toomer 1968, pp. 123 ff.). In al-Zīj al-Muqtabis, Ibn al-Kammād compiled tables from different sources, and so it is perhaps natural to expect that his star list included in that zij also derives from a previous work. We have not succeeded in finding a clear ancestor for it, but there is little doubt that the origin of this table lies in al-Andalus. To be sure, there are very few comparable texts prior to Ibn al-Kammād, and some of them still need to be studied. According to Samsó (1992, p. 157), a star

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table for the beginning of the Hijra was included in the Tabulae Jahen, the zij of Ibn Mucādh al­Jayyānī (d. 1093). This text is only known through a Latin translation printed by Joachim Heller in Nuremberg in 1549 under the title Scriptum cuiusdam saraceni, continens praeterea praecepta ad usum tabularum astronomicarum utilissima. Unfortunately, none of the astronomical tables of Ibn Mucādh is extant. The relevant passage at the end of chapter 11 of his canons reads as follows: Indeed, I have already made tables for the fixed stars, which show the positions for the best known [stars] at the beginning of the Hijra (ad initium repulsionis), as well as tables for the annual and monthly motions. Then take the position of whatever star you wish at the beginning of the Hijra (ad initium repulsionis), and add. … We have not found any other Latin text where repulsionis is used to translate “Hijra”, but F.S. Pedersen brought to our attention an example where expulsionis is so used. In On the Solar Year, extant both in Arabic and Latin (usually ascribed to Thābit Ibn Qurra, but see now Morelon 1987, pp. xlviii ff.), we find the following parallel passages: Then later we measured the autumnal equinox that took place in year 215 of the years of the Hijra (min sinī al-hijra), [which corresponds to] year 199 of the years of Yazdegerd … translated from the Arabic text in morelon 1987, p. 33

Post autem hoc comparauimus equinoctium autumpnale quod fuerit in anno 215 annorum Expulsionis in anno 199 Iezdegerd … carmody 1960, p. 68

To summarize our results: in addition to the version preserved in the Latin translation of Ibn al-Kammād’s zij, this star list is found in the Tables of Barcelona (in Hebrew mss), the tables of Joseph Ibn Waqār (in a Hebrew ms), the tables of Juan Gil (in a Hebrew ms), and the zij of Ibn al-Bannāʾ (in Arabic mss). Closely related to this list are versions preserved by al-Marrākushī (in Arabic) and Abraham Zacut (in Hebrew and Latin). We have not found the ancestor for this list, but we think it likely to have been compiled in Islamic Spain either by Ibn al-Kammād himself or by one of his predecessors in the circle of Maslama al-Majrīṭī.

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Acknowledgements We are most grateful to M. Comes (Barcelona), P. Kunitzsch (Munich), Y.T. Langermann (Jerusalem), F.S. Pedersen (Copenhagen), and J. Samsó (Barcelona) for their many helpful suggestions.

Note Added in Proof After the completion of this article, we found another list of 31 stars, with associated planets but no longitudes or latitudes, in ms Paris, BnF, lat. 7324, f. 51v. This list has the same stars in the same order, with a few variants, as in the list of Ibn al-Kammād.

References Beaujouan, G. 1969: “L’astronomie dans la péninsule ibérique à la fin du moyen âge”, Revista da Universidade de Coimbra, 24: 3–22. Reprinted in G. Beaujouan 1992, Science medieval d’Espagne et d’alentour. Aldershot (Hampshire). Cantera Burgos, F. 1931: “El Judío Salmantino Abraham Zacut”, Revista de la Academia de Ciencias Exactas, Físicas y Naturales de Madrid, 27: 63–398. Cantera Burgos, F. 1935: Abraham Zacut. Madrid. Carmody, F.J. 1960: The Astronomical Works of Thabit b. Qurra. Berkeley and Los Angeles. Chabás, J. 1996: “Astronomía Andalusí en Cataluña: las Tablas de Barcelona”, in J. Casulleras and J. Samsó (eds.), From Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof Juan Vernet. Barcelona. Pp. 477–525. Chabás, J., and Goldstein, B.R. 1994: “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for the History of Exact Sciences, 48: 1–41. Comes, M. 1991: “Deux échos andalous à Ibn al Bannāʾ de Marrakush”, in Le patrimoine andalou dans la culture arabe et espagnole. Tunis. Pp. 81–93. Comes, M. 1994: “La primera tabla de estrellas documentada en al-Andalus”, in J.M. Camarasa, H. Mielgo, and A. Roca (eds.), Actes de les i Trobades d’Història de la Ciencia i de la Técnica, Barcelona. Pp. 95–106. Fischer, K., Kunitzsch, P., and Langermann, Y.T. 1988: “The Hebrew Astronomical Codex ms. Sassoon 823”, Jewish Quarterly Review, 78: 253–292. Goldstein, B.R. 1981: “The Hebrew Astronomical Tradition: New Sources”, Isis, 72: 237– 251. Goldstein, B.R. 1985a: “Scientific Traditions in Late Medieval Jewish Communities”, in G. Daban (ed.), Les Juifs au regard de l’histoire. Paris. Pp. 235–247.

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Goldstein, B.R. 1985b: “Star Lists in Hebrew”, Centaurus, 28: 185–208. Goldstein, B.R. 1994: “Historical Perspectives on Copernicus’s Account of Precession”, Journal for the History of Astronomy, 25: 189–197. Grunebaum, G.E. von 1970: Classical Islam: A History 600ad–1258 ad. Chicago. Kunitzsch, P. 1959: Arabische Sternnamen in Europa. Wiesbaden. Kunitzsch, P. 1966: Typen von Sternverzeichnissen in astronomischen Handschriften des zehnten bis vierzehnten Jahrhunderts. Wiesbaden. Kunitzsch, P. 1986: Der Sternkatalog des Almagest: Die arabisch-mittelalterliche Tradition, i: Die arabischen Übersetzungen. Wiesbaden. Kunitzsch, P. 1990: Der Sternkatalog des Almagest: Die arabisch-mittelalterliche Tradition, ii: Die lateinische Übersetzung Gerhards von Cremona. Wiesbaden. Millás, J.M. 1943–1950: Estudios sobre Azarquiel. Madrid–Granada. Millás, J.M. 1959: Libro del cálculo de los movimientos de los astros de R. Abraham bar Ḥiyya ha­Bargeloni. Barcelona. Millás, J.M. 1962: Las Tablas Astronómicas del Rey Don Pedro el Ceremonioso. Madrid– Barcelona. Morelon, R. 1987: Thābit ibn Qurra: Oeuvres d’astronomie. Paris. Nallino, C.A. 1903–1907: Al-Battānī sive Albatenii Opus astronomicum. Milan. Pingree, D. 1968: The Thousands of Abū Macshar. London. Robbins, F.E. 1964: Ptolemy: Tetrabiblos. London and Cambridge, Mass. Rubió i Lluch, A. 1908: Documents per l’Història de la Cultura Catalana Mig-Eval. Vol. l. Barcelona. Samsó, J. 1992: Las Ciencias de los Antiguos en ai-Andalus. Madrid. Samsó, J. 1994: Islamic Astronomy and Medieval Spain. Aldershot (Hampshire). Sédillot, J.-J., and Sédillot, L.-A. 1834: Traité des instruments astronomiques des Arabes. Paris. Repr. Frankfurt a/M 1984. Sezgin, F. 1984: Comprehensive Collection of Principles and Objectives in the Science of Timekeeping by al-Marrākushī. Reproduced from ms 3343, Ahmet iii Collection, Topkapı Sarayı Library, Istanbul, 2 parts. Frankfurt a/M. Toomer, G.J. 1968: “A Survey of the Toledan Tables”, Osiris, 15:5–174. Toomer, G.J. 1984: Ptolemy’s Almagest. New York-Berlin. Vernet, J. 1952: Contribución al estudio de la labor astronómica de Ibn al-Bannāʾ. Tetuán. Vernet, J., and Català, M.A. 1965: “Las obras matemáticas de Maslama de Madrid”, Al-Andalus, 30: 15–47; reprinted in J. Vernet, Estudios sobre Historia de la Ciencia Medieval. Barcelona-Bellaterra 1979. Watt, W.M. 1953: Muhammad at Mecca. Oxford.

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Astronomical Activity in Portugal in the Fourteenth Century* Madrid, Biblioteca Nacional, ms 3349 (formerly at the Convento de Santo Tomás de Ávila, Spain), is a codex written in the first half of the fourteenth century, for the most part in Portuguese, containing a miscellaneous series of tables and short texts (ff. 1r–12v) and a copy of the Almanac of 1307 (ff. 13r– 55r). Back in 1932, the eminent scholar, Jaime Cortesão, called the manuscript “Almanaques Astronómicos de Madrid”,1 but, as we shall see, this characterization does not render justice to it, given its content as well as the fact that it has nothing to do with Madrid, except for being preserved in a library there at the present time. It is the only known evidence for astronomical activity in Portugal during the fourteenth century or, in Fontoura’s words, “é o único documento escrito, até hoje conhecido, que evidencia o elevado grau da cultura astronómica portuguesa dos principios do século xiv”.2 In 1949 Millás briefly described the contents of this manuscript.3 One section of it has received closer attention, namely, that containing the Almanac of 1307, which is also extant in Latin, Catalan, Castilian, and Hebrew, in addition to this Portuguese version.4 The folios preceding this almanac include a mixture of tables and short texts, the contents of which concern calendaric, astronomical, and astrological matters, that reflect elementary familiarity with these medieval disciplines. It is perhaps noteworthy that nothing is said about the motions of the planets.

* Journal for the History of Astronomy 41 (2010), 199–212. 1 J. Cortesão, “Cultura: Capítulo i—Influência dos Descobrimentos dos portugueses na história da civilização”, in D. Peres (ed.), História de Portugal 4 (Barcelos, 1932), 179–240. See also G. Beaujouan, L’astronomie dans la péninsule ibérique à la fin du moyen âge (Coimbra, 1967), 7. 2 A. Fontoura da Costa, A marinharia dos descubrimientos, 4th ed. (Lisbon, 1983), 80. 3 J.M. Millás, “Almanaques catalanes y portugueses del siglo xiv, de origen árabe”, in J.M. Millás, Estudios sobre historia de la ciencia española (Barcelona, 1949), 387–397. According to Millás, the material preceding the almanac belongs to at least two separate sets: ff. 1–2 and 9–10 on the one hand, and ff. 3v–8 and 11–12 on the other. But neither the contents nor the handwriting supports Millás’s suggestion. 4 J. Chabás, “El almanaque perpetuo de Ferrand Martines (1391)”, Archives internationales d’histoire des sciences, xlvi (1996), 261–308; see espec. pp. 263–264.

© koninklijke brill nv, leiden, 2015 | doi: 10.1163/9789004281752_014

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The first page (f. 1r) was initially left blank, but a later hand has drawn a table with no headings for the columns, assigning various integers to the 12 signs of the zodiac. This table, entitled “Table for fortune tellers” (Tavoa diçentedores na boa ventura), most probably for astrological purposes, has the date 12 September 1410 in the same hand. On f. 1v there is a table, headed “Table for the reckoning of the days in a Roman year [arranged according to] the months of the year” (Tavoa do conto dos dias do ano romano nos meses do ano), in which each day of the year is numbered consecutively such that the first entry is 1 for March 1, the second entry is 2 for March 2, and so on. The table ends abruptly with the final day of the sixth month: the last entry is 184 for August 31. The rest of the table is not found in this manuscript. Then follows a “Table for the solar altitude for those places where the latitude is 40°” (Tavoa da altura do sol nos logares que sa ladeza he 40 grados) (f. 2r). For various days of each month (day 1 and days 3, 6, 9, …, 30) we are given, in integer degrees, the meridian altitude of the Sun. The maximum altitude in the table, 75°, occurs on June 9–15, and the minimum, 27°, on December 6–12. From these entries we can deduce the latitude, φ, for which the table is valid, by means of the equation: φ = 90 – (hmax + hmin)/2. The result is φ = 39°. To determine the place, we may look in the table for geographical coordinates on f. 11r, where we find the following latitudes of Portuguese cities: Lisbon, 40;0°; Santarem, 39;30°; Coimbra, both 39;40° and 39;50°. On f. 2v the heading of the table, “Table of the samt in Burgos” (Tavoa dos çomutes en Burgos), mentions the name of a Spanish city, far from Portugal. Its latitude, 42;18°, is attested by several fourteenth- and fifteenth-century authors, including Juan Gil (see Madrid, bn, ms 23078, f. 148b) and Abraham Zacut,5 but it is quite different from the value embedded in the preceding table. Note that the term, çomutes, in the heading of the table is probably a rendering of the Arabic sumūt, the plural of samt (direction, azimuth, zenith). There are very few known examples of this table and they are associated with solar eclipses. For each degree of each zodiacal sign we are given a value in degrees and minutes;

5 F. Cantera Burgos, “El judío salmantino Abraham Zacut”, Revista de la Academia de Ciencias Exactas, Físicas y Naturales de Madrid, xxvii (1931), 63–398; espec. p. 365. See also A. Zacut, Almanach perpetuum, ed. by J. Vizinus (Leiria, 1496), 168r.

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the maximum value, +23;51°, occurs at Vir 29°–Lib 1° and the minimum, –23;51°, at Psc 29°–Ari 1° (see Table 1 and Figure 12.1). The entries in this table are not distributed symmetrically, for zero occurs at Sgr 10° and Cnc 20°. This table gives the declination of upper midheaven (i.e., the intersection of the meridian and the ecliptic above the local horizon) as a function of the longitude of the horoscope (i.e., the intersection of the ecliptic and the eastern horizon), and can be recomputed according to a 3-step procedure.6 A similar table, with different entries, computed for a latitude of about 36° and an obliquity of 23;51° (rounded from Ptolemy’s value of 23;51,20°), is found in the zij of Yaḥyā ibn Abī Manṣūr (9th century) with the heading “The table of samt for determining solar eclipses” ( jadwal al-samt li-cilm kusūf al-shams), as well as in Ibn al-Kammād’s al-Zīj al-Muqtabis and the Tables of Barcelona.7 The table reproduced below seems to be adjusted for an obliquity of 23;51° and a latitude of 42°–43°. The 3-step procedure to recompute the entries requires the use of 3 tables: one for the solar declination (as in Almagest i.15), another for the longitude of the horoscope (as in al-Khwārizmī’s zij),8 and a table for oblique ascensions for the appropriate geographical latitude. After a blank page (f. 3r) there is a text explaining how to determine some features of the ecclesiastical calendar such as the dominical letter, the golden number, and the date of Easter (f. 3v). The dominical (or Sunday) letter refers to the first Sunday of January for a given year, where the letters a, b, c, …, g, correspond to January 1, 2, 3, …, 7, respectively, and a is assigned to January 8, and so on. The golden number (aureo numero) is an integer, g, ranging from 1 to 19, which is assigned to each civil year, y, according to the rule: g ≡ y + 1 (mod 19).

6 E.S. Kennedy and N. Faris, “The Solar Eclipse Technique of Yaḥyā b. Abī Manṣūr”, Journal for the History of Astronomy, i (1970), 20–38, espec. pp. 21–24. Reprinted in Studies in the Islamic Exact Sciences ed. by D.A. King and M.H. Kennedy (Beirut, 1983) 185–203; this procedure was reconstructed by O. Neugebauer as indicated in a footnote on p. 24. 7 J. Chabás and B.R. Goldstein, “Andalusian Astronomy: al-Zīj al-Muqtabis of Ibn al-Kammād”, Archive for History of Exact Sciences, xlviii (1994), 1–41, espec. pp. 14–17; J.M. Millás, Las Tablas Astronómicas del Rey Don Pedro el Ceremonioso (Madrid–Barcelona, 1962), espec. p. 236; J. Chabás, “Astronomía andalusí en Cataluña: Las Tablas de Barcelona”, in From Baghdad to Barcelona: Studies in the Islamic Exact Sciences in Honour of Prof. Juan Vernet, ed. by J. Casulleras and J. Samsó (Barcelona, 1996), 477–525, espec. pp. 510–511. 8 H. Suter, Die astronomischen Tafeln des Muḥammad ibn Mūsā al-Khwārizmī (Copenhagen, 1914), 171–173.

392 table 1

chapter 12 Table of samt in Burgos

(º)

(º)

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

Lib n 23;51 º 23;50 23;49 23;48 23;45 23;40 23;37 23;33 23;25 23;17 23; 9 22;59 22;40 * 22;40 * 22;40 * 22;41 * 22;11 * 21; 1 * 21;50 21;40 21;28 21;17 21; 6 20;55 20;34 20;14 19;55 19;36 19;18 18;58 n Vir

Sco n

Sgr n

Cap s

Aqr s

Psc s

18;40 º 18;22 18; 4 17;46 17;28 17;10 16;48 16;26 15;59 15;34 15; 9 14;44 14;22 13;57 13;29 12;59 12;30 12; 1 11;32 11; 3 10;34 10; 4 9;35 9; 5 8;31 7;56 7;21 6;46 6;11 5;36

5; 0 º 4;25 3;50 3;15 2;39 2; 4 1;33 1; 2 0;32 0; 0 0;24 0;48 1;16 1;44 2;12 2;40 3; 8 3;37 4;12 4;48 5;23 5;59 6;59 7;10 7;40 8;10 8;37 9; 3 9; 6 9;50

10;12 º 10;24 10;56 11;18 11;39 12; 1 12;25 12;49 13;13 13;37 14; 1 14;25 14;46 15; 6 15;35 15;42 16; 0 16;18 16;33 16;48 17; 3 17;18 17;32 17;46 18; 2 18;18 18;34 18;50 19; 5 19;20

19;30 º 19;40 19;50 19;59 20; 8 20;17 20;26 20;35 20;43 20;51 20;59 21; 7 21;14 21;22 21;29 21;37 21;44 21;51 21;56 22; 1 22; 7 22;12 22;16 22;20 22;25 22;30 22;35 22;40 22;46 22;53

22;58 º 23; 2 23; 6 23;10 23;14 23;18 23;21 23;24 23;26 23;28 23;30 23;32 23;34 23;36 23;38 23;39 23;40 23;42 23;43 23;44 23;46 23;47 23;48 23;49 23;49 23;49 23;50 23;50 23;51 23;51

n Leo

s Cnc

s Gem

s Tau

s Ari

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* The entries for Lib 13 to 18 seem to be corrupt since the differences are not smooth. In particular, although the manuscript reading is secure, the entry for Lib 18 should be 22;1 (instead of 21;1), and this accounts for the “singularity” appearing in Figure 12.1.

figure 12.1 Table of samt in Burgos ( from Lib 1° to Ari 1°)

The text gives an example for year 1339 which is probably its date: “divide 1339 by 19 and we obtain [a remainder of] 9; add 1, and we obtain 10, which is the golden number of the current year.” The usage of the dominal letter and the golden number is described below. This text is accompanied by a drawing of two hands with numbers on them to be used in reckoning other features related to the ecclesiastical calendar (see Figure 12.2). Instructions for using fingers and their joints for similar purposes are found in Bede’s De temporum ratione (On the Reckoning of Time), chap. 55, and in calendaric notes by Thomas Harriot.9 The drawing of hands for this purpose does not appear explicitly in Bede’s work, but it is found in later works, where it is called “the hand of Bede”.10 The hand on the left corresponds to the outside of the right hand and has 15 consecutive numbers on it, 3 on each finger. This might be an illustration 9

10

On Bede, see F. Wallis (trans.), Bede: The Reckoning of Time (Liverpool, 1999), 137–139; on Harriot, see J.D. North, “Thomas Harriot’s Papers on the Calendar”, in The Light of Nature: Essays in the History an Philosophy of Science presented to A.C. Crombie, ed. by J.D. North and J.J. Roche (Dordrecht, 1985), 145–174, espec. pp. 163–167. J.E. Murdoch, Album of Science. Antiquity and the Middle Ages (New York, 1984), 80.

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figure 12.2 Calendaric matters (Madrid, ms 3349, f. 3v). Photograph courtesy of the Biblioteca National de España (Madrid)

of the indiction, an integer ranging from 1 to 15 assigned to each civil year in a cycle of 15 years, introduced in the fourth century.11 Although the indiction is not directly related to computing the date of Easter (the main subject of this section), it appears together with the date of the Paschal full moon in a calendaric table by Dionysius Exiguus (sixth century).12 The hand on the right corresponds to the inside of the left hand and is intended to aid in the computation of the date of Easter. We are given a sequence of 19 integers: 5, 25, 13, 2, 22, 10, 30, 18, 7, 27, 15, 4, 24, 12, 1, 21, 9, 29, 17. These numbers refer to dates in March or April when the Paschal full moon occurs as a function of the “golden number”, i.e., the year in the 19-year cycle (where 19 civil years are

11 12

See Wallis, Bede (ref. 9), 339–340; O. Neugebauer, A History of Ancient mathematical Astronomy (Berlin and New York, 1975), 1062. G. Teres, “Time computations and Dionysius Exiguus”, Journal for the History of Astronomy, xv (1984), 177–188, espec. pp. 180–182.

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assumed to be exactly equal to 235 synodic months). The text indicates that when the number is greater than 20, these dates are in March, whereas when 20 or less, these dates are in April. So we may rewrite the sequence (where m means March, and a means April): a5, m25, a13, a2, m22, a10, m30, a18, a7, m27, a15, a4, m24, a12, a1, m21, a9, m29, a17. The underlying basis for the computation of these dates is the following. The Paschal full moon is the first full moon after the vernal equinox, set at March 21 (although medieval astronomers knew that it was not astronomically correct). An ordinary civil year is 365 days and a leap year is 366 days. Since 12 lunar months are a little over 354 days, the difference between 12 lunar months and an ordinary civil year is 11 days. So, in general, the Paschal full moon in a given year comes 11 days earlier than in the previous year. But if that date falls prior to March 21, a lunar month of 30 days is added. For example, the Paschal full moon in year 1 of the 19 year cycle occurs on April 5; thus the Paschal full moon in year 2 occurs on March 25 [= April 5–11 days]. Similarly, the Paschal full moon in year 3 occurs on April 13 [= March 25 – 11 days + 30 days]. Finally, the Paschal full moon in year 19 occurs on April 17, and the Paschal full moon in year 20, which is year 1 of the next cycle, is April 5 [= April 17 – 12 days]. Easter is then defined as the Sunday that immediately follows this full moon, and can easily be determined from its date and the dominical letter that applies to it. Then follow various tables with headings in Latin. The table on f. 4r–v is for the motion of the lunar node. The title is Tabula capitis draconis meses et dies and has the names of the months as well as marginal notes in Portuguese. This table gives the daily progress of the lunar node (in signs, degrees, and seconds) in a year and is identical to the table in the Almanach perpetuum compiled for the meridian of Montpellier in Southern France by Jacob ben Makhir Ibn Tibbon (1236–1304) and originally written in Hebrew.13 The table on f. 5r entitled, “Table for the lunar and solar motion in an hour and the number of hours for the half-length of daylight at Montpellier” (Tabule cursus lune et solis in huna hora et horarum medietatis diei in Monte Pesulano), seems to have the same origin. This table is presented as two sub-tables: one is for the hourly lunar velocity, in minutes and seconds, given at 6°-intervals from 0s 0° (the corresponding velocity is 0;30,18°/h) to 6s 0° (ms: 0;35,4°/h, instead of 0;36,4°/h); the second is for the hourly solar velocity, in minutes and seconds, given at 6°-intervals from 3s 0° (0;2,23°/h) to 8s 24° (0;2,33°/h), and for the

13

J. Boffito and C. Melzi d’Eril, Almanach Dantis Aligherii sive Profhacii Judaei Montispessulani Almanach Perpetuum, ad annum 1300 inchoatum (Florence, 1908), 113–114.

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half-length of daylight, in hours and minutes, given at 6°-intervals from 3s 0° (7;37h) to 8s 24° (4;25h). Among the tables in the Almanac perpetuum of Jacob ben Makhir in Hebrew is a table for the hourly velocities of the Moon and the Sun: see, e.g., Paris, Bibliothèque nationale de France, ms Heb. 1064, f. 76a, where the hourly lunar velocity has a minimum value of 0;30,18°/h and a maximum of 0;36,4°/h, and the hourly solar velocity has a minimum of 0;2,23°/h and a maximum of 0;2,34°/h. This table is not included in the edition of the Latin version edited by Boffito and Melzi d’Eril. The entries for the lunar and solar velocities ultimately derive from al-Battānī’s zij and they are the same as those in the Toledan Tables and the Almanac of Azarquiel.14 The longest and shortest half-length of daylight in this table, 7;37h and 4;25h, add up to 12;2h, instead of 12h, suggesting that the missing entry for 9s 0° was 4;23h. The modern formula for the length of the longest daylight, m, as a function of the obliquity of the ecliptic, ε, and the geographical latitude, φ, is: –cos m/2 = tan φ tan ε. With m = 15;14h and the standard values of ε used at the time, 23;51° (Ptolemy) and 23;33° (Azarquiel), we obtain φ = 42;54° and φ = 43;18°, respectively. No places in Portugal and very few places in Spain with this latitude are plausible candidates whereas Montpellier, whose latitude is 43;0° according to Jacob ben Makhir,15 is likely. The two tables on f. 5v are identical to two tables in the Almanach perpetuum,16 and have the same headings, respectively: the first is for lunar latitude (with a maximum of 5;0°), and the second for lunar eclipses at mean distance. Then follows a short text in Portuguese on the limits of eclipses (f. 6r). Folio 6v displays four tables. The central one is in Latin, and the other three that surround it, probably added at a later stage, are in Portuguese. The central table, headed “Table to determine the lunar position for each day of the year beginning at midnight” (Tabula de loco lune inueniendo omni die anni a media nocte sui incepto) consists of 14 × 12 cells. Each cell is assigned a zodiacal sign,

14

15 16

On the zij of al-Battānī, see C.A. Nallino, Al-Battānī sive Albatenii Opus Astronomicum (2 vols, Milan, 1903–1907), ii, 88; on the Toledan Tables, see F.S. Pedersen, The Toledan Tables: A review of the manuscripts and the textual versions with an edition (Copenhagen, 2002), 1412; on the Almanac of Azarquiel, see J.M. Millás, Estudios sobre Azarquiel (Madrid– Granada, 1943–1950), 174. See Boffito and Melzi d’Eril, Almanach (ref. 13), 2. See Boffito and Melzi d’Eril, Almanach (ref. 13), 115.

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and the 12 columns represent the 12 months of the year. On the right the argument is “age of the Moon” (etas lune) and runs from 1 to 30 days; the days are grouped mostly in pairs making a total of 14 rows. On the left the argument is “degree of the sign” (gradus signi) and the entries in this column are 13° or 14° apart according to the sequence: 14, 27, 10, 23, 6, 20, 3, … This table is identical to the “lunar table” by Peter (Philomena) of Dacia, fl. 1292–1303, probably compiled in Paris.17 In the next table at the left for determining “the degrees of altitude” (os grados de alteza), the entries are given in degrees at 5-day intervals for each month of the year beginning in July. As in the table on f. 2r for the same purpose (but with a different presentation), the maximum altitude is 75° corresponding to June 15, and the minimum is 27° corresponding to December 15. Again, the underlying latitude is 39°, but we cannot assign a unique place to this value. In the table located at the right for determining the “equal hours” (horas yguaes), the entries are given in hours and minutes for the 10th, 20th, and last day of each month of the year, beginning in January. The longest daylight is 15;29h corresponding to June 10–20 whereas the shortest daylight is 8;31h corresponding to December 10–20. In this case, the figures add up correctly to 24h. If we compare the value for longest daylight with twice the half-length of longest daylight given on f. 5r, we find that the corresponding values differ (15;29h vs. 15;14h). Again, using the modern formula that relates the length of the longest daylight, the obliquity of the ecliptic, and the geographical latitude, we obtain φ = 44;53° for ε = 23;51° (Ptolemy) and φ = 45;18° for ε = 23;33° (Azarquiel), both values suggesting places north of the Iberian Peninsula. The table located below (on the same page) is peculiar indeed and it is aimed at determining the “number of hours and parts of an hour when the moon is visible above the horizon” (quantas horas da noite e partes das horas esta a luna en cada noite). The entries are given in hours and parts of hours, here represented as a sequence of dots, from none to a maximum of four dots. A comment associated with this specific table reads: “Note that where this almanac was made the Sun rises one hour and a fifth [of an hour] earlier than in Coimbra” (Nota que en a terra en que foy feito este almenaque ergese ante o sol huna ora et quinta que en esta de Coymbra). As pointed out by Millás,18 this difference in longitude of 18° (= 1;12h) seems to correspond to the distance between Montpellier and Coimbra. Some scholars have taken this sentence to mean that Coimbra was the place where an almanac was compiled and for this reason they called these 17 18

F.S. Pedersen, Petri Philomenae de Dacia et Petri de S. Audomaro, Opera quadrivialia (Corpus philosophorum Danicorum medii aevi, 10.1–2, Copenhagen, 1983–1984), 360. Millás, “Almanaques catalanes” (ref. 3), 391.

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tables the “Almanac of Coimbra”.19 However, we suggest that Coimbra was only the place where this astronomical material was copied and gathered together, and that the almanac mentioned here refers to some version of the Almanach perpetuum by Jacob ben Makhir. The short text in Portuguese on f. 7r consists of an explanation of the 14 × 12 lunar table and is similar to the canon for this table by Peter of Dacia.20 The two tables on f. 7v are relatively rare but they appear in a number of manuscripts, some related to the Toledan Tables.21 In the first table the argument is given in months, and the entries in days, hours, and minutes. In the second table, the argument is in days, whereas the entries are in degrees and minutes. The headings are “Table of the months” (Tavoa dos menses) and “Table of [lunar] progress” (Tavoa do poiamento), respectively. The term, poiamento or pojamento, no longer in use in Portuguese, derives from the verb pojar, meaning to climb, to increase, to progress. See Tables 2 and 3. table 2

Table of the months

(Month)

(d)

(h)

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

28 56 84 112 140 168 196 224 252 280 309 337 365

2;57 4;35 6;52 9;10 11;27 13;45 16;11 18;20 20;37 22;55 1;12 3;30 5;47

19 20 21

See Cortesão, “Influência dos Descobrimentos” (ref. 1), 196–205. See Pedersen, Petri de Dacia (ref. 17), 333. B.R. Goldstein, J. Chabás, and J.L. Mancha, “Planetary and Lunar Velocities in the Castilian Alfonsine Tables”, Proceedings of the American Philosophical Society, cxxxviii (1994), 61–95, espec. pp. 89 and 93. See also Pedersen, Toledan Tables (ref. 14), 1586–1588.

astronomical activity in portugal in the fourteenth century table 3

Table of [lunar] progress

(d)

(º)

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

13;13 27;46 41;39 55;52 69; 5 83;17 97;30 111;43 124;56 138; 9 152;22 161;35 18[ ];48 194; 1 208;14 222;26 236;40 247;53 263; 5 277;18 291;31 305;44 312;57 333;10 347;23 0;36 14;49 28; 2

28

30;11

399

(Recomp.)

23 24 25 26 27 28 [29 30

13;53 27;46 41;39 55;32 69;25 83;17 97;10 111; 3 124;56 138;49 152;42 166;35 180;28 194;21 208;14 222; 6 236; 0 249;53 263;45 277;38 291;31 305;24 319;17 333;10 347; 3 0;56 14;49 28;42 42;35] 56;27

In Table 2, called Tabula mensium or Revolutio mensium in Latin manuscripts, we are given multiples of 28d 2;17,30h (despite the first entry), up to a total of 365d 5;47,[30]h. This seems to be the length of a tropical year consisting of 13 such “months”. As far as we know, the closest value for this length of the year is

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table 4

Domiciles, exaltations, and dejections of the planets

Domiciles Exaltations Dejections Saturn Jupiter Mars Sun Venus Mercury Moon Ascending node Descending node

Cap, Aqr Sgr, Psc Ari, Sco Leo Tau, Lib Gem, Vir Cnc

Lib Cnc Cap Ari Psc Vir Tau Gem Sgr

17° 15° 28° 19° 27° 15° 3° 3° 3°

Ari Cap Cnc Lib Vir Psc Sco

that in al-Battānī’s zij (365d 5;46,24h, which is obtained by dividing 360° by the daily mean motion of the Sun, 0;59,8,20,46,56,14 º/d).22 In Table 3, called “Lunar progress in the zodiac” (Diete lune in circulo signorum) in Latin manuscripts, we are given multiples of about 13;52,55°, despite the error for the first entry (13;13° instead of 13;53°). We have no explanation for the choice of this peculiar lunar parameter, but this table agrees (except for scribal errors) with the table we published in 1994 (see ref. 21). [Note that this problem has been solved: see B.R. Goldstein and J. Chabás, “Planetary Velocities and the Astrological Month”, jha, 44 (2013), 465–478.] It is quite surprising to find another copy of this peculiar table with the same heading and a different set of scribal errors. In Table 3, the column labeled “(recomp.)” displays the correct multiples of 13;52,55° rounded to minutes. On f. 8r there is a list, in Portuguese, of the domiciles, exaltations, and dejections of the five planets, the Sun, the Moon, and the two lunar nodes. We have gathered this astrological information in Table 4. The Sun and the Moon have one zodiacal sign as its domicile, and each of the five planets has one zodiacal sign as its domicile on the lunar side and one on the solar side, as shown in Figure 12.3. The lunar nodes are treated here as if they were real planets in so far as exaltations are assigned to them. Note that, according to al-Bīrūnī (d. 1048), the dejection of a planet is 180° from its exal-

22

E.S. Kennedy, A Survey of Islamic Astronomical Tables, (Transactions of the American Philosophical Society, ns 46.2 (Philadelphia, 1956)), 156.

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figure 12.3 Domiciles according to al-Bīrūnī

tation.23 For example, Saturn’s exaltation is Libra 17° and its dejection is Aries 17°. These astrological doctrines ultimately derive from Ptolemy’s Tetrabiblos i.17 and 19, but specific data for the domiciles and exaltations are the same as those given by al-Bīrūnī, indicating the Arabic origin of the material used by the compiler of this manuscript. The list of exaltations agrees with al-Bīrūnī’s, except for Saturn where al-Bīrūnī has Lib 21° (as does Segovia, Catedral, ms 110, f. 84v).24 There follows a list of 35 stars with the title and headings in Latin, supplemented by some notes in Portuguese (f. 8v). The title is “Table for the positions of the fixed stars with their latitudes from the ecliptic, their declinations from the equator, and their mediations, verified for the year 577ah” (Tabula locorum stellarum fixarum et earum latitudinum ab ecliptica et declinationum ab equinoctio et grade cum quo mediant celum anno arabum 577 verificata). The names of the stars are transliterated from Arabic and no Latin names are given. Each star is associated with its ecliptic coordinates (longitude and latitude, in degrees and minutes, together with an indication n or s) and its equatorial coordinates (“declination” and “mediation”, together with an indication n or s). This is the star list associated with the Toledan Tables,25 where the increment in the longitudes with respect to those in Ptolemy’s star catalogue is 14;55° or 15;7°. 23 24 25

R.R. Wright (ed. and trans.), The book of instruction in the elements of the art of astrology, by al-Bīrūnī (London, 1934), 258. For domiciles according to al-Khwārizmī, see Suter, Al-Khwārizmī (ref. 8), 231. P. Kunitzsch, Typen von Sternverzeichnissen in astronomischen Handschriften des zehnten bis vierzehnten Jahrhunderts (Wiesbaden, 1966), 87–94; Pedersen, Toledan Tables (ref. 14), 1494–1497.

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The year 577ah appearing in the title corresponds to 1181–1182, but it has been suggested that the true date of this star list is perhaps 1132–1133.26 The first table of this group is on f. 9r and its title reads: “Table for reckoning of Jewish and Arabic ordinary [lit. non-leap] years, and the days of their months, and their months” (Tavoa do conto dos anos ebraycos e arauygos non bisestiles e o conto dos dias de seus meses e o conto de seus meses). The names of the months are transliterated from Arabic and Hebrew, and an integer number is assigned consecutively to each day of the year. The table begins with month 7: its first entry is 178 for Nisan 1 and Rabīc ii 1, and it ends with 354 for Elul 29 in the Jewish calendar without indicating an equivalent in the Arabic calendar. Note that this is not the continuation of the truncated table appearing on f. 1v. Then follows a table for much the same purpose as the preceding one (restricted to the Jewish calendar), but for leap years, “Table for reckoning the days and Jewish leap years, and the days of their months, and their months” (Tavoa do conto dos dias e dos anos bisextilles ebraicos e la conta dos dias e de seus meses en la conta de seus meses) (ff. 9v–10r), which begins with day 1 (Tishri 1) and ends with day 384 (Elul 29 in a leap year). On f. 10v there is a table of astrological “terms” according to Ptolemy. The underlying principle is the following: each zodiacal sign is divided into 5 unequal parts, called terms, and each of the 5 planets is the lord of one term in each zodiacal sign. For each zodiacal sign the sum of the terms must equal 30°. In the Tetrabiblos Ptolemy lists terms in two systems, one associated with the Egyptians (Tetrabiblos, i.20), and another introduced by Ptolemy in his own name (Tetrabiblos, i.21).27 ms 3349, f. 10v, follows the system according to Ptolemy, with a few variants (where ms 3349 differs from the Tetrabiblos, an entry is given in the row labeled [Ptol.]): see Table 5. The list of geographical coordinates has a title in Portuguese and gives the longitudes and latitudes of some 70 places and regions, most of them in Islamic territories (f. 11r). Despite the addition of some Portuguese localities, the table is essentially the same as that found in the Toledan Tables.28 It is noteworthy that two different prime meridians are used, called “of the land” (da terra) and “of the water” (dagoa), at a distance of 17;30° as pointed out by Millás,29 although a marginal note in Portuguese indicates that the distance between them is 17°. The “meridian of water”, was a prime meridian for geographical longitude, 26 27 28 29

See Pedersen, Toledan Tables (ref. 14), 1489. F.E. Robbins (ed. and trans.), Ptolemy: Tetrabiblos (London and Cambridge, Mass., 1940), 97 and 107. See Pedersen, Toledan Tables (ref. 14), pp. 1512–1513. Millás, “Almanaques catalanes” (ref. 3), 391–392.

astronomical activity in portugal in the fourteenth century table 5

Ari [Ptol.] Tau Gem Cnc [Ptol.] Leo [Ptol.] Vir Lib Sco Sgr Cap Aqr Psc

403

Astrological terms according to ms 3349 and Ptolemy

Jup

6

Ven Merc Mars Mars Sat Jup Merc Sat Mars Jup Ven Sat Ven

8 7 7 6 6 6 7 6 6 8 6 6 8

Ven Ven Merc Jup Jup Jup Merc

3 8 7 6 6 7 7

Ven Ven Ven Ven Merc Merc Jup

6 5 7 6 6 6 6

Merc

7

Mars

5

Jup Ven Merc

7 7 7

Sat Mars Ven

2 6 7

Mars Sat Jup Merc Jup Merc Jup Ven Merc

5 6 5 5 8 5 7 8 6

Ven

6

Sat Jup Merc Sat Sat Jup Mars

6 8 6 6 6 5 5

Sat Sat Mars Sat Sat

9 4 6 4 3

Jup Mars Mars Mars Sat Mars Mars Mars Sat

6 5 6 6 3 5 5 5 5

usually considered to be 17;30° to the west of Ptolemy’s prime meridian, the Canary Islands, which he called the “Fortunate Islands”.30 On f. 11v, just after a short text, there is a circular table (or “rose”) with entries from 1 to 5 for the astrological dignities of the planets faces, terms, triplicities, exaltations, and domiciles.31 We are aware of only one other example of a table for astrological dignities that uses the codes 1 to 5 in this way.32 But the same information is presented differently (i.e., without these codes) in Segovia, Catedral, ms 110, f. 84v, where each row of the argument is a zodiacal sign, each column is for one of the five dignities, and the entries are names or symbols for the planets, that is, a planet, or several planets, are “lords” of each dignity. For a transcription of the table on f. 11v, see Table 6 (where the circular form has been replaced by a rectangle).

30

31

32

J. Samsó, Las Ciencias de los Antiguos en al-Andalus (Madrid, 1992), 90; M. Comes, “The ‘Meridian of Water’ in the Tables of Geographical Coordinates of al-Andalus and North Africa”, Journal for the History of Arabic Science, x (1994), 41–51. For an explanation of this code see Wright, The book of instruction (ref. 23), 258; see also J. Chabás and B.R. Goldstein, Astronomy in the Iberian Peninsula: Abraham Zacut and the Transition from Manuscript to Print (Philadelphia, 2000), 72 and 87. Pedersen, Toledan Tables (ref. 14), 1599, Table re21.

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table 6

Sat. Jup. Mars Sun Venus Merc. Moon

Astrological dignities

Ari

Tau Gem Cnc Leo Vir

Lib

Sco Sgr

Cap

Aqr Psc

2 2,3 1,5,2 3,4,1 2,1 2 0

1,2,3 2 2 0 2,5 1,2,3 4,1

1,3,2,4 1,2 2 0 5,2 2 1

2 2 1,5,3 1 1,2 2 3

5,2 1,2 4,1,3,2 1 3,2 3,1 0

5,2 2 2 0 3,1,2 3,1,2 1

2 2, 1 2, 1 1 3,2 5,2 1

2 4,2 2,3 1 1,2 1,2 5, 1

1,2 3,1,2 1,2,3 5,3 2 2 0

3,2 2 2 1 1,2,3 4,1,2,5 3

1,2 5,2,3 3,2 3 2 1,2 1

1,2 1,2,5 1,2,3 0 2,3,4 2 3

Gem, Moon: 1. Read: 0 (as in Pedersen 2002, p. 1599). Cnc, Sun: 1. Read: 0 (as in Pedersen 2002, p. 1599). Lib, Sat.: 3,2,4. Read: 1,3,2,4 (Pedersen 2002, p. 1599: 4,2,1). Cap, Merc.: 3,1. Read: 3,2 (as in Pedersen 2002, p. 1599). Sco, Mars: 1,5,3. Read: 1,5,2 (as in Pedersen 2002, p. 1599). Psc, Venus: 2,3. Read 2,3,4 (Pedersen 2002, p. 1599: 4,2).

In Table 6 dignity 1 represents the lords of the faces. According to al-Bīrūnī: “Each third of a sign—10 degrees—is called a ‘face’ and the lords of these faces according to the agreement of the Persians and the Greeks are as follows: the lord of the first face of Aries is Mars, of the second the Sun, of the third Venus, of the first face of Taurus, Mercury, and so on in the order of the planets from above downwards till the last face of Pisces.”33 Dignity 2 represents the lords of the terms, as in Table 5. Dignity 3 represents the lords of the triplicities. The 12 zodiacal signs are assigned, in groups of 3, to one of the 4 elements: fire, earth, air, and water. The “standard” list for the triplicities is given in al-Bīrūnī, Astrology,34 and it agrees with the list given by Abraham Zacut in his Ḥibbur.35 But the entries for the triplicities in ms 3349 are inconsistent and there are not always three planets, as expected. Where the entries for zodiacal signs in the same triplicity are inconsistent, entries in Table 7 for successive zodiacal signs in the same triplicity have been put in separate rows. Some of these inconsistencies are

33 34 35

See Wright, The book of instruction (ref. 23), 26. For a table of lords of the faces, see Wright, ibid., 263. See Wright, The book of instruction (ref. 23), 259. See Chabás and Goldstein, Abraham Zacut (ref. 31), 87.

astronomical activity in portugal in the fourteenth century table 7

405

Astrological triplicities

fiery (Ari, Leo, Sgr): [Leo] [Sgr] earthy (Tau, Vir, Cap): [Vir] [Cap] airy (Gem, Lib, Aqr): [Lib] [Aqr] watery (Cnc, Sco, Psc): [Sco] [Psc]

ms 3349

Toledan t.

Zacut

Jup Sun/Jup/Mars Sun/Jup/Mars Merc/Sat Sat/Ven Mars/Ven Ven Sat Ven/Merc Mars Mars Mars/Ven

Sun/Jup/Mars Sun/Jup/Sat

Sat Sat Sat/Merc Ven

Ven/Moon/Mars

Sat/Jup/Merc

Moon

Ven/Mars/Moon

shared with the list in the Toledan Tables,36 and we have no explanation for them. For the lords of the triplicities in ms 3349 compared with those in the Toledan Tables,37 and those in Zacut’s Ḥibbur, see Table 7. In Table 6 dignity 4 represents the exaltations, as in Table 4. For example, the exaltation of the Moon is Tau 3; hence the entry “4” in the row for the Moon under Taurus. Dignity 5 represents the domiciles. For example, the domicile of Mars is Aries; hence the entry “5” in the row for Mars under Aries.38 The last table in this group gives the position of the lunar node on a yearly basis for 93 consecutive years (f. 12r–v). For each year there are two entries, headed medius motus and verus motus, in signs, degrees, minutes, and seconds. In most cases the two entries correctly add up to 12s. On f. 12v there is a text in Portuguese explaining the use of the table, and on f. 12r we are also given entries for two specific years only: 1299 and 1300. This table agrees with the corresponding table in the Almanach perpetuum by Jacob ben Makhir, which also gives entries for the same two years, and no others.39

36 37 38 39

For the “standard” list, see Pedersen, Toledan Tables (ref. 14), 1592, Table ra22. See Pedersen, Toledan Tables (ref. 14), 1599. For a list of planetary domiciles, see Wright, The book of instruction (ref. 23), 256. See Boffito and Melzi d’Eril, Almanach (ref. 13), 112.

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In sum, the material in ms 3349, ff. 1–12 is a mixture of tables of diverse origins and intended for different purposes, some of which are strictly astrological. A significant feature is that there are no tables for planetary motion. Thus, this collection of heterogeneous material is certainly not an almanac (despite claims in the secondary literature, cited above), for it does not contain a set of tables giving the daily true positions of the Sun, the mean positions of the Moon, and the true positions of the five planets for periods that differ for each celestial body. Hence, it makes no sense to call it an “almanac of Coimbra”, although it is clear that these tables were compiled at Coimbra around 1339. We can also say that this material belongs, for the most part, to the tradition of the Toledan Tables, since almost all its tables are directly connected with this zij or copied from Jacob ben Makhir’s Almanach perpetuum which, in turn, is based on the Toledan Tables. Finally, in this material there are various references to the Jewish and Arabic calendaric and astronomical traditions, a feature which is shared by many astronomical manuscripts of the time in the Iberian Peninsula.

Index Abraham Bar Ḥiyya 385 Abraham Ibn Ezra 200 Access and recess 102, 115, 260, 278–279; see also trepidation Adelard of Bath 2 Adrano 339 Agrigento 364 Ibn al-Aʿlam 102 Alexander vi, pope 172 Alfonsine corpus 21, 28, 57, 60, 68–69, 105, 115, 150–175, 336 Alfonsine Tables Castilian Alfonsine Tables 42, 75–76, 94, 157–158, 227, 229, 257, 261, 266, 280, 310–311, 325 Editio princeps 10, 15–17, 21, 24, 28, 78, 86–87, 90, 105, 146–147, 158, 160, 170, 172, 175, 251, 260, 264, 275, 283, 311, 314, 316, 324 Parisian Alfonsine Tables 3–4, 9, 10, 16–18, 22, 36, 45–46 et passim Alfonso x, king 4, 10, 53, 157–158, 227, 250–251, 282–283, 308, 310, 312, 315–317, 321, 326, 332, 336 Alfonso de Córdoba 172–173 Almagest 1, 3, 9, 12, 74–76, 78, 82, 85– 88, 94–95, 124, 126, 130, 137 et passim Almagest i 391 Almagest iii 104 Almagest iv 13, 332 Almagest v 16, 110–111, 216 Almagest vi 12, 22, 41, 146, 201, 246 Almagest xi 150–151, 159, 217, 263–264, 266, 270 Almagest xii 130, 156, 218 Almagest xiii 73, 76–77, 79–81, 83– 85, 87–88, 145, 216, 271, 274, 277, 325 Arabic versions of 380–381 Almanac 174, 327, 406 Almanac of 1307 389 Almanac of Azarquiel 14, 190, 201, 214–216, 221 Almanach Perpetuum (A. Zacut) 47–48, 61, 64, 77, 79, 174, 355, 384, 396

Almanach perpetuum (J. ben Makhir) 395–396, 398, 405–406 Almanach Planetarum 309 al-Amad ʿalā al-abad 181–182, 207, 373 al-Andalus 158, 160, 179, 260, 340–341, 345, 368 Apollonius 104 Arin 231 Aristotle 341 Ascendant 195, 198 Ascension (right and oblique) 198, 215, 222, 391 Asher ben Yeḥiel 338 Astrology 282, 368, 400–404 Auctores calendarii 166, 309 Ávila 389 Azarquiel 14, 180, 183, 204, 207–208, 211–212, 214, 220–221, 240, 368, 381, 396–397 Ibn ʿAzzūz al-Qusanṭīnī 74, 95 Baghdad 2, 189, 240, 336 al-Baghdādī 163, 336 Ibn al-Bannāʾ 75, 89, 157, 161, 211–212, 264–266, 382–383, 386 Barcelona 95, 308, 385 Basel 52 al-Battānī 2, 4, 14–15, 17, 41, 52, 58, 74–75, 89 et passim Bayeux 228–229 Ibn al-Bayṭār 163 Bede 393 Bernardus de Virduno 183 Bianchini, Giovanni 50, 52, 73, 79–95, 170–171 al-Bīrūnī 200, 401, 404 Botarel, Moses Farissol 40 Bowen, A.C. 38, 95 Brahe, Tycho 53–54 Brugge 254 Burgos 43, 74, 167, 338, 383, 390, 392–393 Caesar, J. 233 Cairo 336, 342 Calendar Christian 64, 166, 341 Ecclesiastical 391, 393

408 Egyptian 207 Hijra (or Arabic or Muslim) 166, 207, 212, 402, 406 Jewish 64, 357, 402, 406 Julian 207 Roman 233 Camariotes, Matthew 340 Canary Islands 365, 403 Castile 4, 10, 57, 157–158, 173, 230, 261, 308, 310, 312, 321, 326, 329, 336, 338 Climates 215, 221 Coimbra 389–390, 397–398 Cologne (Köln) 338 Comes, M. 387 Commentariolus 94, 172 Conjunction 3, 9, 11–12, 14–17, 20, 36–38, 40–41, 47 et passim; see also syzygy Constellation 281–282 Coordinates ecliptic 374 geographical 364, 390 sidereal 342, 361 tropical 236, 342, 361 Copernicus, N. 1, 53, 73, 79, 90–91, 93–95, 172, 263, 270 Córdoba 4, 167, 180, 182, 195, 211, 215, 220, 222–223, 373, 385 Cortesão, J. 389 Cracow 53, 73, 79–80, 83, 90, 95 Creation 357–360, 362–364 Curtze, M. 79 Cyclical radices 147, 171 van Dalen, B. 95 Day additional 233 astronomical 231, 341 civil 231, 235, 341 equinoctial 231 sexagesimal 159, 241 De imaginibus coelestibus 341 De revolutionibus 1, 53, 90, 94, 270 De temporum ratione 393 Dejection (astrological) 400–401 Dignities (astrological) 403–404 Dionysius Exiguus 394 Displaced tables 4, 99–148, 168, 170–171, 175 Domicile (astrological) 400–401, 403, 405

index Dominical letter 391, 393–394 Double argument tables 41–42, 44–45, 47, 49–52, 64, 67–68, 77–79, 85 et passim Du Hamel, P. 45 Easter 391, 394 Eclipse 9, 17, 50, 229, 339, 341, 383 area digits 145, 200 colors 200–201, 350 diameters 349 disks 103, 145 half-duration 206, 351, 357 limits 103, 145, 341–342, 356, 394 linear digits 145, 200–201, 206, 359 lunar 3, 53, 199–201, 221, 341–342, 349–350, 355, 368, 396 magnitude 199–200, 206 mid-eclipse 54 samt 195–198, 390–393 solar 3, 17, 195–198, 200–201, 341–342, 356, 362, 368 totality 199, 351 Ecliptic 120, 145, 195, 209, 391 Egidius de Tebaldis 282 Eighth sphere 116–117, 237–238, 313, 342 Elongation 11, 13, 16, 41–42, 44, 46–47, 107–111, 114, 167, 194, 205, 245–246 Ephemerides 174 Epoch 230 Equation of time 17, 46, 51, 102, 107, 218, 326 Equinox 158, 363, 383, 395 Erfurt 9, 17, 25, 30, 48, 57 Escorial 195 Exaltation 400–401, 403 Excess of revolution 219–220 Expositio 158, 162, 309–310, 336 Face (astrological) 403–404 al-Fazārī 199 Federico de Montefeltro 339, 341, 358–359, 361, 368 Ferrara 79, 170 Fez 74 Firmin of Beauval 167 First point of Aries 207–208, 381 Fixed stars 4, 115–116, 207, 228–229, 277–303, 333–387, 401 Fontoura da Costa, A. 389 Fortunate Islands 365, 403

409

index Gascon, A. 342, 358–359, 368 Geniza 69 Geoffrey of Meaux 230, 233 Gerard of Cremona 9, 88, 283, 304 Gislén, L. 313 Golden number 391, 393–394 Granada 47, 157 The Great Composition 61, 74, 174, 338; see also ha-Ḥibbur ha-gadol Ḥabash al-Ḥāsib 156, 183, 213, 261 Ibn al-Hāʾim 180–181 al-Ḥajjāj 381 Handy Tables 1, 74, 145, 150, 152, 154–157, 159–160, 165, 175, 195, 201, 215–218, 352–353 Harriot, T. 393 Ibn al-Ḥātim 341 Heller, J. 386 ḥeleq (ḥelaqim) 357 ha-Ḥibbur ha-gadol 47, 61, 68, 77, 174, 328, 338, 355, 384, 404–405 Ibn Hibintā 216 Hijra 213, 219, 375, 382, 386 Hipparchus 381 Horizon 391 Horoscope 391 Immanuel ben Jacob Bonfils 47, 51–52, 55–56, 102, 167, 340, 349–351, 367–368 Incarnation 313 Indiction 394 Isaac ben Elia Kohen 40 Isaac Ibn al-Ḥadib 4, 40, 47, 338–369 Isaac ben Sid 157, 227 Isabella, Queen 173 Isḥāq (version of the Almagest) 381 Ibn Isḥāq al-Tūnisī 75, 157, 161, 163, 212, 250, 253, 373 Jacob ben David Bonjorn 10, 18, 47, 64, 200, 340, 368 Jacob ben Makhir Ibn Tibbon 395–396, 398, 405–406 Jaén 199, 222 Jerusalem 57, 61, 64, 68, 74 Joan i, king 338 Johannes Angelus 172 Johannes Virdung 171

John of Dumpno 42, 167, 181–182, 373 John of Genoa 16–17, 21–22, 24–25, 28, 229 John of Gmunden 49–53, 55, 77, 171–172 John of Lignères 15–16, 21, 78, 85, 87, 93–95, 104, 107, 147, 163–166, 172, 227–230, 237, 248, 283, 309–310, 320, 329, 367 John of Montfort 16–17 John of Murs 4, 16, 40, 101, 147, 158, 162–163, 166–167, 227–230, 323–233, 308–337 John of Saxony 9–10, 13, 15–17, 21, 25, 28, 36, 46, 53–55, 57, 229, 233, 309, 311, 313, 367 John of Spira 228, 248 John Vimond 4, 76, 87, 94–95, 102, 104–105, 117, 140, 147, 157–164, 166, 170, 173, 175, 227–305, 309–310, 318–321, 323–326, 336 Joseph Vizinus 384, 390 Joseph (or Yosef) Ibn Waqār 44–45, 55, 167, 180, 367, 383, 386 Juan Gil 43–44, 74, 167, 180, 190, 207, 383, 386, 390 Judah ben Asher ii 167, 338 Judah ben Moses ha-Cohen 157, 227, 283 Judah ben Verga 248, 340 Kabbala 339 Ibn al-Kammād 4, 42–47, 55, 74–75, 167, 179–223, 341, 343, 350, 357, 359, 362, 364, 367–368, 373–387, 391 al-Kawr ʿalā al-dawr 181–182, 222, 373 al-Kāshī 89 al-Khāzinī 89, 95 Kennedy, E.S. 2, 4, 99–101, 179 Kepler, J. 3 al-Khwārizmī 2, 4, 41, 58, 74, 183, 189–191, 198, 201, 204–207, 212–213, 215, 218–219, 221–222, 264, 271, 362, 364, 391 Kitāb al-ʿAmal biʾl-Asṭurlāb 200 Kitāb al-hayʾa 385 Kremer, R.L. 313, 333, 337 Kunitzsch, P. 304–305, 369, 373, 375, 383, 387 Lamarca, M. 95 Langermann, Y.T. 69, 369, 387 Latitude, geographical 107, 147, 198, 215, 352–355, 364, 390–391, 396–397, 402 Leiria 47, 61, 77, 174, 384 Lejbowicz, M. 336

410 Length of daylight 102, 107, 215, 349, 353, 395–396 Levi ben Gerson 10, 46–47, 51, 55, 64, 102, 167, 248, 261, 340, 367 Liber motu octave sphere 207 Libro de las estrellas de la ochaua espera 283 Libro de las estrellas fixas 283 Libro de las Tablas Alfonsíes 200 Libro de las xlviii figuras de la viii spera 283 Libro del astrolabio plano 158 Liechtenstein, P. 172 Lisbon 164, 330, 332, 390 Lisieux 229 London 254 Longitude, geographical 101, 107, 147, 364–365, 402 Luminarum atque planetarum motuum tabulae octoginta quinque 52 Lunation 238–239, 251, 317 Madrid 75, 389 Maestlin, M. 53 Maghrib 75, 102, 158–160, 260 Maimonides 360 Mancha, J.L. 38, 310 Marrākesh 157 al-Marrākushi 180, 207, 219–220, 382–283 Abu Maʿshar 221, 223 Maslama b. Aḥmad al-Majrītị̄ 207, 212–213, 221–222, 384–386 Melchion de Friquento 100 Mercier, R. 56, 152, 340 Meridian 195, 391 Meridian of water 365, 402 Mestres, À. 223 Method of sines 204 Midheaven 195, 355, 391 Milan 73 Millás, J.M. 389, 402 Minhāj 161, 382 Mithridates, Flavius 338–369 Molad Tohu 357 Month astrological 5, 400 synodic 222, 231–232, 236, 239, 241, 318–319, 332, 358–361, 363–364, 395 Montpellier 395–397 Morella 229

index Moon astrological 5, 399–400 anomaly 11, 13–15, 19–20, 25, 48–49, 51–53, 58, 64–69, 108–110, 112, 191, 212 et passim apogee 61, 217, 354 argument of latitude 145, 199–200, 204, 317, 330, 349, 351, 355, 364 correction 10, 13, 42, 47, 61, 64 cycle 316–317, 332, 359–364, 394 deferent 110 eccentricity 22 equations 13, 22, 52, 102, 109–112, 114, 163, 191, 216–217, 243–244, 313, 318, 342, 345–346, 348, 366 increment 109, 111–112, 114 latitude 102, 110, 145, 199, 204, 216–217, 355–357, 396 motion 47, 102, 107–110, 167, 213–214, 237, 239, 317–318, 332, 347, 360, 362–364, 366 longitudo 13, 15 models 11, 13, 16, 22, 46, 48, 53, 110 nodes 103, 115, 145, 213, 236–238, 240–241, 320–321, 341–342, 355, 364, 395, 405 orb 145, 216 perigee 61, 64–67, 354 positions 13, 45, 48, 108–111, 113–115, 230, 237, 247–248, 316, 332, 334–335, 348, 396 progress 398–399 variation 54 velocities 12–18, 19, 21, 24–25, 41–44, 46–48, 52–53, 58, 167, 190–194, 206, 244–248, 317, 332–334, 347, 355, 357, 395–396 Ibn Muʿādh 199, 222, 386 Muḥammad 382 Ibn al-Muthannā 200 al-Muqtabis 42, 179–223, 373, 381, 385, 391 Nallino, C.A. 2 Naples 73, 100, 339 Nativity 222 Nebulous 304 Neugebauer, O. 2, 53, 198, 350 Newton, I. 2 Nicholaus de Heybech 3, 9–38, 48–50, 52–53, 55, 57–69, 333, 367

index Nicholaus de Reichenbach 50 Ninth sphere 236, 342 Nissim Abū l-Faraj 339, 368 Nonagesimal 355 Normandy 229, 309 North, J.D. 161, 179, 305, 308, 336 Nuremberg 73, 94, 386 Obliquity of the ecliptic 107, 195, 198, 205, 211, 214–215, 391, 396–397 On the solar year 386 Opposition 3, 7, 9, 29–31, 36–38, 40–41, 47, 49–52, 57, 69, 190, 231, 234–236, 245–247, 330–333, 338, 358, 361, 368; see also syzygy Oraḥ selulah 47, 338, 341, 368 Oxford Tables 77, 79, 85–86, 167; see also Tabule anglicane Palermo 42, 167, 181–182, 373 Parallax adjusted 201–204, 206, 352–353 in latitude 201 in longitude 205–206 lunar 201, 353–355 solar 201 Paris 3–4, 9–10, 45, 58, 69, 73, 76, 78, 101, 107–109, 119, 157–158, 163 et passim Patefit ex Ptolomei 310, 317–319, 330, 334, 336 Pedersen, F.S. 95, 305, 387 Pedersen, O. 261, 342 Pere iii (el Cerimoniós), king 338, 383 Perpignan 47, 107, 200 Perugia 80 Peter Nightingale, of Dacia 230, 397–398 Peter of St. Omer 328 Peurbach, Georg 52–54, 172 Pico della Mirandola, G. 339 Planets age 163 anomaly 77, 79, 83, 86, 104, 118–120, 124, 128, 130, 134 et passim apogees 81–82, 116, 120, 124, 126–131, 135–136, 139 et passim argument of latitude 81–82 associated 283–305, 374, 383–384 conjunctions 162–163, 167, 311–315, 322, 329–330, 336 center 77, 83, 86, 118–120, 122, 124–125, 135, 137 et passim

411 deferent 81–82, 90, 120, 156, 269, 312, 314–315, 321–323 eccentricity 151, 155–156, 161, 264–265, 269 deviation 73, 75–79, 83, 85–87, 89–90, 92, 94–95, 146–147, 271, 325 epicycle 120, 126, 130–131, 137, 151–154, 261, 269–270 equant 120, 152, 155, 269 equations 103, 117, 119–127, 130–133, 135, 137–140 et passim inclination 73, 76–77, 82–87, 92, 271, 274–275, 324 latitudes 3, 73–95, 101, 103, 144, 146, 272–276, 322–326 longitudes 73, 79, 99, 312 models 151–152 motion 102–103, 117–119, 155, 162, 213–214, 249–250, 252–253, 259, 313, 316, 336, 406 nodes 82, 216, 400 perigee 154–155, 254 phases 144 porcio (portio) 104, 118, 120–121, 123, 142–143 positions 4, 120, 141, 144, 152, 154–155, 157, 162–163 et passim slant 73, 76–77, 82–87, 92, 146, 271, 274, 277, 324–326 stations 100, 117, 120, 128–130, 135–137, 156, 158 et passim unequal motion 78 velocities 160, 167, 257, 266, 310 visibility 103, 144 Plato of Tivoli 282, 304 Polonius, Nicholaus 60 Porres, B. 305, 369 Poulle, E. 38, 330 Precession 103–104, 115–116, 174, 211, 375, 381, 384–385 Prime meridian 365, 402–403 Priores astrologi motus corporum 78, 94, 104, 147 Prosdocimo de’ Beldomandi 170 Provence 64 Prugnerus, N. 52 Prutenic Tables 1, 53 Prowe, L. 79 Ptolemy 1–4, 10–13, 16, 25, 28, 40–41, 46–48, 53–55 et passim

412

index

al-Qānūn al-Masʿūdī 200 al-Qaṭṭān 384–385 Quadripartitum 282–283; see also Tetrabiblos Quran 341 al-Qusanṭīnī (al-Qusanṭaynī) 212 Rabat 215 Radix 230 Ibn al-Raqqām 47, 157, 211, 341 Ratdolt, E. 1, 172 Recueil des plus celebres astrologues Regiomontanus, J. 52–53, 171–172 Reinhold, E. 1, 53, 90 Rome 73, 172, 339 Rosińska, G. 79–80, 83

229

Salamanca 48, 57, 60, 64–65, 68, 77, 174, 328, 355 Salé 215, 222 Saliba, G. 99 Samsó, J. 95, 305, 387 Samuel ben Nissim 339, 368 Sanctification of the New Moon 360 Santarem 390 Santritter, J.L. 78, 87, 90, 93–95, 172 Saros 332 Scriptum cuiusdam saraceni 386 Seasonal hour 102, 107 Seleucid era 207 Sevilla 167, 172, 180 Ibn al-Shātịr 206 Shifted tables 101 Ibn Sina 341 Sicily 4, 47, 338, 364, 368 Signs (physical and zodiacal) 10, 50, 100, 104, 147, 159, 164, 236, 281, 304, 311, 323, 326, 330, 332, 334 al-Sijzī 221 Simon de Phares 229 Six Wings 50, 349, 351, 368 Solstice 363 al-Ṣūfī 283 Sun altitude 219, 390 anomaly 11, 19–20, 48–49, 51–52, 58, 64, 66, 117 et passim apogee 64–65, 104–105, 151, 156, 160, 183, 212 et passim

center 101, 103–106, 109, 212 correction 10, 21, 42, 47, 61, 64 cycle 316 declination 195, 198, 214, 391 diameter 356 eccentricity 22, 182–184 equation 18, 20, 52, 102, 105–106, 151, 155–156, 158, 182–191, 217, 313, 318, 342, 344, 346 longitudo 13, 15 model 155 motion 21, 47, 66, 100–105, 117, 212, 214, 220, 237, 239–241, 252, 313, 330, 349, 359, 362–363, 400 porcio (portio) 104 positions 13, 45, 48, 68, 106–107, 117, 212, 215, 230 et passim velocities 11, 14–15, 17, 19–21, 24–25, 41–45, 47, 52, 167 et passim Sunrise 107, 230–231, 327–328, 353 Sunset 107, 231, 353, 357 Superatio 14 Suter, H. 2 Swerdlow, N.M. 53, 73, 95 Syracuse 40 Syzygy 3, 9–38, 40–56, 57–58, 60, 64 et passim; see also conjunction, opposition Tables for the Seven Planets 4, 99–148 Tables of 1321 163, 167, 308–337 Tables of Barcelona 43, 74, 167, 180, 190, 195, 198, 200, 203–207, 219, 383–384, 391 Tables of Toulouse 166, 190 Tabula primi mobilis Joannis de Monteregio 52 Tabulae astronomiae 171 Tabulae breviores 172 Tabulae directionum 53, 90 Tabulae eclypsium 52–53 Tabulae maiores 172 Tabulae permanentes 167 Tabulae resolutae 60, 171 Tabule anglicane 167; see also Oxford Tables Tabule astronomice Elisabeth Regine 172 Tabule frequentine 100 Tabule Jahen 386

413

index Tabule magne 104, 164, 166, 248 Tabule Verificate 60–65, 68, 328, 355 Tannstetter, G. 52 Tarascon 51, 102, 167, 340, 369 Term (astrological) 402–404 Tetrabiblos 282–283, 304, 375, 401–402; see also Quadripartitum Thābit Ibn Qurra 207, 386 Theoricae novae planetarum 94 Theorica planetarum 9 Thorndike, L. 9 Thoren, V.E. 54 Tihon, A. 152, 340 Toledan Tables 14–15, 17, 42, 156–157, 159–160, 166 et passim Toledo 4, 54, 69, 100–101, 108, 119, 227, 231–231 et passim Toomer, G.J. 151, 156 Toulouse 107 Treatise on the motion of the fixed stars 207, 381 Trepidation 102–103, 115–116, 207, 209–211, 237, 240, 278, 280; see also access and recess Trigonometric functions 218–219 Triplicity 403–405 Tübingen 53 Tunis 157, 373 Tunisia 75 Ujjain 231 Urbino 339, 341, 358–359, 361, 368 Vatican 73 Venice 73, 79, 147, 158, 171–172 Verger, N. 95 Vernal point 152, 207, 381 Vienna 49–50, 52, 73, 77, 171

waḍʿī 101, 168 Weekday 68, 361 William Batecombe 77, 87, 95, 167 William of Saint-Cloud 230, 233, 309 William Raymond de Moncada 338–339, 341, 360–361, 364 Yaḥyā ibn Abī Mansụ̄ r 74, 180, 183, 195, 203–204, 206, 216, 391 Yaʿqūb ibn Ṭāriq 199 Yazdijird era 207, 386 Year Arabic 211 Civil 391, 394–395 Collected 103 Egyptian 117, 207 Expanded 103 Leap 232–233, 361, 395 “Mighty” 221, 223 Roman 231 Sidereal 220 Solar 219–221, 240 Tropical 239, 399 Yuçaf Benacomed 181 Ibn Yūnus 41, 336 Zacut, A. 47–48, 60–61, 64–65, 68–69, 77, 87, 95, 174, 320, 328, 338, 340, 355, 384, 386, 390, 404–405 Ibn al-Zarqālluh see Azarquiel Zenith distance 203–204 Zīj (zījāt) 42 al-Zīj al-Khāqānī 89, 95 al-Zīj al-Mumtaḥan 74, 180, 214, 216, 382 al-Zīj al-Ṣābiʾ 155; see also Battānī al-Zīj al-Sanjarī 89, 95 Zīj al-Shāh 155 al-Zīj al-Sindhind 155, 199 Zodiac 281