Ibn al-Muthanna commentary on the astronomical tables of al-Khwârizmî

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YALE STUDIES IN THE HISTORY OF SCIENCE AND MEDICINE, 2

Ibn al-Muthanna’s Commentary on the Astronomical Tables of al-Khwarizml

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Copyright © 1967 by Yale University. Designed by Anne Rajotte, set in Electra and Hebrew Peninim No. 217 type, and printed in the United States of America by The Vail-Ballou Press, Inc. and Maurice Jacobs, Inc. Distributed in Canada by McGill University Press. All rights reserved. This book may not be reproduced, in whole or in part, in any form (except by reviewers for the public press), without written permission from the publishers. Library of Congress catalog card number: 67-13434 Published with assistance from the Hickox Fund.

To Hanina and Rachel

Preface One of the foremost intellectual currents that preceded the develop¬ ment of modern science, and indeed greatly contributed to its founda¬ tion, was a long and fruitful tradition in astronomy and mathematics. It is particularly fortunate that in the late nineteenth century a major effort was made to produce reliable editions of Greek scientific texts. More recently, and in large measure due to the work of Otto Neugebauer, the sources for Babylonian mathematics and astronomy have been added to the literature of the exact sciences in antiquity. The medieval period has not received the same devoted attention, and we suffer from the paucity of published texts. With a few notable excep¬ tions, the medieval writers were excessively verbose, endlessly belaboring well-known material, with the result that all too often the author’s original contribution is obscured. The many medieval manuscripts that rest in the great libraries are of varying scientific quality and of different degrees of historical interest, so that one should be judicious in selecting a text for intensive study. Nevertheless, it is only with the publication of relevant texts and adequate commentaries on their scientific content that our knowledge of the period can be put on a firm basis. Because of the unfailing efforts of many scholars, Moritz Steinschneider foremost among them, we possess a number of useful guides to the vast library holdings of Hebrew, Arabic, and Latin scientific manuscripts, yet few investigators have taken full advantage of these pioneering labors. Ibn al-Muthanna's astronomical work, though not of high technical quality, is related to a number of published texts of great historical interest. In the search for details of the early stages of astronomical activity in the Islamic world, with which this family of texts is con¬ cerned, we are handicapped by an almost complete lack of contem¬ porary documents. In a larger sense, the whole process of the medieval revival of learning cannot yet be understood, because our knowledge of the emergence of Islamic science is incomplete and fragmentary. The texts concerning the astronomical tables of al-Khwarizml (ca. 840)

vii

PREFACE

seem, at present, to be as close as we can get to the early development of Islamic astronomy in the late eighth century, when Hindu astrono¬ mical procedures were first introduced. Ibn al-Muthanna’s commentary on the astronomical tables of al-Khwarizmi, which dates from the tenth century, is not directly helpful in solving this historical problem, but it does give us some idea of the extent to which the Hindu procedures were understood in the Islamic world. Ibn al-Muthanna generally assumed

that the Hindu and the Greek approaches to

astronomy were virtually identical, and he attempted to explain the Hindu procedures by appealing to the Greek models. Since the two methods are quite distinct, his comments often reveal that his under¬ standing was both faulty and incomplete. But this account of a major early astronomical text cannot be ignored, for it is one of the most detailed reports that antedate the surviving Latin version of al-Majriti’s recension. Ibn al-Muthanna’s commentary does not survive in Arabic; only the Hebrew and Latin versions remain. In the course of this study of the two Hebrew versions of Ibn alMuthanna’s commentary on the astronomical tables of al-Khwarizmi, I have had the benefit of advice from several scholars in related fields. I wish to thank Dr. Isadore Twersky and Dr. Harry A. Wolfson of Harvard University, Dr. Abraham Sachs of Brown University, and Dr. Asger Aaboe of Yale University for their many helpful suggestions in the preparation of this book. Special thanks are due to Dr. Otto Neugebauer of Brown University, who supervised the doctoral dissertation on which this publication is based. To my wife, Pauline, I am indebted for constant encouragement and her willingness to type successive drafts. Finally I wish to thank the Trustees of the Bodleian Library and of the Biblioteca Palatina di Parma for their cooperation in this study and their permission to publish Hebrew manuscripts from their collections. Spring Glen Hamden, Connecticut June, ig66

B. R. G.

vm

Contents

Preface

vp

Introduction: The Transmission of Hindu Astronomy to Islam Transcription of Hebrew Letters Appearing in Figures

3 14

Translation of the Michael Version Introduction I.

Chronology

II.

16

Planetary Theory

26

Mean motion; Radix; Apogee; Node; Equation of center and anomaly; Maximum equation of anomaly; Computation of the longitude of an inner planet; Explanation of the tables; Table for stations; Radix III.

Trigonometry

49

Sine function; Declination; Mural quadrant; Declination com¬ puted; Interpolation for sine and declination; Versine function; Finding latitude of cities; Finding noon altitude of the sun; Right ascension; Oblique ascension IV.

Seasonal Hours and Gnomons

81

Daily arc; Seasonal hours; Time reckoned from solar altitude; Converting seasonal hours to equal hours; Exact determination of time from solar altitude; Finding the ascendant; The twelve astrological houses; Gnomon; Finding the altitude of the sun from its gnomon shadow V.

Planetary Latitude

89

Latitude of the moon; Latitude of the planets VI.

Conjunction and Opposition

Mean syzygy; True syzygy

IX

94

CONTENTS

VII.

First Visibility of the Lunar Crescent

96

Day of new moon; Polar longitude computed; Right ascen¬ sional arc corresponding to lunar latitude; Oblique ascension between sun and moon—Digits of light; Rising time and set¬ ting time—Projection of first visibility VIII.

Diameters of Sun, Moon, and Shadow

104

Sun and moon; Shadow; Explanation for the sizes of the sun, moon, and shadow IX.

Lunar Eclipses

109

Maximum argument of latitude; Llalf-duration of eclipses; Finding lunar latitude at beginning and end of eclipses; Cal¬ culation of eclipse limits; Colors of lunar eclipses X.

Solar Eclipses

120

Elements for solar eclipses; Parallax in longitude; Parallax in latitude; Area digits; Duration and eclipse phases; Projection of solar eclipses XI.

Excess of Revolution

Translation of the Parma Version

143 147

Notes to Ibn al-Muthanna’s Commentary on the Astronomical Tables of al-Khwarizmi

185

Notes to the Introduction of the Parma Version

243

References and Bibliographical Abbreviations

247

Index

253

Hebrew Section Indices of Personal Names

262

Glossaries

274

Parma Text

302

Michael Text

404

Contents

408

Introduction

*

The Transmission of Hindu Astronomy to Islam

For many years scholars have labored to determine the sources of Islamic astronomy. It has long been known that the original impetus to study astronomy came from India toward the end of the eighth cen¬ tury 1 and that the Hindu procedures were later displaced in large measure by Greek-Ptolemaic methods. Both the Greek and the Hindu approaches to the solution of astronomical problems were highly suc¬ cessful, and either one can be used to compute the position of the sun, the moon, and the planets at any given time, as well as to predict solar and lunar eclipses. Their methods, however, are quite different and can be distinguished easily. The task of reconstructing the details of the transmission of Hindu astronomy to Islam is made difficult by the almost complete lack of suitable textual material dating from the latter half of the eighth century. There are two main sources for our knowledge of this trans¬ mission: the accounts of the historians—all of whom lived several cen¬ turies after the events they tell us about—and gleanings from the body of astronomical material itself for evidence of borrowed Hindu pro¬ cedures. A typical account of the transmission dates from the twelfth century, or about four centuries after the event. Ibn Ezra, a twelfth-century Spanish Jew, gave this account (in Hebrew) which may be summarized as follows (cf. Parma text, introduction). i. Recently E. S. Kennedy and B.L. van der Waerden (cf. “World-Year,” pp. 324 ff.) have raised the question of the intercession of Persian sources between the Hindus and the Arabs. It seems that at least some Hindu astronomy entered the Islamic world via a translation from the Pehlevi of the Zij-i Shah in the eighth century.

*■>

;>

IBN AL-MUTHANNa’s COMMENTARY

The first Abbasid khalif (ca. a.d. 750) had heard of the sciences of the Indians and wished to have some of their books translated into Arabic, for “profane,” i.e. nonreligious, sciences were un¬ known to the Arabs at that time. He was not sure that this was religiously permissible until the angel of dreams assured him of it. He sent for a Jew, and told him to go to Arin (i.e. Ujjain) to bring back an Indian scholar, which by some subterfuge the emis¬ sary managed to do. The Jew served as an interpreter between the Indian, Kanka, and the Arab who translated the book of Indian astronomy. Only traditional facts were presented in this book; the reasoning behind them was omitted. The last remark is very important for it indicates that only the rather complicated procedures were transmitted to the Arabs, and that the ex¬ planation and the meaning of these procedures remained unknown to them. The Hindu texts that are available to us are written in verse and indicate the procedures very cursorily. All explanation was left to the commentaries, which were not transmitted to the Arabs, who had to reconstruct the meaning of the verses as best they could. Let us note in passing that Arin, which in medieval Europe became the “navel of the earth,” was a real city in India and, in fact a center for astronomy. A somewhat different account of the transmision is given by an eleventh-century Muslim, al-Biruni, who had actually visited India and studied Hindu astronomy from its Sanskrit sources. According to al-Biruni, Hindu astronomy entered the Islamic world by way of a trans¬ lation from Sanskrit called the zij al-Sindhind (not extant), written ca. 770, which was a translation of the Brahmasiddhanta of Brahma¬ gupta (a.d. 628) made at the court of the khalif al-Mansur.2 Zij, an Arabic word borrowed from the Persian, denotes a set of astronomical tables (Greek: kanon) together with precepts, with some occasional remarks explaining the theories according to which the tables were con¬ structed. His account agrees with Ibn Ezra’s above in most of its essential features, although there is a difference of some twenty years

2. Kennedy, Survey, p 129; al-Biruni, RascPil, part al-Biruni, India, Sachau, II, 15 and passim.

4

3, p. 27

(trans., p.

31);

INTRODUCTION

in the date when the Hindu scholar is said to have arrived at the court of the khalif. The divergence between the sources is, however, not too serious, and the resultant dating is far better than that obtainable for previous transmissions of astronomy—for example from Babylonian to Greek or from Greek to Indian. This study presents one text in the Islamic astronomical literature, that tries to explain the Hindu procedures. Although the text is some¬ what removed from the time of transmission, it is about as close as we can come to the Hindu methods that were adopted by the Arabs. This text (from the tenth century) is a commentary to the astronomical tables of al-Khwarizimi, the original Arabic text of which, composed ca. 840, is lost. It survives only in a Latin translation (presumably written by Adelard of Bath) of the Arabic version by al-Majriti. The text of Adelard’s translation with the accompanying tables was pub¬ lished with a German commentary by Suter,3 and more recently an English translation of the Latin text with a new commentary to the text as well as the tables was published by Neugebauer.4 It has been pointed out that the text of Adelard is quite removed from the original zij, since many changes, the details of which are not known, were intro¬ duced by Maslama ibn Ahmad al-Majriti (fl. 1000).5 Besides being a distant witness to the Khwarizmi text, the Latin includes little in the way of explaining the tables other than telling us how to use them in making various computations. We turn therefore to the commentary of Ibn al-Muthanna for more information about the original Khwarizmi tables and to find out more about the Hindu influence on them. Unfortunately, the text of Ibn al-Muthanna is also lost, though it is preserved in both Hebrew and Latin versions. Moreover, another difficulty is posed by the intrusion of al-Farghani (ninth century) between the Khwarizmi zij and the com¬ mentary of Ibn al-Muthanna. Al-Biruni noted the existence of a book by al-Farghani, criticizing the zij of al-Khwarizmi,6 and further evidence

3. H. Suter, Khwdrizmi, Copenhagen, 1914 4. O. Neugebauer, Khwdrizmi, Copenhagen, 1962. 5. Ibid., p. 8. 6. Kennedy, Survey, p. 128; al-Biruni, Rasd'il part 1, pp. 128, 168.

5

IBN AL-MUTHANNa's COMMENTARY

of the existence of that text is found in Ibn al-Muthanna’s introduction. In fact, our author asserts that he wrote his book to clarify the ob¬ scurities of al-Farghani, and one may therefore ask whether his work is not really a slightly expanded version of al-Farghani’s commentary. Since the Farghani text has not survived, it is difficult to answer that ques¬ tion definitively. Our text, which is presented in the form of questions and answers, often includes part of the commentary in the question, and sometimes consecutive questions form a continuous unit, while the answers are irrelevant (see the eclipse theory, for example).

The Contents of Ibn al-Muthannas Text The text itself furnishes evidence of the influence of Hindu astron¬ omy. First we note that the arrangement of the chapters follows the Hindu texts rather than the Almagest of Ptolemy. For example, planetary theory precedes the discusion of eclipses. The first section deals with problems of chronology. We are told how to transform a date reckoned with respect to one epoch into a date with respect to another epoch. Three epochs and their corresponding calendars are discussed: the era of Alexander, i.e. the Seleucid era, Syrian norm; the Hijra era—the strictly lunar Arabic calendar; and finally the Persian era, dating from Yazdegird, the last Persian king be¬ fore the Muslin conquest, with a calendar using Egyptian years of ex¬ actly 365 days each. The original Khwarizmi tables were arranged for the Persian era, but the surviving version is arranged for the Hijra era. This is the first of many changes which were made in the text of the Khwarizmi tables. None of these eras was used in India, and thus the transformation from the Hindu calendar to the Muslim calendar had already taken place by the time of al-Khwarizmi. This is not too sur¬ prising, since the transmission took place late in the eighth century, whereas al-Khwarizmi was writing some years later, early in the ninth century. T he second section begins the discussion of the determination of planetary positions in longitude for any given time. The Hindu pro-

6

INTRODUCTION

cedure for finding planetary longitude consists of a number of successive approximations. Although it is possible to give a geometrical justifica¬ tion for the computation, it is presented as an arithmetical device in the Hindu sources. Ibn al-Muthanna is quite familiar with the details of the procedure, but he has great difficulty in explaining why it works. His explanation hinges on a geometrical model better suited for describing Greek than Hindu procedures. We have here a verification of Ibn Ezra’s remark, quoted earlier, that only the procedures were transmitted, not the reasons for them. Lacking an alternative, our author tried unsuc¬ cessfully to explain these Hindu methods with Greek-Ptolemaic models for planetary motion, which were widely known in his day. The next section concerns trigonometry and contains a curious mix¬ ture of Hindu and Ptolemaic methods. As in most medieval texts, Ptolemy’s table of chords of a circle is replaced by a table of the sine function. Our text goes into much detail about computing this func¬ tion, all based on simple applications of the Pythagorean theorem. But in computing right and oblique ascension, the text makes use of the theorem of Menelaus—the fundamental theorem for solving spherical triangles—which was unknown to the Hindus. For computing ascen¬ sions, the obliquity of the ecliptic is taken as 24°, the standard Hindu value, and the unit radius is taken as 150, a value found in a work of Brahmagupta (a seventh-century Hindu) which was certainly one of the texts transmitted to the Arabs. On the other hand the values for dec¬ lination are taken from Ptolemy’s table of declinations, which is based on an obliquity of 23551,20° (i.e. 23 degrees, 51 minutes, and 20 seconds; this notation will be used for all sexagesimal fractions), rather than the 24° of the Hindus. Moreover, our author is aware of the better Islamic value for obliquity, namely, 23533° which was found from observations shortly before the Khwarizmi text was written, but he does not use it in any way. Thus even when better values are ob¬ tained

from observation, they are not necessarily introduced into

theoretical astronomy; traditions are not readily discarded. The next section is concerned with various problems involving the time of day. One part deals with determining the time from the solar altitude, and uses a well-known Hindu method. We are also told how to find the solar altitude from the gnomon shadow, a device known both

7

IBN AL-MUTHANNa’s COMMENTARY

to the Greeks and the Hindus. The proof that the gnomon shadow allows us to find the solar altitude parallels that of Ptolemy except that sines replace chords, and the length of the gnomon is taken to be twelve digits, as in the Hindu theory, rather than sixty parts as in Ptolemy’s Almagest. In the discussion of planetary latitude, again we find Hindu pro¬ cedures described. For example, the maximum lunar latitude is taken to be 4V20 as in the Hindu theory', rather than the 50 of the Almagest. Our author does not attempt to explain the procedures for finding planetary latitudes from a geometric model, as he did for planetary longitudes, but merely expands on the details of the computation by which one arrives at the planetary latitude. As in the Hindu sources, our text concludes with a lengthy discus¬ sion of the procedures that lead up to the determination of first visibility of the lunar crescent, and the prediction of solar and lunar eclipses. The Hindu procedures are systematically followed, so much so that the order of the questions in our text is often that of the corresponding verses in the work of Brahmagupta. However, we find that the explana¬ tion of these procedures is inadequate and incorrect, once again lending credence to the theory that the procedures arrived in the Islamic world without proper commentary. In fact, sometimes our author has mis¬ understood and misrepresented Hindu procedures in his attempt to make them conform to principles of Greek-Ptolemaic astronomy. Unfortunately for Ibn al-Muthanna, the two systems are quite distinct, and the Greek model does not apply. Much still remains to be learned about the transmission of Hindu astronomy to Islam, yet this text confirms some of the medieval tra¬ ditions about the transmission, and elaborates on the procedures and what they meant to the Muslims who applied them.

Sources The commentary to the zij of al-Khwarizmi by Ibn al-Muthanna is preserved in both Hebrew and Latin versions, although the original Arabic text is lost. The two Hebrew versions, written in Rabbinic script, are: (1) Ms Parma (Bib. Palatina 2636, olim De Rossi 212),

8

INTRODUCTION

ff- 1-13V>7 ar)d (2) Ms Michael 400 (Bodleian Library, Oxford), ff. 4^74b of which f. 49 is completely blank. The Parma manuscript is much shorter than the Michael text and ends abruptly in the middle of a sentence. There is an added introduction in the Parma text concerning the transmission of Hindu astronomy to the Arabs, which is not found in the Michael version, whereas the first chapter of the Michael text (on chronology) is missing in the Parma version. Moreover, although the Parma version is explicitly attributed to Ibn Ezra (d. 1167),7 8 9 and is dated

a.d.

1160 (see Parma p. 150), there is no reference to the author

or the date of composition in the Michael text. Ibn al-Muthanna (probably tenth century) is known only to have written this book; the one reference to it may be found in the Tabaqat al-Umam of Sacid al-Andalusi where we learn that Ibn al-Muthanna wrote a work entitled: Taclil zij al-Khwarizmi? In the text of Ibn alMuthanna, reference is made to the Mumtahan zij (M, p. 63) and the Habash zij (M, p. 109). The text also mentions Muhammad b. c Abbas, the computer, whom I have not been able to identify. Steinschneider, who was the first to study this text, assumed that the author of the Michael text was the same as the author of the Parma version—namely, Ibn Ezra—and this authorship has never been ques¬ tioned in the subsequent literature.10 However, it is my opinion that a close reading of the two texts negates this claim. In the Parma text there are many parenthetical remarks introduced by the expression

7. A transcription of the Parma manuscript was kindly sent to me by Prof. J. M. Millas Vallicrosa. 8. For a discussion of Ibn Ezra’s contribution to Hebrew letters, see Stein¬ schneider, Gesammelte Schriften, “Abraham Ibn Ezra,” pp. 407-98. 9. Tabakdt al-Umam, trans. Blachere, p. 113: “Signalons Ahmed ibn alMuthanna ibn cAbd al-Karim, 1’auteur du TaHil zij al-Khwdrizmi.” Before this refer¬ ence to Ibn al-Muthanni was noticed by Millas Vallicrosa (cf. Azarchiel, p. 26), it had been suggested that Muthannfl was identical with Biruni (cf. Suter, “Der Verfasser des Buches”). 10. Steinschneider, “Zur Geschichte der Uebersetzungen aus dem Indischen ins Arabische und ihres Einflusses auf die arabische Literatur,” ZDMG, 24 (1870), 325 ff.; 25, 420 ff. An English translation of Steinschneider’s Hebrew text of the introduction to the two manuscripts was published by D. E. Smith and Y. Ginsburg, “Rabbi Ben Ezra and the Hindu-Arabic Problem,” The American Mathematical Monthly, 25 (1918), 99-108. Cf. J. M. Millas Vallicrosa: “cAb6datS shel R. Abraham

ibn

cEzra

be-hokmat

hatekuna,”

9

Tarbiz,

9

(1938),

306-22;

Las

IBN AL-MUTHANNa’s COMMENTARY

“Abraham [Ibn Ezra] said,” while the Michael version omits both the remarks and the attribution to Abraham. Differences in style between the two versions abound. To illustrate this, one may compare the five following pairs of sentences where the sense is the same in both texts.

Parma

Michael

'Oia p nana nax nab 9:x3

nmsaxn ibx on na 4:a47

amxaxn

eras? naa mmba aman

a^abnan

1

’anian

mmba a^xsaa

Tp mbxn *o aaa?m an ^ nn 6:a47

man naaan aa?n 'a as?t?m

naaba nan ns?aia>n a^aaaan maa Tm

amnaa naaapaa avnwa m?au?n

pna anbty 'brim aaana a^aaaan'na

p»m

pbna

amana

i2:x3

2

naaapaa

nba biaa

nbaa ptrma anman im np^n pas? na 28:a47

anau?

ttnarcn

as?a

na

i9:a3

3

bab anso nbnna ,,oaa |a nana

aaaa *?a>b> mamba

ma?a 19:X48

avni^ab tm 'a nr as?ai 5:a4

sam cnaaan) baban ana a'baba 'a

ms?’© ntaaan baba ana o-’baba

babaa

naartw

mnu? pn ba?aan baba msrwa

aaaa naaa mna? xbx nabtan

aaaa ntaaa

nrcpn a'aaaab ,a nr pasa naann

baban

aaa

7:a50

•’yxaxn *]bnan 'a naron 2i:x7

a^arban

*]bnaa mna a^arbana nnx bab

tana *|bna mm a?aa?n -|bna s?xaxa

pma *|bna mna a>au> ba? •’ysax

napn babaa aaax aapm

ppn babaa aabna pn ppn baban

a^aaaan mna

aaax

aabna

naaarnn

s?xax

nan

4

5

The first example illustrates the difference in the questions. The form in the Michael version is generally: na^as? naa ... an na which is not found in the Parma version. In the second example we see many divergences in technical vocabulary, such as different words

traducciones orientates en los manuscritos de la Biblioteca Catedral de Toledo, Madrid, 1942, p. 192; Estudios sobre Azarchiel, Madrid-Granada, 1943-50, pp. 2526; El libro de los fundamentos de las Tables astronomicas, Madrid-Barcelona, 1947, p. 13; “Autenticidad,” Isis, 54 (1963), 114-15.

10

INTRODUCTION

for planet, apogee, node, and minute. The Parma version generally uses the word Q17D for reason, whereas the Michael text uses p3S7 or rDD . In the third example we see that the Michael text uses “)p,’S7 instead of liHItP , and mn instead of 2PD . In the fourth sentence there is a difference in syntax between the two versions in addition to differences in vocabulary. The Parma text uses where the Michael version has nann bibj ; and the Parma text uses p*i where the Michael text has the more common rVr. It has been noted that this uncommon use of pn is a stylistic peculiarity of Ibn Ezra.11 In the fifth example, we again see several noteworthy differences. The Parma text has blV), ]Bpn where the Michael text has HDpn . Moreover, the Parma text uses the word pSl» where the Michael version has the more common T3“ia. The word p22T?3 is, however, common in the writings of Ibn Ezra.12 Although both texts often use more than one technical word to ex¬ press the same sense, many words and phrases are unique to one text or the other. For example, the word K3*” for sine is found only in the I arma text, and the word S7p3 for sine only in the Michael version. Likewise the Parma text uses rvmp 13 for eclipse, whereas the Michael text usesmpb. Glossaries to the two texts, where all the technical terms may be found, are appended. If, as I have tried to prove, the two texts were written by different authors, it is not surprising that the introduction to the Parma text is missing from the Michael version, since it was added by Ibn Ezra and did not form a part of Ibn al-Muthanna’s text. There is another treatise by Ibn Ezra which includes many quotations from and remarks parallel to the commentary of Ibn al-Muthanna. This work, extant in Latin, was published by J. M. Millas Vallicrosa with the title: El libro de los fundamentos de las Tablas astronomicas (Madrid-Barcelona, 1947), and the parallel passages are indicated in the notes. The Latin translation of Ibn al-Muthanna’s work was written by

11. C. Rabin, “Abraham bar Hiyya and Medieval Hebrew,” Metzudah, 3 (1945), 163. 12. Cf. Ibn Ezra’s commentary to Koheleth 3, 15; Book of Reasons, ed. Naphtali ben-Menachem (Hebrew text), Palestine, 1941, p. 9. 13. Ibn Ezra translated a treatise by Mashallah called Sefer be-qadrut; cf. Steinschneider, “Abraham Ibn Ezra,” p. 493 ff.

11

IBN AL-MUTHANNA’s COMMENTARY

Hugo Sanctallensis14 and is preserved in three manuscripts: Bodleian, Arch. Selden B 34; Bodleian, Savile 15; Cambridge, Gonville and Caius College 456. Recently, an edition of the Latin text was published by Millas Vendrell (Madrid-Barcelona, 1963). The independence of the Latin from the Hebrew versions may be seen from the fact that the Latin preserves many technical expressions from the Arabic which are not found in either Hebrew version. On the other hand, the Hebrew versions retain the question-and-answer form of the Arabic, which Hugo abandoned. In the translations that follow, the diagrams have been reconstructed from the instructions in the texts. Figures 1-7, 9, 11-13 appear identi¬ cally in both translations. Figures 8 and 10 are alike in both translations except for the lettering. The diagrams found in the two manuscripts are included without alteration in the edited Hebrew texts. Although illus¬ trations in the Parma text are clearly drawn, those in the Michael text often seem to have been done hastily and without a straightedge, and these have been modified slightly for purposes of clarity. The manu¬ scripts are denoted: P for Parma 2636 (De Rossi 212) M for Michael 400 Se for Arch. Selden B 34 In the edited texts of the two Hebrew versions, the folio and line numbers of the manuscripts have been preserved. In order to facilitate the identification of a passage in the Michael text with its English trans¬ lation, the questions have been numbered [Qi], [Q2], etc. In the margin of the Parma manuscript the questions are numbered from one to seventeen. In the edited text of the Parma version, the questions are numbered according to the corresponding passage in the Michael text. In translating the Hebrew texts, I have felt free to rephrase some passages whose sense was clear from the context. Parentheses are used only where the context does not seem unequivocal. Square brackets indicate that the Hebrew text has been emended.

14. Cf. C. H. Haskins, Studies in the History of Medieval Science, Cambridge (Mass.), 1927, pp. 67-81.

12

Translation of the Michael Version

Transcription of Hebrew Letters Appearing in Figures

k a l 7

n T

A B G D E Z

n u D

*? a 3

H T K L M N

o » b s p

S O F C Q

This book, by Ahmad b. al-Muthanna b. cAbd al-Karim for his brother Muhammad b. cAli b. Ismacil, was composed to explain the tables of al-Khwarizmi. You mentioned, may God bring you success, some contradictions that you noticed in the tables and that the authors of those tables did not ad¬

5

duce proof or argument for the rules which they instructed us to follow, and they did not make known to us why they instructed us as they did. They left these matters to us and presented them as something handed down by tradition, without any argument. Now when treatises are com¬ posed in this way, they lead to misunderstanding because of the concise¬

10

ness which was required. A fault-finding reader would say that the author heard certain statements that he did not understand. Another reader who tried to excuse the author’s brevity would say that the author had high regard for the esteemed discipline of astronomy and did not want to reveal it plainly. Indeed, we have already seen that some scholars

15

outside this branch of science, in whose competence there is no doubt, followed a similar course. For example, al-Akhfash composed a treatise in the art of grammar, called The Mean (al-’Awsat), which was so brief that the masters of grammar declared the book was suitable for neither master nor student.

20

You further mentioned, may God bring you success, your discovery that al-Khwarizmi was among those who practice brevity and withhold explanations in their treatment of most subjects. You mentioned, may the Creator reward you, that you found the book ascribed to al-Farghani, which concerns the tables of al-Khwarizmi, lacking completeness and insufficient for the purpose you wished; and that you found he explained things which were clear, well known, and easy, but he neglected to discuss difficult and incomprehensible matters. Since you asked me to explain the reasons to you, I will do so in order

*5

25

IBN AL-MUTHANNA S COMMENTARY

that nothing be hidden. I have already revealed to you the meaning of all that you asked about. This will please you and will benefit your understanding, and similarly it will please all men of science and geometry, may I receive aid from God. You mentioned, may the Creator

5

help you, that you found the explanations in al-Farghani’s treatise lack¬ ing completeness: I too have read it and found it thus. But from reading other books by al-Farghani, it was clear to me that he was a learned scholar, so it came to my mind that he was engaged in writing his com¬ mentary when death prevented him from finishing the book. But some¬

10

one copied his notes and claimed that they represented a complete com¬ mentary. A second possibility is that al-Farghani completed his commen¬ tary before he died, but that part of it perished in the transmission of the text. A third possibility is that a miserly person purposely deleted passages from his commentary. But God forbid that we doubt his wis¬

15

dom, for truly, no one could comprehend these subjects unless he had an understanding of geometry. I now present to you my commentary on selected subjects, arranged in question and answer form to facilitate your understanding of these tables, and to help you learn and retain this knowledge. I hope, by the

20

Creator, that this book satisfies your request.

I [Or] Question: Why did Muhammad b. Musa al-Khwarizmi say con¬ cerning the derivation of the months of the Arabs: take the complete years of the years of the Arabs, and put them in two places and keep them; multiply one of the two places by 354 and keep it; then multiply

25

the other place by 11 and divide it by 30 and add the result to the other place. Answer: The reason is that the years of the Arabs are lunar years, which, according to the opinion of al-Khwarizmi, have 334 days plus a fifth and a sixth. When he wished to convert the years of the Arabs

30

into days, he instructed that one multiply one of the two places by 354 and the other by 11, which is a fifth and a sixth, for it is the least that is found of a fifth and a sixth from 30, and he instructed that one multiply

16

TRANSLATION OF THE MICHAEL VERSION

it by 11 and divide it by 30, which is the [sum] 1 of a fifth and a sixth, so there would result for him the fraction which is a fifth and a sixth complete days. When he multiplied the years by 354 he had already made them days. Then he added what resulted from multiplying the years by 11 and dividing it by 30, for thus this fraction is made into

5

complete days. If the remainder is greater than 15, take one day for it, for, when there remains 15, it is a half day; and when there remains more than half a day make it a day. But if it is less than 15, which is half a day, disregard it. [Q-z] Why did he say that the total is the radix?

10

Answer: He put these years into days, that is, the number of complete days from the beginning of the days of the Arabs until the day with which one is concerned. And this he called the radix. [Q3] Why did he say: and when you wish to know the beginning of any month, add 5 to the radix and cast out sevens so that the remainder

15

is less than 7; beginning from Sunday, the day which you reached is the first day of Muharram. Answer: The reason is that the year from which the Arabs begin counting had, as the first day of Muharram, Thursday. But the week begins on Sunday, according to the opinion of al-Khwarizmi, so he

20

added 5 to the radix, i.e. the number of days from the beginning of the Arab years until the day being considered, and cast out sevens from that, since there are seven days to the week. As for the remainder, he began counting from Sunday. Whoever follows his reckoning will find the

25

first day of Muharram. [Q4\ Why did he say: if you wish any other month, add to the nota of the year two days for every full month and one day for every other month, the general rule being three days for every two months. Answer: This results from the method of approximation, that is, the lunar month is 29I/2 days and a small fraction, so that when you cast out sevens, there will remain a day and a half and a small fraction, or three

1. Ms: difference; marginal gloss, bo, meaning sic (?).

l7

30

IBN AL-MUTHANNA’s COMMENTARY

days and a small fraction for every two months. He did not wish to say for every month a day and a half, so instead he instructed that one add three days for every two months in agreement with the remainder for two months. He disregarded the small fraction because he wished to ease

5

matters for the student. Indeed the number of days in a lunar month, according to the opinion of Ptolemy, is 29;31,50,50,44,33 days.2 [Q5] Why did he say in deriving the Persian years and months from the Arabic: take the radix and add to it 30 or 29 days for each month of that year that has gone by, and subtract 3,624 from that number,

10

dividing the remainder by 365. Answer: The reason is that the (Persian) era, on which his tables are based, dates from the time of King Yazdegird, and between the era of the Arabs and King Yazdegird are 3,624 days, so that when you subtract that number of days from the days of the Arabs, the remainder is the

15

days of the era of Yazdegird. When this number is divided by 365 days, which are the number of days in a Persian year, the result is the number of complete years. Further, he says that the remainder which is less than 365 should be put into months, for we have seen that what is not a year is months. Also he says, what does not reach this amount (i.e. a

20

month) is days, as we have mentioned. [Q6] Why does he say: when you wish to know which is the current month of the gentiles, and how many years have passed from Dhu'lQarnain, take the radix and add 287 to it, then add to it the months

25

and days of the year which have gone by; then multiply the total byfour. Answer: The reason is that the first day of the years of the Arabs took place in the year 933 of the era of Dhu’l-Oarnain, and an additional 287 days had already gone by. Therefore he said, add 287 to the radix. Further, he had to transform them into Roman years of 365lA days in

30

order to add them to 932, which is the number of complete years of Dhu’l-Qarnain. He instructed that one multiply those days by four because every four years one day is intercalated, and he said also to

2. The Latin version reads: 29531,50,5,44,33d (Se i4r:i4).

18

TRANSLATION OF THE MICHAEL VERSION

divide that number by 1,461 because, by multiplying by four one had calculated quarter days, and it is necessary to put the days of the years into quarters also. Since a year has 365V4 days, one needs 1,461 quarter days in order to express the result in Roman years, which inter¬ calate one day every four years. As for the remainder, which is less than

5

a year, divide it by four, because the days were multiplied by four and it is necessary to return the remainder into days. He said, further, in order to put the result into months, cast out the appropriate number of days for every month, beginning with Tishrin I which is the first month of the Roman year; the remainder, which does not reach a month, gives us the day of the month; and it is thus.

10

[Q7] Why did he say: and if the remainder of the years when divided by four is two, the year is intercalary, i.e. Shvat3 has 29 days. Answer: The reason is that the first day of King Dhu’l-Qarnain was in the second year after the intercalation and two years later there was

15

another intercalation. For that reason he said: and if the remainder is two, the year is intercalary. The Romans wished their year to agree with the solar year which, according to Ptolemy, is 365*4 days, less one part m 300 of a day. Other ancients disagreed with this number of days by a small amount and differed in their observations. The Romans add one

20

day for the quarter day every fourth year, intercalating by making Shvat 29 instead of 28 days. The Persians also intercalate in order to make their years approximate solar years, but they add 30 days in each 120 years. The reason is that the Sagosiim [Soghdian?] peoples believe that one of the angels rules

25

every day of the month. Five days of every year are ruled by other angels, so that no angel rules more than twelve days within a year. (Al-Khwarizmi) said that Ormazd rules the first day of the month; after him comes Bahman, and so on throughout the days of the month. He also said that Dai be-Adar, Dai be-Mihr, and Dai be-Din 4 are called by these names because the first rules the heavens above, which we do not

3. Marginal note in Ms: called February by the Romans. 4. Dai be-Adar is the eighth day of the Persian month; Dai be-Mihr is the fifteenth; Dai be-Din is the twenty-third. Ms: Dai be-Himr and Dai be-rbd.

*9

30

IBN AL-MUTHANNA’s COMMENTARY

perceive; the second rules what we see from the heavens to the earth, but we do not perceive him; and the third rules what we perceive and see. Thus they intercalate 30 days every 120 years. Indeed, they think they do this in order to equalize between the angels, for if they inter¬ 5

calated only one day, they would confuse the angel for that day. It is generally thought that this was their intention. [Q8] Why did he say: when you wish to know the beginning of any Persian month, already knowing the number of complete years of Yazdegird which have gone by, always add three to them and cast out sevens

10

from the sum, and the remainder less than seven is the nota of the year. Answer: The reason he said always add three to them is that the: beginning of the first year of the era of the Persians was a Tuesday, and between it and Sunday are three days. Further, since a year has 365 days, when you cast out sevens, only one day remains. So it suffices to

15

take one day for every simple year and add three to it; the total is days, for the remainder from every year is one day. When one casts out sevens from this sum, the remainder is counted from Sunday, which is the beginning of the week, and the day you reach is the nota of that year, that is, the first day of Farwadin-meh.5

20

[Q9] Why did he say: if you wish to know the beginning of any other month, add two days to the nota of the year for every month, but for Aban-meh do not add anything to it. Answer: The reason is that the Persian months each have 30 days so that when one casts out sevens, two remains for every month. But

25

Aban-meh is the last month, and when one casts out (sevens),6 nothing remains of it. [Q10] Why did he say: when you wish to know when any Roman month begins, take the complete years of Dhu’l-Oarnain and add to them a quarter of them and always add two more, then cast out sevens, so

30

that the remainder is the nota of the year. Answer: The Roman year has 36514 days. When one casts out sevens,

5. Ms: Afrudin-meh. 6. Ms: thirties.

20

TRANSLATION OF THE

MICHAEL VERSION

there remains one and a quarter. Thus one need only take the number of )-ears and add a quarter of them, for he already made the quarter day into complete days. And he said: add two to it, because the first day of Fishrin I of King Dhu 1-Qarnain was Monday, so one must add two days in order to begin from Sunday, and you cast out sevens for the days of the week.

5

[Q11] Why did he say: if you wish to know the beginning of a month other than Tishrin I, add to the nota of the year two days for every 30-day month that has gone by, and three days for every 31-day month that has gone by, but do not take anything for Shvat unless the year is intercalary, in which case take one day for it.

10

Answer: The reason for this is that some of the Roman months have 30 days and some have 31 days. When you cast out sevens, the re¬ mainder is two days from a month of 30 days, and three days from a month of 31 days. Shvat has 28 days when the year is not intercalary,

15

so that when sevens are cast out from it, nothing remains. But when the year is intercalary, one day remains. [Q12] Why did he say: if you wish to know the intercalary year, cast out fours from the years of Dhu’l-Qarnain, and if the remainder is two the year is intercalary, but if the remainder is greater or less than two, the year is not intercalary.

20

Answer: As we mentioned previously, the year in which Dhu’l-Qarnain began to reign was two years after an intercalation. When you cast out fours, which is the number of years between intercalations, from the years of Dhu’l-Qarnain, the year is intercalary if the remainder is two.

25

[Q13] Why did he say: if you wish to find out the Arabic month, how many of its days have gone by, and which Arabic year corresponds to the year of Dhu’l-Qarnain, take the number of complete years of Dhu’lQarnain and subtract 932 from them; multiply what remains by 365V4 days and subtract 287 days, and then add to the remainder the total number of days for the months which have gone by. Begin from Tishrin I so that the number you reach is the number of days from the epoch of the Arabs until the day you have in mind. Answer: He told you to subtract 932, for it is the number of years be-

21

30

IBN AL-MUTHANNA S COMMENTARY

tween Dhu 1-Qarnain and the era of the Arabs. He said to multiply the remainder by

days in order to put the years into days. He ashed

you to subtract 287, for it is the number of days which had gone by of the year of Dhu’l-Qarnain, before the beginning of the era of the Arabs, 5

and to add to this the number of days in the months which had gone by, so that the sum would agree in days with the total number of days of the Arab reckoning. [Q14] Why did he say: if you wish to put these days into years, multiply them by 30 and divide by 10,631 so that the result is the number of com¬

10

plete Arab years which have gone by until the year you have in mind. As for the remainder, divide it by 30 so that the result is days, and put them into months, some of which are 30 days, and some 29 days, starting from al-Muharram. And when you reach the month you are in, the re¬ maining number of days are the days which have gone by of this month.

15

Answer: He said to multiply them by 30, and divide by [10,631].' Multiply these days by what is common to the fifth and the sixth, which is 30. Further, multiply the years of the Arabs, which have 354 days plus a fifth and a sixth, by what is common to a fifth and a sixth, namely 30. He considered the days of the years as a number which includes the fifth

20

and the sixth in it, and then he divided 10,631 into the product of the days times 30 which is he least common denominator of a fifth and a sixth. He told you to do this so that the result of the reckoning would be lunar years, and the remainder of the division would be a number having a fifth and a sixth. You divide this by 30, which is what is common

25

to a fifth and a sixth, so that the result is days, and he had them put into months. [Q15] Concerning the derivation of Arab months and years from the Persian, why did he say: if you wish to derive Arab years from Persian years, take the number of complete Persian years and multiply them

30

by 365, and then add the months and days which have gone by since Farwadin-meh. Add to this 3,624 so that the total is the number of days from the beginning of the Arab years until the day you have in mind.

7. Ms: 10,080.

22

TRANSLATION OF THE MICHAEL VERSION

[Answer: There were 3,624 days between the epoch of] 8 the Arab years and the epoch of the Persian years. He said to multiply the years by 365 to put them into days. (Al-Khwarizmi's) instructions allow us, then, to understand the tables for transforming a date from one calandar to another.

5

[Q16] Why did he put this table for the derivation of the beginnings of the months at the beginning of his book, rather than leave it for the end of his book where the derivation of the beginning of the months is presented in its complete form? Answer: This is a shortened version of the table. He has already in¬

10

formed us that lunar years are, in his opinion, 354 days plus a fifth and a sixth, and the lunar month is 29 days and a fraction. He wished to give an approximate value without departing from the truth. I will ex¬ plain the information he presents so that [these] 9 tables will not be difficult. He takes 30 days for some months and 29 for others, and casts

15

out sevens, so that two days remain for Muharram; this number is placed next to Safar, whereas zero is placed next to Muharram. A number is also put next to the beginning of every single year, i.e. days which are to be added to the radix. You cast out sevens from the number of days of Safar so that one remains which is then added to the number next to

20

Safar, and put the sum next to Rabh I and continue doing so until you finish with all the months. Cast out sevens from a year, i.e. 354 days plus a fifth and a sixth, so that four days remain plus a fifth and a sixth. So add four days to the radix and throw away the fifth and the sixth. This amounts to nine days for the second year. Cast out seven from them so

25

that two days remain and put this number next to the second year, and so on until you reach thirty years. He put nothing next to al-Muharram nor to the beginning of the year. He instructed you to consider the year which is not complete and the month which is not complete. He made this table in order to approximate this matter,10 but it is not sufficient

8. This phrase has been restored on the basis of the Latin version. There is no indication in the Hebrew manuscript that several words are missing. 9. Ms: a thousand. 10. First visibility of the new moon.

23

30

IBN AL-MUTHANNa's COMMENTARY

until he informs you of the numbers at the end of the book by which you can find the first day of the months in its complete form. Know the numbers which are placed next to the single years, of which there are thirty. He put the radix, which is five, next to the first year, and then 5

added 354 days plus a fifth and a sixth for each year in turn, casting out sevens from the sum, so that for the third year, for example, six and approximately three-quarters remain. He put them into whole days, of which there are seven, and put the seven next to the third row, and similarly until the end. The sum which must be put next to the

10

thirtieth year is 5 days and 38 minutes which makes six days; and simi¬ larly compute for the collected years, all this by way of approximation. Most of these numbers agree with the truth, but sometimes they do not agree, so trust only the explanation at the end of his book, where he calculated the appearance of the moon most accurately.

15

[Q17] Why did he use this era in the tables and put the days in ranks called first parts, second parts, and third parts? Answer: The era is in the radix. Days are counted from the creation of the world, or from the birth of someone, or from the reign of a king, or from some event on a certain day which men have fixed as the nota

20

from which they count, and they enter a new year when the days of the year have been completed. Before our era time had been reckoned in many ways. The Persians counted by the good kings who ruled them; the most famous example being ben Dara the Lesser.11 The Arabs counted from the building of the [Kacba] 12 and from the year of [the

25

Elephant].13 The Romans reckoned time in much the same way. The Hindus are said to have counted by the abundance of gold which they used to store in a mountain. But we should not believe all of their re¬ ports, and it is more fitting to count time some other way. These are the eras found in the tables of al-Khwarizmi: the Arab, the

30

Persian, and the Roman. He converted their years into days and put them in ranks on analogy with third parts, seconds, minutes, and degrees,

11. Era of Darius the Lesser. 12. Ms: Kafba. 13. Ms: all.

24

TRANSLATION OF THE MICHAEL VERSION

but in this case the unit of the first rank is one day. Whatever adds up to 60 days should be transferred as one to the second rank, where every unit is 60 days. When the second rank reaches 60, you transfer one to the third rank, where every unit is 3,600 days. Next to each month he put the number of days which have gone by from the first day of the

5

year until that month. Similarly the number of days since the beginning of the era is put next to the years. He indicated the total number of days for collected years, as well as the number of days for simple years, in order to make it easy for anyone who wishes to derive a date in one calendar from another. Therefore he said, concerning the derivation of

10

the number of days for eras: if you wish to put it into days, multiply the upper gate 14 by 60 and add the second gate to it which is written after it, and multiply the total by 60; and so on for the gate which is written after that one, if there are more gates. [Q18] What is the meaning of his remarks concerning the periods be¬

15

tween the eras: add or subtract the period from the era when one era divides another into two. Answer: They are the number of days between the years of Dhu’lQarnain and the years of the Arabs and the difference between the years of the Arabs until King Yazdegird, which he arranged in their

20

proper ranks. He further instructed us to subtract this when it is proper to subtract it, or to add it when it is proper to add it. [Q19] Why did he put this number of days next to the first month of the Roman and Persian years, but did not put it next to Muharram which is the first month of the Arab months? Answer: If he had done so, no sin would have come upon him. Indeed

25

he wished to explain the difference between the years of the Arabs and of the Romans and Persians. For this reason, he instructed that in de¬ riving the era of the Arabs from the tables, you take the days for the year which is not complete and the month which is not complete. The era of [the Sindhind] 15 which he mentions in the tables is like the eras which I have explained, and also has ranks of months, days, and years. 14. Gate: sexagesimal place. 15. Ms: al-Kalsa.

25

30

IBN AL-MUTHANNa’s COMMENTARY

II [Q20] What are the mean motions which he put in his table and what do they mean? Answer: The mean motions preserved in the tables of al-Khwarizmi, the men of the East and others, including the scholars of India, derive 5

from the Elindus, who believe that God created the seven stars, i.e. sun, moon, and the five planets, as well as their apogees and nodes, in the first minute of Aries, and then commanded them to move, and so they moved. Each one has a fixed motion until they all reach the place where they started simultaneously, and then God will judge and decide what

10

to do with them. The fixed number of days from the beginning of their motion to the time when they will reach the point where they began to move is i,577?91^’450’000 days, and this number of days was deter¬ mined and preserved by the Hindus. The number of rotations of every planet during these days, i.e. until their return to the point where God

15

created them, is fixed. When it was desired to know their mean motion without excess or deficiency, I mean their unequated motion, the num¬ ber of days was divided by the number of rotations of each planet, so that the amount of their motion per day resulted. Then they put these days into hours and divided them into their rotations so that the result

20

was the amount of their motion for an hour. These amounts indicate how much the planet moves in a day or an hour. This procedure yields their degrees, minutes, seconds, third parts, and up to tenth parts. They arranged them and put them in the tables, including only degrees,

25

minutes, seconds, and third parts; and they did the same for their apogees and nodes. When you wish to know the positions of the planets for the radix, note the number of days of the Hindus which have gone by until the day you wish, and multiply that number by the number of rotations of the planet whose position you wish to know. Divide the product by the

30

days °f the Hindus so that the number of rotations of the planets re¬ sults. Disregard the complete rotations that have gone by, because you have no need of them. But note the remainder that is less than a rota-

26

TRANSLATION OF THE MICHAEL VERSION

tion and multiply it by 12 and divide it by the same number as before so that the result is zodiacal signs, and keep them. As for what is less than a zodiacal sign, multiply it by 30 and divide by the same number as before so that the result is in degrees, and keep them. Multiply the remainder by 60 and divide it by the same number as before so that the

5

result is in minutes. Multiply this remainder by 60 and divide it by the same number as before so that the result is in seconds. Third parts, fourth parts, and so forth, to whatever precision you wish for the calcu¬ lation, are found in the same way that we derived the seconds. Count the result from the beginning of Aries so that the position which is

10

reached in the zodiac is the position of the planet in zodiacal signs, de¬ grees, minutes, and seconds, for the day you had in mind. This is the radix preserved in the tables. [Q21] Why did he not establish the motion of the apogee and the

15

nodes in the tables? Answer: The apogee and nodes of a planet deviate from their posi¬ tions by only a small amount which is barely perceptible even after many years. Thus other astronomers who composed tables, besides al-Khwarizmi, neglected them because these motions cannot be perceived even in many years; yet they are noticeable in longer periods, and can

20

be restored by dividing the total motion by the time that has elapsed. However, the apogee and the node of the moon move perceptably, and so astronomers establish their motions in tables. [Q22] What is the meaning of the radix which he put into his table for every planet? Answer: All of the astronomers composed tables for the mean motion

25

of these planets according to some epoch which has gone by, just as alKhwarizmi composed his tables according to the epoch of the Yazdegird era. In order to find the position of the planet for the day on which the era began, he divided the motions of the planets, as we have explained in the previous section. Their positions in the zodiac are given in zodiacal signs, degrees, and minutes, and put in the first row of each of his tables, which he called the radix. If the tables had been composed according to an era other than that of Yazdegird, he could have made

27

30

IBN AL-MUTHANNA’s COMMENTARY

a calculation similar to the one for which he gave us instructions. For example, the radix of the sun in his tables is Gemini 26514,35,34° 16 on the first day of the Yazdegird era. Further, he added tables of the mean motions for days, months, and years. 5

[Q23] What is the apogee and what does it mean? Answer: It is the position of the planet where no correction to the mean motion is needed; i.e. the planet’s mean motion is the same as its corrected motion. Every one of the five planets has fixed spheres, ac¬ cording to the opinion of the Hindus, among them a sphere like the

10

sphere of the zodiac which rotates about the center of the Earth and another sphere called the sphere of apogee whose center is removed from the center of the earth. For the sun and moon the correction de¬ pends on the distance between the center of the earth and the center of the sphere of apogee. For the other planets, this distance determines the

15

correction of center. We will illustrate this with a figure [Fig. 1]: the A

Fig. 1

16. Latin: Gemini 26;!^,340.

28

TRANSLATION OF THE MICHAEL VERSION

circle of the sphere of the zodiac is represented by circle ABGD, with center E, the center of the earth. The diameters, which intersect at E, are AG and BD. Then we take an amount on line EA corresponding to the distance between the center of the earth and the center of the sphere of apogee for every planet. We put Z at this point and construct

5

circle HTKL with radius ZA whose radius is less than the radius of the sphere of the zodiac by the amount between their centers. Point H coincides with point A in the figure. The sun goes around on the circle of apogee, starting from point A. When it has gone 90° in the sphere of the zodiac, the lines from the two centers to the center of the sun

10

intersect the sphere of the zodiac at different points. When it is at points H or K, the lines from the two centers coincide. Point A is far¬ ther from the earth than point K, and at both these points the sun needs no correction. Point A, the farthest place from the earth is called the apogee, whereas point K, the closest place to the earth, is called the

15

perigee, or in Arabic al-hadid. These two words are both Persian.17 [Q24] What are the nodes and what do they mean? Answer: Node is also a Persian word in Arabic, namely jawzahar, and we will explain the meaning of this term. Each of the six planets has an inclined sphere whose radius is the same as the parecliptic except that

20

it inclines from it. The inclination of each of these spheres is the amount of greatest latitude of that planet, except the sun, which has no such sphere and must remain in the zodiac, never having any latitude. These two different spheres intersect in two opposite places, each inter¬ section being called a node, or the head or tail of the dragon; it is also

25

called the knot. When a planet is at one of these intersections, it has no inclination from the sphere of the zodiac; i.e. it has no latitude. But when the planet reaches the northernmost or southernmost point of its inclined sphere, its latitude equals the amount of inclination of this sphere to the sphere of the zodiac. For each of the five planets, the center of the epicycle is attached to the inclined sphere, and this is its diagram [Fig. 2].

17. Perigee and apogee.

29

30

IBN AL-MUTHANNA.’s COMMENTARY

[Q25] What is the meaning of the corrections which he mentioned? Answer: The correction is the difference between the position of mean motion and the position of true motion. The sphere of the zodiac, which is the sphere of the fixed stars, is the largest and highest one.

5

The sphere of each planet lies beneath this sphere, so that in our view the point on the zodiac corresponding to the position of these planets varies according to the anomaly in its own sphere and according to the amount of its motion. All this refers to our line of sight to the sphere. If the point of the ray which proceeds from our sight to the

10

zodiac is behind the mean position of the planet, subtract the correction from the mean position; but if it surpasses it, add the correction to the mean position. [Q26] What is the correction, in (al-Khwarizmi’s) opinion and those who incline to his opinion?

15

Answer: For the correction of the sun and the moon take half the

3°

TRANSLATION OF THE MICHAEL VERSION

distance between the center of the sphere of the zodiac and the sphere of apogee and take its arc. As for the five planets, the correction of their centers is also half the distance between the centers of the sphere of the zodiac and the sphere of apogee, considered as a sine and taking its arc; so the arc is the correction of the center. The correction for anomaly is

5

the radius of the epicycle compared to the sphere of the zodiac. Let us illustrate this with a figure so you may see it clearly [Fig. 3]. For the sun

and moon we rotate a circle to represent the sphere of the zodiac, circle ABGD with center E which is the center of the earth. We construct two diameters intersecting at E, AG, and BD. On line AE we take an

10

amount corresponding to the distance between the centers, in the figure line EZ. We consider point Z as another center and rotate circle HTKL with radius ZA so that H and A coincide. This circle is called the sphere of apogee, and H is the apogee of the sun. The mean motion of the planet proceeds in the zodiac. If it is in the quadratures of the sphere of apogee,

31

15

IBN AL-MUTHANNA’s COMMENTARY

the ray from the center of the earth to the center of the sun differs from the ray from the center of the sphere of apogee to the center of the sun. Both rays go through the sun, but they meet the sphere of the zodiac at different points, in this figure EM 18 and ZN. The two points, M and 5

N, differ for the sun, and arc MN is the correction for the moon as well, except for the fact that the distance between the centers differs for the two cases, as is seen in the figure. The figure for the five planets is different because their spheres of epicycle are attached to the inclined sphere which inclines to the sphere

10

of apogee. The center of the inclined sphere is the same as the center of the sphere of apogee, but it inclines from it by an amount of the [maximum latitude] 19 of every planet, and the sphere of epicycle is attached to it. We will illustrate this with a figure [Fig. 4] which will show you the correction for the five planets. We rotate a circle, repre¬

15

senting the sphere of the zodiac, ABGD with center E, which is the center of the earth. Then we construct the two diameters, which inter¬ sect at E. We take an amount on line EA for the distance between the centers for every planet, in our diagram, EZ. We consider point Z as center and rotate circle HTK with radius ZA so that point A coincides

20

with point H. This circle is called the sphere of apogee, and point H is the apogee of the planet. Then we consider T as center and rotate a circle with radius TL to represent the sphere of epicycle, circle LMN. We consider point E as center again and rotate a circle to represent the sphere of the sun, and we construct lines [EO, ZS, EF].20 We assume

25

that the planet lies in the same direction as the sun, degree for degree, and then let their spheres move, the sun in its sphere, and the planet in its sphere of epicycle. The planet moves to point N and the sun to point [Q] 21 and, as we mentioned, [OF] 22 is the correction for anomaly. Arc SO is the correction of center, and half of arc OF is half the correction

30

which al-Khwarizmi instructed us to subtract from the apogee of the

18. Ms: ES. 19. Ms: distance. 20. Ms: ES, EO, ZO, ZF. 21. Ms: T. 22. Ms: LN.

32

TRANSLATION OF THE MICHAEL VERSION

planet or to add to it. In the figure, we let line [ED] 23 pass through the first minute of Aries. Thus the mean longitude of the planet will be approximately arc TD, and [DA] 24 of the apogee and arc [QD] 25 the mean longitude of the sun. All these circles are divided into 360 parts. Outer

Planet

When we subtract the mean motion of the planet from the mean motion of the sun, there remains arc [OQ],26 which is less than six signs. We take half the correction of the result and it [corresponds] 27

23. Ms: EBD. 24. Ms: GA. 25. Ms: QC. 26. Ms: OE. 27. Ms: is equal to.

33

5

IBN AL-MUTHANNa’s COMMENTARY

to MN on the parecliptic, or on the sphere of the zodiac it is equal to [CA].28 We subtract arc CA from the apogee so that the remainder is arc [CD].29 Then we subtract this from arc [TD]30 so there remains arc [TC] 31 which is the centrum,32 [TC] 33 is less than six signs. We 5

subtract the correction of center from arc [CO],34 and we add this to the anomaly which is arc [OO],35 so that the anomaly is arc [OQ plus OS] 36 Find the entire correction for arc LN on the epicycle, using the tables of the correction for anomaly, and this amount is arc [FO] 37 on the great sphere. We put the planet at point [F] 38 on the sphere of the

10

zodiac so that the line which proceeds from the center of the earth will pass through the planet. This is what we wished and this is its diagram. [Q27] Why were the maximum corrections for the sun, the moon, and the five planets put opposite different degrees, all greater than go, and the maximum correction of some put opposite degrees less than [120] 39

15

and some opposite degrees more than [120]? Answer: The maximum correction of the mean motion of the planet is in this case, as we explained, a section of the sphere of the zodiac compared with the [radius]40 of the sphere of epicycle. You cannot know the exact size of this arc except by finding the angle which it forms

20

at the center. This angle reaches its maximum when the planet lies at a point on its epicycle such that a ray from the center of the earth to the planet is tangent to the epicycle. Euclid informed us in the first book of his work that the largest angle is opposite the largest side. Since this

28. Ms: DA. 29. Ms: GE.

3°. 3132333435-

Ms: TK. Ms: ADB. The angle from the planet to its apogee. Ms: GK. Ms: ZO. Ms:

36. Ms: 37- Ms: 38. Ms:

oc. sc. so. 0.

39- Ms: 180; Se 19T3 reads 125. 4°. Ms: diameter.

34

TRANSLATION OF THE MICHAEL VERSION

is so, it is proper that this arc should be the greatest of all the arcs of correction. The line that proceeds from the center is tangent only to the circle on which the center of the planet travels, when the planet has already moved more than a quarter or less than three quarters on its epicycle from the time when it separated from the sun.

5

The two tangents to the planet on the epicycle divide the epicycle into two unequal halves. The half facing the center of the earth is smaller than the outer half. At that time the distance of the sun from the planet will be more than 90° and less than 270°. The outer planets come into conjunction with the sun at the apogee of the circle of the

10

sphere of epicycle, and they separate from it. The inner planets also have conjunctions with the sun in the revolutions of their spheres, and their distance or separation from the sun varies according to the revolu¬ tions of their spheres until the time of correction. It is necessary to know the distance of the sun from the planets or the distance of the

15

planets from the sun for the points on the spheres of epicycle. He put the maximum correction opposite that distance, which for each planet depends on the amount of its motion in its sphere of epicycle and the size of its sphere of epicycle. We will show you a diagram [Fig. 5] so that you will comprehend this matter. Construct circle ABGD to

20

represent the sphere of the zodiac, with center E the center of the earth. We draw two diameters which intersect at E, and we take an amount, ZE, from EA for the difference between the two centers. Considering Z as center, we construct a circle, HT, with radius ZA. We consider H as center and construct the circle of epicycle, KLM,41 with radius HK.

25

Then about center E we draw the circle of the sun, NOS. We draw lines EN, ES, EO, which pass through the center of the planet, reaching a point on the sphere of the zodiac.42 In our example we first put the planet on point K and the sun at S so that the planet is in the circle of the sphere of epicycle, and they both move in their spheres. The sun moves in its sphere to point B, which is more than a quarter of its sphere from the time of conjunction with the planet, while the planet

41. Ms: kulam. 42. Ms: point O.

35

30

IBN AL-MUTHANNa’s COMMENTARY

moves to point L in its sphere of epicycle. At that moment the planet is on line EO which proceeds from the center of the earth and which is tangent to the circle of the sphere of epicycle. The planet moves until it seems to be at point O, as we explained, and Outer Planet

5

SO is the total correction. When the planet is at some other point than L on arc LQ, the line that proceeds from the center to it will seem to be at some point on the sphere of the zodiac other than O toward point S. There is no doubt about this, because it is the [maximum] 43

43. Ms: radix.

36

TRANSLATION OF THE MICHAEL VERSION

correction. Similarly, when the planet is at point M, which is less than 270°, it is on one of the lines which proceed from the center of the earth to the center of the planet. When it moves on arc [MK],44 it will fall on arc NS from point N toward point S. As Euclid informed us in his

Optics: when the eye looks at a circle or a cylinder from a

point which is far from the observed object, it will see less than half of it. To prove this, we draw [see Fig. 6] two diameters of the circle

of the sphere of epicycle which intersect at point H at right angles, AB and

QS, which is the sine of 30° or the sine of Aries.45 We also draw

44. Ms: QSB. 45. Cf. Se 20r (right margin), LM is the sine of 30°; P 7r: 13, we take a chord of 30°.

37

5

IBN AL-MUTHANNA’s COMMENTARY

ML. In every circle, the longest line is one that passes through the center of the circle. Only arc LQM, which is smaller than a semi¬ circle, is visible, because, if we consider point E as the position of the

line

eye, the ray that goes out from our eye is tangent to the sphere of

5

epicycle at two points, M and L, which define an arc of the sphere of epicycle less than a semicircle, as we sought to explain. [Q28]

Why did he say for the correction of the outer planets: we should

subtract the mean motion of the planet from the mean motion of the sun so that the difference would be the anomaly, but for the two inner

10

planets he instructed us to subtract the mean motion of the sun from their mean motion.

Answer: The mean motion of the outer planets is less than the mean motion of the sun. Their mean motion is the motion of the center of their epicycle, while their motion in the sphere of epicycle de¬

15

termines the amount one adds to or subtracts from the mean motion. Since the motion of the sun is greater than the mean motion of the planets, the sun separates from the planets until it is in conjunc¬ tion with them again, and the planets have different phases in the sphere of epicycle according to their distance from the sun. It is neces¬

20

sary to subtract the mean motion of the planet from the mean motion of the sun in order to know the distance between them. One adds where it is proper to add and subtracts where it is proper to subtract. As for the inner planets, the center of their sphere of epicycle moves with the mean motion of the sun, although their mean motion is

25

greater than that of the sun. One should compute their mean motion from the motion of the centers of the spheres of the planets. Since the mean motion of the sun is less than that of the inner planets, he instructed us to subtract the mean motion of the sun from that of the planet, so that the mean position of the sun becomes the center of its

30

epicycle, and we compute from it so that the computation is from the center. [Q29]

Why did he say for the correction of the inner 46 planets: if the

anomaly is less than six signs, subtract half the correction from the apogee so that the remainder is the corrected apogee. Furthermore, sub46. Ms: outer planets.

38

TRANSLATION OF THE MICHAEL VERSION

tract the corrected apogee from the mean motion of the sun so that the remainder is the centrum. And he instructed us to note: if the anomaly is greater than six signs, add half the correction to the apogee so that it becomes the corrected apogee. Furthermore, he said: if the centrum is less than six signs, subtract the correction of center from the centrum

5

and add it to the mean motion. Then enter the table of correction with this mean motion as argument. If it is less than six signs, add the total correction to the corrected apogee, but if the anomaly is greater than six signs, subtract it. After the addition to the apogee, you add or sub¬ tract the centrum.

10

Answer: Know that in his opinion, and that of all those who followed the procedure of the Hindus, Persians, and the men of the East, the correction is made up of two corrections, including that for the anomaly. When the planet has moved a certain amount in its epicycle from the time of its separation from the sun, one takes half of the amount given

15

in the table, which is the necessary correction for the position of the planet, because it is half the arc of the sine of the distance on the circle of the sphere of epicycle. One always adds or subtracts this. He instructed us to subtract it from the difference between the mean motion of outer planets and the mean motion of the sun or, in the case of

20

inner planets, from the mean motion of the planet. The reason is that it was necessary to determine the distance between the two lines, i.e. the line that proceeds from the center of the sphere of apogee passing through the center of the planet and the line through the center of the sphere of epicycle; this distance is the correction for

25

anomaly. When you have added or subtracted it from the apogee, subtract this from their mean motion so that what remains on the spheres of the planets is the distance from their apogee to the line that proceeds from the center of the sphere of apogee to the center of the planet. When you have done this, you have corrected for anomaly,

30

just as it was previously necessary to correct the center. The corrected motion of the planet results after adding or subtracting this correction of center which depends on the difference between the two centers, i.e. the center of the earth and the center of the sphere of apogee, as we have explained. This is the corrected motion in the tables for every planet. He instructed us to subtract the correction from the centrum and to

39

35

IBN AL'MUTHANNA’s COMMENTARY

add it to the anomaly if the centrum is less than six signs but to add it to the centrum and to subtract it from the anomaly if the centrum is greater than six signs. The reason for this is that when the distance of the center of the sphere of epicycle from the corrected apogee is less 5

than six signs, the line that proceeds from the center of the sphere of apogee through the center of the sphere of epicycle will precede the line that goes out from the center of the earth through the center of the sphere of epicycle. So it is necessary to subtract the amount which is between them in order to restore the planet to the point on the great

10

sphere where the line that proceeds from the center of the earth meets the great sphere. When this distance is greater than six signs, the line going out from the center of the earth through the center of the sphere of epicycle precedes the line going out from the center of the sphere of apogee through the center of the sphere of epicycle. Therefore, it is

15

necessary to add this amount to the centrum to restore it to the point on the great sphere where the line that goes out from the center of the earth meets the great sphere. He mentioned that when one subtracts this correction from the centrum, it is added to the anomaly, and that when one adds it to the centrum, one subtracts it from the anomaly. The

20

reason is that he has corrected the anomaly by lines that go out from the center of the sphere of apogee through the center of the planet and has restored it by this increment or diminution to the line that goes out from the center of the earth through the center of the planet. So when this is done, he has corrected both the anomaly and the centrum.

25

Then it is necessary to add the corrected centrum or subtract from it the amount of the arc of the sine of the planet’s distance in its sphere of epi¬ cycle from its apogee. This is the increment or diminution in longitude: diminution when the second anomaly is greater than six signs, and incre¬ ment when the anomaly is less than six signs. When the second anomaly

30

is [less]47 than six signs, the planet is in one half of the circle of the sphere of epicycle. But when the anomaly is greater than six signs, the planet is in the other half of its sphere of epicycle. For this we will con¬ struct a diagram [Fig. 7]. We fix the spheres and write on that of the epi-

47. Ms: greater.

40

TRANSLATION OF THE MICHAEL VERSION

cycle

ABGD. Put the planet on point A and the sun in its sphere on point

T so that the planet has the same degree as the sun, and there is only one correction—the correction of center which is arc [TK],48 which is added or subtracted. The planet separates from the sun, and they

both move in their spheres until the planet reaches point

B at the time

5

of its correction, and the sun reaches point L. Here both corrections must be used. The correction of center is arc

TK, and the correction for the position

of the planet in its sphere of epicycle is arc SO. These two corrections suffice. Then it is necessary to add to, or subtract from, the center of the

48. Ms: MB.

41

10

IBN AL-MUTHANNA’s COMMENTARY

sphere of epicycle the arc of the sine corresponding to the amount the planet has moved in its sphere of epicycle. This equals his calculation for longitude. He instructed us to add to, or subtract from, the center of the epicycle an amount for the arc on the sphere of epicycle—arc 5

AB in this figure. It is necessary that the distance of the planet from the apogee of its epicycle approximate the distance of the sun from the center of the epicycle at the time of correction. So, as we mentioned, it is necessary to add the correction which is the arc of the sine corre¬ sponding to the amount the planet has moved in its sphere of epicycle

10

on one half of its epicycle, namely arc

AH. But one subtracts the

correction when it is on the other half of its epicycle, namely arc HA. And this [Fig. [Q30]

7]

is its diagram.

What is the section called in Persian kardaja which he put in

his tables, as distinct from the preceding tables of correction?

Answer: We have already explained the nature of the correction in

15

this booh. But we will also say that the hardaja is a Persian word meaning a section. One wished to divide the circle into sections in order to make known the correction for each section, as well as for each sign. First the correction for each sign was indicated, and then each sign was

20

divided into sections. For the sun and the moon, it sufficed to divide each sign into two sections, called kardajas. Thus, in the case of the sun and the moon, there are six divisions for every three signs, because their maximum correction occurs when their distance from their apogee is

25

90°.

As for the five planets, the signs were divided into sections which

are fixed in the tables. Thus, when Mars, Venus, or Mercury is in the upper half of its epicycle, there are four sections in each sign so that each section is

but in the lower half of the epicycle each planet

has eight sections, each one being

3;q5°.

But for Saturn and Jupiter

each sign is divided into four sections, so called, in both halves of the 30

epicycle. The maximum correction for each planet is placed next to the degree of anomaly, which is the point on the epicycle where the line, proceeding from the center of the circle on which the epicycle goes around, is tangent to the epicycle, as we explained previously. This is placed at the end of the table of the first correction for each planet.

35

In addition to the simple correction in his tables next to the sections,

42

TRANSLATION OF THE MICHAEL VERSION

he wished to simplify this procedure by indicating the correction next to every degree. Because of the rapidity of the moon's motion, he divided each section into four sections of

3;45°

and noted the corresponding

correction. Then he divided this correction by these degrees in order to find the amount for i°. He composed this table degree by degree until

5

each correction was completed up to six signs. Al-Khwarizmi divided the sun s degrees into sections as previously indicated, following the proce¬ dure of the scholars of India. Their value for the correction of the sun differs from what al-Khwarizmi put into his tables next to the sections of the Hindus, although the difference between them is small. Then for

10

the sun he divided each section into four sections and proceeded as he did for the moon. As for the five planets, he noted the degrees of correction for each section, and then divided the correction according to the degrees of the section. He placed the result, in minutes, next to i° and composed the table at degree intervals. Take the case of Mercury, which has When this is divided by sections, the result is

16

7V2,

120

15

minutes for its first section.

which is the number of degrees in its

minutes, so he put this number next to i° as

its correction. Then he composed the tables from this until he reached its maximum correction, which is next to three signs and

22V20.

After

20

reaching this maximum, he subtracted the result of the division from it until he reached six signs, where no correction remains. [Q31] Why did he say for the correction of the sun and the moon: if this anomaly is up to three signs, use it, but if it is greater, subtract it from six signs and use the remainder. But for Venus, if the anomaly is up to four signs

150,

use it, and if it is greater than four signs

25

150,

subtract it from six signs and use the remainder. In the case of Saturn and Jupiter: if the anomaly is up to three signs and

7V20,

use it, but if

it is greater than this, subtract it from six signs and use the remainder. And he said for Mars: if the anomaly is up to four signs 7V20, use it, but

30

if it is greater, subtract it from six signs and use the remainder. Answer: We have already explained that the maximum correction occurs when the planet’s position on its epicycle is on the line from the center, tangent to the epicycle through the center of the planet, on either side of the epicycle. These two lines divide the epicycle into two

43

35

IBN AL-MUTHANNA’S COMMENTARY

unequal parts, the upper half being greater than the lower half. We also explained that the distance of the planet on its epicycle is approxi¬ mately equal to the distance of the sun from it. These are the signs and degrees which we recorded for every planet, and they are the degrees

5

corresponding to the maximum correction. The amount of half of the upper half of the epicycle is the amount of the anomaly which he re¬ corded for each planet, and this is equal to the distance of the sun from the planet [at the time of maximum]49 correction, and all circles are divided into 360 parts whether they are large or small.

10

[Q32] What are those numbers that he recorded for deriving these cor¬ rections when one uses the second and fourth columns of the tables of subdivision for each planet? This is the number from which he in¬ structed us to subtract the result of the division for every correction. Thus he told us to subtract the remainder from the apogee or to add to

15

it, so that the result is the (corrected) apogee, which was recorded. Answer: We have already explained in this book of ours that each planet has a maximum correction up to which the correction of the planet increases; then it begins diminishing until it reaches the number corresponding to the correction for six signs; i.e. when there is no

20

correction. The maximum correction of Venus is 2,831 minutes, Mercury 1,290 minutes, Mars 2,431 minutes, Jupiter 652 minutes, Saturn 340 minutes. These numbers, as we explained, are the maximum corrections, and they are the distance between the planet and the sun when the planet is on the tangent to the sphere of epicycle. The minutes resulting

25

from the division are subtracted from it, and so the tables are composed. Therefore, we are instructed to subtract the resulting correction from these numbers because the correction always increases until it reaches its maximum, whereupon it begins to decrease. [Q33] What is the meaning of the 900 which he instructed us to use

30

in deriving the correction for the sections of the sun and the moon; that is, the number by which the total correction is divided? Answer: The 900 is one section, or 150 changed to minutes. The

49. Ms: when its correction is next to half the apogee.

44

TRANSLATION OF THE MICHAEL VERSION

ratio of each section to its correction is equal to the ratio of the degrees, which do not complete a section, to their correction. These are four numbers, which form a proportion. The first is the section changed to 900 minutes. The second is their correction which al-Khwarizmi called the diminished section. The third is the degrees that do not complete a

5

section, which he instructed you to change into minutes. The fourth is the correction that corresponds to this number of degrees, and it is the unknown that we seek in this operation. For any four numbers forming a proportion, the product of the first and the fourth divided by the second is the third; or the product of the second and the third divided

10

by the first is the fourth. So when the fourth is sought, he instructed you to multiply the second by the third and divide the product by the first to find the fourth, which is unknown. Therefore, he directed you to multiply the minutes of the section which is not complete by the total correction and then divide the product by 900, which is one section.

15

Similarly, for all five planets, we are instructed to divide by 900, since it is one section for each of these planets changed to minutes, and the procedure to form the ratio is the same. [Q.34J Among the tables of corrections, what is the table that is called the first station for the planets? He said: when you wish to know

20

whether the planet is in direct or retrograde motion, note the equated centrum and take the signs and degrees opposite it in the column of first station, the result is the first station; subtract it from twelve signs so that the remainder is the second station. Then note the second anomaly: if it is greater than the first station and less than the second station, the

25

planet is retrograde, but if the opposite is true, it is moving forward. Answer: We have already explained what the anomaly means. For the outer planets it is the mean motion of the sun less the mean motion of the planet. But for the inner planets, the mean motion of the sun is subtracted from the mean motion of the planet, and the difference is

30

the distance in longitude between the sun and the planet. We also explained that the planets go around in their spheres of epicycle in the direction of the signs. When the planet moves in the upper half of its epicycle, its motion appears to be toward the sun, in the direction of the signs. But when it is in the lower half, its motion appears to be

45

35

IBN AL-MUTHANNa’s COMMENTARY

opposite the order of signs. Therefore, when the distance between the planet and the sun is the amount of the first station, i.e. when the planet has completed a quarter of its epicycle from the time it separated from the sun, the planet appears to be stationary, and its motion is

5

similar to motion on a straight line. So one says that it is stationary, because the lines from the center of the earth to the sphere of the zodiac through the planet stay close to each other on the sphere of the zodiac. When the planet moves in its path to the lower half of its epicycle, its motion is retrograde, i.e. opposite the direction of the signs

10

so that it appears to be retrograde. When it completes its path in the lower half of its epicycle, it seems to stop on one line, so one says that it is stationary. When it is again in this situation, it is at that distance from the sun, i.e. the second station. These two stations are what alKhwarizmi recorded in his tables.

15

Some men who have no knowledge of the science of the spheres and geometry think these planets are bound to the sun with bonds as animals are bound, and that when the sun is far from them or the planets are far from it, the sun pulls them to it by the bonds. This statement is certainly wrong, and shows no knowledge of astronomy. When the

20

planets are in these phases, their locations in the spheres, as we have mentioned, are fixed in corrected degrees. Likewise, at the same time the corrected degrees of the sun are fixed, and so the distance between them is fixed. Since the phases change from situation to situation, it was necessary for the masters of the laws of astronomy to determine the

25

stations of the planet for direct motion or retrogradation. Moreover, it was necessary to know the distance between them in corrected degrees so that their laws about retrogradation would refer to the amounts of the characteristic parameters of the planet. The laws are essential be¬ cause the planets advance only by them, and observation confirms these

30

laws. The makers of these rules had no knowledge of the wisdom of India, but the Hindus informed them that when the distance between the planet and the sun in corrected degrees is such and such, the planet is about to go forward or backward, and (the Islamic astronomers) accepted this rule and called it a bond. When days and years had gone

35

by and they died, a new generation arose after them who did not under¬ stand this matter, nor could they find anyone to explain it to them. So

46

TRANSLATION OF THE MICHAEL VERSION

they thought that the planets were bound to the sun; but if this were so, the planets would neither increase nor diminish from these de¬ grees. We have observed these matters with instruments, and we found that the degrees increase and decrease according to the correction of the sun and the correction of center of the planets. Sometimes they are

5

both increasing, and sometimes they are both decreasing. But when one of the corrections for a bond is a diminution and the other is an incre¬ ment, the effect is so small that al-Khwarizmi did not mention it.

Outer Planet F

Fig. 8 When the second anomaly is equal to the first station, the planet is ready to begin retrograde motion. But when it is equal to the second station, the planet is ready to resume its direct motion. Let us construct a figure for this [Fig. 8]. We draw the circle of the zodiac, upon which

47

10

IBN AL-MUTHANNANS COMMENTARY

are placed ABGD, with center E, the center of the earth. We then draw a circle about center Z for the circle of apogee upon which are HTK. We draw about center H a circle of the epicycle, upon which are LMNS, and we draw about E, as center, the sphere of the sun upon 5

which are FOC. We first put the planet on point L where it has the same degree in its epicycle as the sun, degree for degree. The sun is then at point F on its sphere. The sun moves on its sphere, and the planet moves on its epicycle in the order of the signs, both of them in the same direction. The planet moves in its sphere of epicycle to point M

10

while the sun moves to point O of its sphere, and then the distance between them, i.e. the second anomaly, is equal to the first station that al-Khwarizmi recorded. When the planet begins to move from point M to point N, we see it from

the earth as if it were descending on a

straight line, and one says that it is stationary. When it has descended 15

moving to point N, one sees that its motion is different from its previous motion. In the upper part of its epicycle both the sun and the planet move in the direction of the signs. These phenomena appear to vary on account of the small size of the sphere of epicycle and the fact that it goes around the earth. When the planet moves from point N

20

toward point S, it seems to be in retrograde motion. When it ap¬ proaches point S, it appears to be stationary, moving on a straight line, so one says that it is about to move forward, and this point is called the second station. The distance between the sun and the planets at that time is the amount of the second station that al-Khwarizmi re¬

25

corded. When the planet reaches point S, it moves toward point L in the direction of its mean motion in the sphere of the zodiac. The argu¬ ment of anomaly for its phases in its epicycle is the same as its distance from the sun. The planets do not rest for a moment; and this is its diagram [Fig. 8],

30

To find the time corresponding to the motion which al-Khwarizmi mentioned, he instructed us to divide the degrees between the station and the second anomaly by the mean motion in anomaly in order to learn the direct and retrograde motion. He had already explained for the outer planets that the daily mean motion in anomaly is the daily mean

35

motion of the planet subtracted from that of the sun; whereas in the case of the inner planets, the daily mean motion of the sun is subtracted

48

TRANSLATION OF THE MICHAEL VERSION

from that of the planet. He gave us these instructions so that the dividend would be an amount combining the distances of the planets and the sun; and the distance between the planets and the sun is the second anomaly. He wanted the divisor to be an amount proportional to its motion so that there would be no error in his calculation.

5

He said that he considered noon Tuesday, which is the first day of the reign of Yazdegird, as the radix for the planetary tables, because he made it the epoch for his tables, as has already been explained. He said that the calculation and observation of the men of India is beyond doubt, and if he had said other than this, we would not have accepted

10

it. By saying that their place of observation is at the dome of the earth, they meant that their place of observation has no longitude; i.e. it is 90°

from both east and west. He meant by dome that it is the middle

of the earth, which is inhabited both to the east and west. This place is used because the observations of the Hindus were made there. It was

15

told to me on the authority of al-Farghani that the place of observation is called Tanah and also Arin.50

Ill

[Q35] What are the two tables he presented for the subdivisions of sine and declination, and what do these terms mean? Why did he also in¬ struct us how to derive the sine as if these two tables were not sufficient?

20

Answer: We have already explained the sine and the declination according to the method of Ptolemy. I mean that the diameter of a circle is

120

in geometric proofs, as it is in our book to Muhammad b.

c Abbas the computer. We will now establish the sine and declination according to the method of the Plindus. These two tables are really

25

six tables for the six subdivisions up to the maximum of the sine. These refer to the six sections of the circle. You already know what a section is, and why he put the maximum sine opposite the sixth section. You know that the sine is the perpendicular to the diameter which proceeds from the angle whose sine you wish to know. The perpendicular meets

50. Ms: ^urn.

49

30

IBN AL-MUTHANNA’s COMMENTARY

the diameter at right angles. The length of the perpendiculars from the circumference of the circle to the diameter is less than the length from the quadrant points of the circle. Since the perpendiculars from the quadrant points fall on the center of the circle, these are the farthest

5

points on the circle. We have explained that every circle is divided into 360 parts, so that a quarter of the circle is 90. (Al-Khwarizmi) uses intervals of 150 as he did for the correction of the sun. The maximum sine is opposite six sections because the sine reaches its maximum

10

there. We will draw a figure for this. Let us construct [Fig. 9] circle ABGD, A

with center E, and we construct the diameter DB through the center of the circle. The diameter of a circle divides it into two equal halves. We have already explained that circles are divided into 360° so that the diameter divides the circle into two equal arcs of 180°. We also

15

divide arc DB into two parts at point A so that both arc DA and AB equal 90° or a quarter of the circle. We wish to know the sine of arc DA so we drop a perpendicular from A to diameter DB, i.e. perpendic-

5°

TRANSLATION OF THE MICHAEL VERSION

ular AE. This arc is 90°, and no perpendicular to diameter

DB is longer

than perpendicular AE, which is the maximum sine, the sine of 90°.

E is the center of the circle, AE is equal to EB, both of which are radii of the circle. Since diameter DB is the chord of arc DB, the sine of Since

90 is half the chord of 180, and similarly the sine of any angle is equal

5

to half the chord of the double angle. When we wish to know the sine of 30, we divide arc AD into three parts so that each arc is 30°, namely arcs DZ, ZH, HA. Then we drop perpendicular ZK from Z so that ZK is the sine of 30°. When we wish to know the sine of 60, we drop perpendicular

HT from point H, and HT is the sine of 6o°.

10

We wish to know the amount of the sines and their ratio to the diameter, and we derive the rest of the sines from the sines of the six sections. We have explained that the sine of any arc is equal to half the chord of the double angle, and we wish to know the amount of the sine of 30°, namely line ZK. We extend the perpendicular ZK to point E on arc GD. Since lines AE and ZK are perpendicular to line

15

DB,

they are parallel. The section of arc DG which was cut off by the ex¬ tension of ZK is equal to arc DZ, and it is called DL or 30°. Line ZL is the chord of 60, and line ZK is equal to line KL. Hence both are half of the chord of 6o°, which is what we wish to explain.

20

Since we have presented the proofs, we will now explain how it is possible to find the amounts of these lines; why they considered the total sine 150 minutes, and why they do not use the entire amount of the diameter. I say that there is no proportion at all between arcs and chords. The amount taken for the diameter does not matter to the

25

masters of calculation. The Hindus considered the diameter as 300 parts; whereas others, following Ptolemy, consider the diameter as 120 parts; still others may consider it as more or less than this amount. It does not affect the sine, because the chords are derived from the diameter and have their ratio to it. In computing sines to given arcs,

30

the amount of the diameter does not matter, since the chords are de¬ rived from the diameter and have a fixed ratio to it. In converting sines back to arcs, it also does not matter what value was taken for the diame¬ ter, and the Hindus decided to consider the diameter as 300 parts. This number is divisible by the parts of the circle so that it is easier for them to derive the proportions, most of which are exact. But the men of the

51

35

IBN AL-MUTHANNa’s COMMENTARY

West and Ptolemy considered the diameter as 120 parts, and this number is divisible by the parts of the circle. We have already mentioned in our treatise that the radius of every circle is equal to the chord of a sixth of its circumference. We explained

5

this by a proof in our book to Muhammad b. cAbbas. We further say, in order to establish what we have said, that when two equal circles are so placed that the distance between their centers is equal to their radius, the intersected arc is equal to a third of either circle. To prove B

Fig. 10

this, we consider line AE as the common radius. We construct a circle

10

about E with radius AE, namely ABGD [Fig. 10]. We then divide the (circumference of) the circle into twelve equal parts—AZ, ZH, HB,

TK, KG, GL, LM, MD, DN, NS, SA. Then we consider point A as center, and we draw a circle again with radius EA, namely circle HEN. Al-Khwarizmi explained that the sum of arcs HE and EN is equal to

BT,

15

arc NAH because the distance between their centers is the same line, and it is fitting that each circle should intercept equal arcs on the other, namely arc

NAH which is a third of circle ABGD and arc HEN which

is a third of its circle. I also say that the radius of each of them is equal to the chord of a sixth of its circumference. To prove this we construct

52

TRANSLATION OF THE MICHAEL VERSION

lines

EH, EN, NA, AH. Lines AH and AN are equal to lines EN and

EH, because all of them are equal to EA. Since all of them are equal to the common radius of the two circles, they are equal to each other. Each of these lines cuts off two parts of the circle, and two parts of the circle are equal to a sixth of the circle; thus the chord of a sixth of a circle is equal to its radius.

5

It is clear that the sine of any angle is equal to half the chord of its double angle. The sine of 30° is a quarter of the diameter of the circle A

because it is half the chord of 6o°. Also, the total sine is known to be half the diameter. The sine of 30 0 is known, namely a quarter of the di-

10

ameter. We wish to know the complement of the sines for the end points of the sections. We already know that in any right triangle, the hypote¬ nuse is greater than either leg. We construct circle ABGD and draw two diameters which intersect at point E [Fig. 11]. We divide arc AD into three equal parts so that each of them is 30°, arcs DZ, ZH, HA. We draw lines EZ, ZT, ZK so that we have rectangle there are two right triangles, ZET and

15

TK. In this rectangle

ZKE, with right angles at T and

K, and their hypotenuse is line EZ which is the radius of the circle. Leg ZK is the sine of 6o° as has been explained, and it is equal to leg TE. Leg ZT is a quarter of the diameter, which is the sine of 30°, and it is equal to EK.

53

20

IBN AL-MUTHANNA’s COMMENTARY

When we wish to know the amount of line ZK, which is the sine of 60, we multiply the sine of 30 by itself and subtract it from the product of the radius by itself. We take the square root of the remainder and the result is line ET, the sine of 60. We subtract a quarter of the

5

diameter, which is the sine of 30, or the sine of Aries, namely line EL, from line ET, and there remains the sine of Taurus, namely line

LT.

A

We put these sines in the table opposite Aries and Taurus, respectively. Then we subtract the sine of 60 from the radius so that the remainder is the sine of Gemini, namely arc TD, and we put it next to Gemini.

10

We have thus produced the three sines and this is their table.51 Since you wish to know the sines of the six subdivisions—i.e. when each zodiacal sign is divided into two sections—following the Hindu pro¬ cedure, we construct a figure and draw DZ, the chord of 30, and half of it is the sine of 150 [Fig. 12]. We wish to find line DZ, which is the

15

chord of 30°, so that we can take half of it and put it opposite the first

51. The table does not appear in the manuscript.

54

TRANSLATION OF THE MICHAEL VERSION

section. Line ZD is the chord of right angle T. We multiply DT which is the sine of Gemini by itself, and we add it to the product of the sine of Aries by itself. Then we take the square root of the sum, namely line DZ, the chord of 30°. We take half of it, which is the sine of 150, and we put it opposite the first of the six sections. We put this in the table

5

and subtract it from the sine of Aries so that the remainder corresponds A

to the second section of Aries, and we put it next to it. And this is the form of the table.62 We want you to know what sine corresponds to the third subdivision. We draw a figure again [Fig. 13], dividing arc AD in two halves at N, and draw lines AD, NM, NO. We wish to know the amount of line NO which is the sine of 45°. We have explained in the preceding chapters that the sine of any angle is half the chord of its double angle. We multiply the radius by itself; i.e. we multiply AE by itself and put it

52. See previous note.

55

10

IBN AL-MUTHANNA’S COMMENTARY

down. Then we multiply ED by itself and take the square root of the sum so that the result is line AD which is the chord of 90, and half of it is the sine of 45, i.e. line EM, because

EM is equal to NO and NM.

Since we know the sine of 45, we can subtract the sine of Aries from

5

it, and the remainder is the sine of the third section. Then we subtract the sine of 45 from the sine of Aries and Taurus so that line MT re¬ mains, which is the sine of the fourth section, and we place each sine opposite its section. We wish to explain the sines which belong to the third section, [i.e.

10

from 6o° to 900]. It is necessary to employ the proofs which we will record, and they agree with the preceding part of our work. We con¬ struct circle ABGD and its two diameters which intersect at E at right angles [Fig. 14]. We mark off an arc of 150 from arc AD, namely arc AZ. We draw line ZH from point Z parallel to one of the diameters,

15

and it marks off 30° on the circle. Arc AZ is 150, and we wish to know A

Fig.14

56

TRANSLATION OF THE MICHAEL VERSION

the amount of line ZT which is the sine of 75°- We draw line EZ and we have a right triangle, ZET, in which side ET is the sine of 150, side ZE is the radius, and side ZT is the sine of 750, equal to ply

EK. We multi¬

ET by itself, and we subtract it from the product of the radius by

itself. We take the square root of the difference so that the result is line

5

EK which is equal to ZT. We subtract the sine of Aries and Taurus

from it in order to find the sine of the fifth section, while the remainder of the radius is the sine of the sixth section. We place each sine op¬ posite its subdivision, and this is the rule of the Hindus. If we wish to divide this arc into more sections, we can divide the

10

quarter-circle into twelve subdivisions such that each sign has four parts. To clarify this, we again construct a figure [Fig. 17]. We draw circle

ABGD with center E, and draw its diameters which meet at right angles at point E. We mark off arc AZ equal to 150 so that arc DZ is 750. We draw lines ZH, ZT, ZA so that line ZH is the sine of 150, line

A

Fig. 15

57

15

IBN AL-MUTHANNA.’s COMMENTARY

ZT the sine of 750, line AZ the chord of 150, and line ZT is equal to

EH. We wish to know the amount of the chord of 150, namely AZ, so that we may take half of it, the sine of fA°. Angle H is a right angle. We multiply the sine of 150 by itself, and we add to it the product of line HA, which is one of the six subdivisions, times itself, and we take

5

the square root of the sum. The result is the arc of 15 > half of which is the sine of 7V20, and we put it opposite the first of the twelve

A

sections. We then subtract it from 150 so that the remainder is the sine of the second section, and we put it opposite the second section. When 10

we wish to know the third part of Aries, we draw circle AJBGD, as we said and divide arc AD into two parts at point Z so that each arc is 450 [Fig. 16]. We draw lines ZH, ZT, and

ZD; both ZH and ZT are equal

to the sine of 450, whereas ZD is the chord of 450, since half of it is the sine of 221/2°. Line ZT is equal to EH, and they are equal to lines 15

ZH and ET. Line DH is the difference between the radius and the sine of 450. We multiply the sine of 450 by itself, and we also multiply DH

58

TRANSLATION OF THE MICHAEL VERSION

by itself, and take the square root of the sum. The result is the chord of 450, half of which is the sine of

22V20. We subtract the sine of 15°

from it, so that the remainder is the sine of the third part of the twelve parts, and we put it opposite that part. Further, we subtract the sine of

22V20 from the sine of Aries so that the remainder is the sine of the fourth part.

5

When we wish to know the fifth part of the twelve parts, we again make use of a diagram [Fig. 17]. We mark the sine of 150 on line ED,

namely line EZ, and from point Z we draw a line parallel to AE, namely ZH. Then we draw line

HT equal to EZ, and both are equal to the sine of 150. Arc HA is 150, and arc HD is 750. Line DH is the

10

chord of 750, and line DZ is the difference between the total sine and the sine of 150. Angle Z is a right angle, line ZH is parallel to AE, and we wish to find the chord of 750, half of which is the sine of 371/2°. We multiply the sine of 750 by itself, and keep it. Then we subtract the sine of 150 from the total sine and multiply the remainder by itself. We add the two products and take the square root of the sum. The result is the

59

15

IBN AL-MUTHANNA’s COMMENTARY

chord of 75° which is line

HD. We take half of it and subtract the sine

of 30° from that amount. Then we subtract it from 450, which yields the sixth part. If we wish to find the sine of the seventh part, we again construct a circle [Fig. 18], and we mark off 37V20 from arc AD, namely arc AZ

5

so

that ZD is

521/2°. We draw lines

ZE,

HZ, and ZT. Line

ZH is the sine of 52I/20, line ZT is the sine of 37I/20, equal to EH, and

A

EZ is the radius of the circle. We wish to know the sine of 52V20, so we multiply the sine of 37V20 by itself and subtract it from the product 10

of the radius by itself. We take the square root of the difference, and the result is line ZH which is the sine of 521/2°. We subtract the sine of 45 from it, and the remainder is the sine of the seventh part which we put opposite it. Then we subtract the sine of 52V2 from the sine of Aries and Taurus, the sine of 60, and the remainder is the sine of the eighth

15

part which we put opposite it. When we wish to know the ninth part,

60

TRANSLATION OF THE MICHAEL VERSION

we again construct a circle [Fig. 19]. We mark off 22V20 from arc AD, namely arc DH, and draw lines HT, HK, and HE. Line HK is the sine of 22V2 , line HT is the sine of 6yV2°, and HE is the radius of the circle. Line HK is equal to line ET. We wish to find the amount of HT which is the sine of 67V20. We multiply HK, the sine of 221/2°, by itself and subtract it from the product of the radius by itself. We take the square root of the remainder in order to find line KE, the sine of

5

A

Fig.19

67^2°, and it is equal to HT. We subtract the sine of 6o°, and the re¬ mainder is the sine of the ninth part, which we put opposite its section. We then subtract the sine of 67F2 from the sine of 75 so that the remainder is the tenth part, which we put opposite its section. When we wish to find the eleventh part, we construct a circle again [Fig. 20]. We mark off 71/2° from arc AD, namely arc DZ so that arc AZ is 82V20 and draw lines ZE, ZH, and ZT. Line ZT is the sine of 7I40, ZH the sine of 82V20, and DE the radius; and ZH is equal to ET. To find TE, the sine of 82F20, we multiply ZT which is the sine of 7F20 by itself, and subtract it from the product of the radius by itself.

61

10

15

IBN AL-MUTHANNA.’s COMMENTARY

Take the square root of the remainder, which is line ET, the sine of S2V20. We subtract the sine of 75 from it so that the remainder is the sine of the eleventh part, which we put opposite its section. Then we subtract the sine of 82Vi from the total sine so that the remainder is A

5

10

the twelfth part, and we put it opposite its section. When you wish to make the table finer than this, use more than twelve subdivisions, and the procedure is the same. Al-Khwarizmi, as well as others who indicate the sines by degrees, proceeded in the same way as they did in composing the tables of corrections. Some of them corrected the planets for intervals of ten days53 while others used different amounts. They divided these subdivisions by their degrees and composed tables for degree intervals.

53. Degrees, rather than days, make sense.

62

TRANSLATION OF THE MICHAEL VERSION

The declination which he put into the table of subdivisions is dif¬ ferent from the simple declination in his tables for each degree, because the declinations opposite his subdivisions follow the usage of the Hindus. The obliquity according to the Hindus is 24°, and alkhwarizmi put 240 in his tables of declination. In the simple tables he considered the obliquity as 23^1 °, following Ptolemy. What is obliquity? It is the inclination of the sphere of the zodiac with respect to the sphere of the equator. We have explained this in our book to Muhammad b. 'Abbas, the computer. We say, moreover, that the sun makes a circle in its motion through the heavens which is called the belt of the zodiac because the sun never strays from this path; the other planets incline from this path both to the north and south by a little bit. Day and night change by an increment or diminution for inhabited cities according to their distance north of the equator, but day and night are equal twice a year, whereby one knows that the sun at that time is in its path on the equator. The horizons of inhabited cities are different from each other, as well as different from the horizon of the equator. It is also known that when the day is longer, the sun inclines to the north, whereas when the day is shorter, the sun inclines to the south. One has to know the amount of this inclination for subsequent calculations. The men of India thought this inclination was 240, whereas Ptolemy thought it was 23;5i°. The scholars of the Mumtahan zij who observed for the Khaliph Ma’mun,54 found it to be 23;33°. This entire report was transmitted to us, so one can be amazed at the small disparity between these values, which is not perceived because this amount of minutes is not noticeable with an astrolabe even if the instrument is as large as it can be made. Whoever wishes can construct such an instrument in order to know the truth of these reports. Take the instrument which is mentioned in our book, well-squared, as is required, and its faces level. Then put a point at a given corner and consider it the center, about which is drawn a quarter circle. The larger the instrument, the wider the circle, hence

54. Ms: ben Ma'mun.

63

5

10

15

20

25

30

IBN AL-MUTHANNAS COMMENTARY

it becomes more accurate and freer from error. Whatever the size it is constructed, divide the quadrant into 90 parts, and each part into minutes, as finely as possible, guarding against error. Then construct two pins equal in thickness and of the same length. Put one pin at the

5

center of the circle and the other (removable pin) on the arc of the quadrant, at the place from which you begin the subdivisions. Direct this instrument to the meridian and let a line hang on the pin that is at the center of the circle. If this line falls on the other pin, you know that you have placed the instrument exactly in the meridian

10

plane. Then find the meridian for your city and place the instrument in the meridian properly perpendicular; guard against error. The center of the quadrant is highest, so that the line from which you begin the subdivisions points to the south, and the place of writing on the quad¬ rant is to the west. Then observe the sun at noon on the summer and

15

winter solstices and observe where the shadow of the sun at the center of the quadrant falls, and keep it. Note: if the sun is at the summer solstice, which is the beginning of Cancer, subtract the latitude of the city from the number you kept, so that the difference is the obliquity of the ecliptic. If you direct the instrument to the altitude of the pole,

20

you will not need either increment or diminution. If you wish to use this instrument to learn when the sun enters the equinox, look at the sun when it is at one of the equinoxes, i.e. when it has no declination. There is no deviation from the movable pin, because the shadow of the center pin will fall on the movable pin at the place from which you

25

began the numbering of the subdivisions. This is because the instrument was directed to the altitude of the pole. If it is not so, the shadow of the pin will fall on the amount of the altitude of the pole at your city, which is the latitude of the city. The instrument is shown in the figure [Fig. 21].

30

Our calculation for declination follows al-Khwarizmi’s instructions. When you wish to know the declination of any degree, first consider the total declination as a sine; you know its amount from observation and reports. Also consider the degrees whose declination you wish to know as a sine. Then multiply the sine of the total declination by

35

the sine of the degree and divide the product by the total sine. The result is the sine of the declination of the degree, and the corre-

64

TRANSLATION OF THE MICHAEL VERSION

spending arc is the declination of that degree. This operation was per¬ formed according to the rules of proportion and does not differ from Ptolemy’s, because the ratio of the total sine to the sine of the total declination is equal to the ratio of the sine of the degrees to the sine of the declination. These four numbers form a proportion: the first is

5

the total sine, the second is the sine of the total declination, the third is the sine of the degree, and the fourth is the sine of the declination of the degree, which is the unknown. We have already mentioned that the product of the first number and fourth number divided by the second number produces the third number. If you divide the product of the second and the third numbers by the first number, you find the fourth number, which is the case here. We have explained the meaning of the sine and the declination and the manner of deriving them by

65

10

IBN AL-MUTHANNA.’S COMMENTARY

degrees. If you wish to derive the declination as you derived the sine, you should follow my instructions. Know that everything al-Khwarizmi mentioned up to this point—I mean until the declination for each degree—follows the method, observations, and derivations of the Hin¬

5

dus; this was also true for the correction. The derivation of the declination, as well as what follows in al-Khwarizmi’s tables, follows the procedure of Ptolemy, except that sometimes al-Khwarizmi follows only the principles which Ptolemy expounded. We will explain the contents of each section just as Ptolemy would have explained them,

10

God willing. [Q36] W7iat is this 900 by which he instructed us to divide the result of multiplication in deriving the sine and declination? Answer: We have explained that these are the 900 minutes which equal one section; i.e. they are the number of minutes in 150. This

15

operation is similar to the one he gave us for deriving the corrections of the sun and the moon. [Q37] What is the versine? Answer: We have said that the sine is the line from the degree whose sine we wish to find to the diameter, and the sine is perpendicular

20

to the diameter. This perpendicular, the sine, marks off a section of the diameter that faces the circumference. The remainder of the diameter is the versine [see Fig. 22a]. This is the case when the degree sought is less than 90. If it is 90, its plane sine—i.e. the line that falls on the center of the circle—is half the diameter, and half the diameter is also the

25

versine [see Fig. 22b]. When the degrees surpass 90, the plane sine is less than half the diameter while the versine is greater than half the diameter, because the plane sine marks off more than half the diameter for the versine until the degrees reach 180, at which point the versine is equal to the entire diameter. This amount is greater than the

30

maximum amount of the plane sines, namely half the diameter. But the versine may reach the amount of the entire diameter [see Fig. 22c]. [O3S] Why did he say in connection with finding the latitude of cities: know the altitude of the equator for your city, i.e. when the sun is at the beginning of Aries or Libra, subtract it from 90, and the

66

TRANSLATION OF THE MICHAEL VERSION

remainder is the latitude of your city. If the sun is not in either of these places, subtract the declination of the sun from the altitude in the case of the southern signs, and add it to the altitude in the case of the northern signs. Then subtract the result from 90 so that the remainder is the latitude you wish to ascertain.

5

Answer: When the sun enters the first minute of Aries or Libra, it

Fig.22 has no declination, so that its altitude observed from the equator is 90°, and it will be at the zenith there. For cities which incline from the equator, the celestial equator inclines from the zenith by an amount equal to the altitude of the

10

pole at their location. Subtract from 90 the altitude of the pole which is equal to the latitude of the city. If the sun is not in the beginning of Aries or Libra, you also should give it a second inclination according to the amount of the declination of the degree of the sun. If the sun is in the southern signs, i.e. from the beginning of Libra to the beginning of

15

Aries, this declination is to the south, and it is subtracted from the alti¬ tude of the equator which passes through the beginning of Aries and Libra. If the sun is to the north, i.e. from the beginning of Aries to the beginning of Libra, this declination is to the north, and it is greater than the altitude of the equator by the amount of this declination which belongs to the degree of the sun. For this reason al-Khwarizmi said: note, if the sun is in a northern sign, subtract the decimation from the altitude of the sun; but if it is in a southern sign, add the declination to its altitude. The result is the

33>49° 054,40,20 O

°;2,3M9° 014,40,23

Nn

30° 60° 90°

Newminster

Latin

Hebrew

052,34,49° 054,40,23 °;5>3L34

We note that for 90° the Hebrew and Latin agree against the Newminster, for 6o° the Hebrew and the Newminster against the Latin, and for 30° all three differ in the seconds place. All the tabulated numbers are derived from the expression: (150 tan 8)/i2. The Newminster text has the preferred values, since they are derived from a fixed obliquity of 23551°. The rising time difference from longitude 90° in the Ibn al-Muthanna text leads to an obliquity of 23553°, a value which is hardly significant.20 Q45 (81:4). Again the rules for linear interpolation are presented, here for the entire interval of 30°.

18. Neugebauer, Khw., p. 52. Almagest, II, 7. 19. Neugebauer, Khw., p. 55. Latin from Se 35r. 20. Cf. Lesley, “Birum on Rising Times and Daylight Lengths,” p. 127.

206

IV. Seasonal Hours and Gnomons i. Daily Arc . Ql6 (81:14) • The daily arc is the path of the sun in the sky from sun¬ rise to sunset, expressed in equatorial degrees. This is equivalent to the total rising time of the six zodiacal signs above the horizon at sunset. There are, of course, always 180° of the ecliptic above the horizon, for all great circles on a sphere bisect one another (cf. Millas Vallicrosa, Fund., p. 156).

2. Seasonal Hours Q47 (81.18). Since 560 equal one day or 2q hours, one hour equals 15 equatorial degrees. A seasonal hour is defined as %2 of the daily arc and can be greater or less than an equal hour (cf. Millas Vallicrosa, Fund pp

i56-57)-

3. Time Reckoned from Solar Altitude Q48 (82:1). The rule stated in the text is correct only at an equinox. From the projection onto the meridian plane (see Fig. C-6) we see that sin a, sin t

sin an —-— = sin cp

zenith

or:

207

(1)

IBN AL-MUTHANNA’s COMMENTARY

Sin a, _ Sin an Sin t

(2)

R

where an is the noon altitude of the sun, a, the altitude of the sun at time t, and t is measured in equatorial degrees after sunrise or before sunset.1 For any other time we must refer to Figure C-7, and equation (1) no longer ap¬ plies. (82:25). Changing to versine does not improve matters, unless the whole formula is revised (see 5, below). zenith

4. Converting Seasonal Hours to Equal Hours Q49 (83:3)- Seasonal hours are given in equatorial degrees, every 15 of which equal one equal hour.

5. Exact Determination of Time from Solar Altitude Q50 (83:11). Let c be the daily arc, a, the altitude of the sun at time t. Then the rule stated in the text is: 2 Vers 9, = Vers (c/2)

Vers (c/2) Sin a, Sin On

(H

1. Cf. Millas Vallicrosa, Fund., p. 158; Ms Par. Gr. 2425, sections 42, 43. 2. Cf. Nadi Nadir, Abu al-Wefd on the Solar Altitude,” for a discussion of several solutions to the problem at hand, including this one. Cf. K-K, Hi, 13-16, where the gnomon hypotenuse h is defined as h - 12/Sin a, so that the’formula there is equivalent to our formula above. Cf. Ms Par. Gr. 2425, sections 65, 66.

208

NOTES

where 9, is the time since sunrise or before sunset expressed in equatorial degrees. Dividing by 15 changes the degrees into equal hours. This formula is correct and not equivalent to the previous rule (3 above) as our author would have us believe.

6. Finding the Ascendant Q51 (84:1). The ascending degree of the ecliptic may be found if the equatorial arc from the horizon to the sun is known (cf. 5, above), since: p(A.h — \q ) = Qt

(1)

The inverse function of oblique ascension yields the longitudinal difference between the sun and the ascendant and, since the longitude of the sun is known, the longitude of the ascendant may be deduced.3 The bulk of the paragraph has confused this problem with the problem of changing seasonal hours into equal hours.

7. The Twelve Astrological Houses

QS2 (84:14)- The houses are arranged so that the first house begins at the ascendant, the fourth house at the culminating point of the ecliptic be¬ low the horizon (i.e. lower midheaven), the seventh house at the setting point, and the tenth house at the culminating point above the horizon (i.e. midheaven). Each house rises in two seasonal hours. The table beginning with Capricorn is the same one we discussed in Section III: 9 above. It tabulates a'(\) a'[\(M)] = a[A(M)] + 90 = p[A(H) ]

(1)

where A(lVf) is the longitude of midheaven and A(H) is the longitude of the ascendant. From this table one can find A(M) directly from A(H). (84:31). The meridian crossing at any latitude is equal to the rising at sphaera recta, since the meridian is a great circle perpendicular to the equa¬ tor, just as the horizon is at sphaera recta. Q53 (85:4)- Since the tenth house begins at midheaven, the eleventh house, the House of Hope, begins at the degree of the zodiac that corre-

3. Cf. Ms Leiden Or. 199.3, which includes a table for 0t as a function of the noon altitude of the sun and the observed altitude of the sun for the express pur¬ pose of calculating the ascendant. For an analysis of this table, see Bernard R. Goldstein, “A Medieval Table for Reckoning Time from Solar Altitude,” Scripta Mathematica, 27 (1964), 61-66.

209

IBN AL-MUTHANNA S COMMENTARY

sponds to two seasonal hours before midheaven, and the twelfth house, the House of Enemies, begins another two seasonal hours before the eleventh house, etc.4 (86:11). The two adjacent quadrants are unequal, as stated in the text, and the problem here is to find the difference between them. The text should perhaps read: “take the degrees of six seasonal hours and subtract them from 90. If this difference is added to the other six seasonal hours [i.e. on the other side of the ascendant or the setting point], the sum will be 90. The difference between one diurnal house and one nocturnal house is equal to the difference between the degrees of two houses and 6o°. The sym¬ metries mentioned for the quadrants are correct.

8. Gnomon O54 (87:1). The gnomon is taken to be 12 digits long, as in the Hindu theory.5 Let the shadow of the gnomon be called s and the altitude of the sun a, then: s _ 12 Sin (90 - a) __g_ Sin a tan a

/

x

In Figure 26 (p. 88), T is directed toward the sun, EG is the gnomon, EKG is the altitude of the sun, and line KG is the shadow. (87:17). The text reads: one digit is equal to •% of 60 parts, and one digit is equal to %5 °f 150 minutes, rather than the inverse of these frac¬ tions, as we would expect.

9. Finding the Altitude of the Sun from Its Gnomon Shadow Q55 (88:4). The text uses the formula: Sin

(90 — a) = (150 • s)/(s2 + 144)1/2

(1)

where the denominator is the hypotenuse of the shadow (line EK in Fig. 26, p. 88), and 150 is the total sine or radius. (89:22). Here we are given the alternative formula: Sin a = 150 X 12/(s“ + i44)1/2

(2)

where 12 is the length of the gnomon. 4. Neugebauer, Khw., pp. 76, 77. Cf. Ms Par. Gr. 2425, section 55. Cf. Rasd’il (Beirut ed., 1376/1957) for the Arabic names of the houses. 5. Cf. K-K, Hi, 9, for example. Ptolemy (Almagest, II, 5) considers the gnomon equal to 60 parts. Ikhwan al-Safd5 I, 136

210

V. Planetary Latitude i. Latitude of the Moon Q56 (89:29). Let a) be the argument of latitude for the moon, A„ the longitude of the node, and \m the longitude of the moon. Then: — Am

An

(^)

Let f3 be the latitude of the moon. Then according to the text: 1

P=

(9/5) Sin

(2)

Equation (2) is based on the assumption that the maximum latitude of the moon is 4:30°.2 The latitude is found from the solution of a spherical triangle (see Fig. C-8): Sin /3mai = R Sin p/Sin

(2)

max

or Sin p = (Sin /?max Sin W)/R For small angles, sin p is approximately equal to p. Tlierefore, if p is ex¬ pressed in minutes:

1. Cf. Ms Par. Gr. 2425, section 60-62, 4. 2. K-K, i, 31; S-S, i, 68. Ptolemy took the maximum latitude to be 50. See Almagest, Manitius, I, 285, 299.

211

IBN AL-MUTHANNA S COMMENTARY

/3 *=* (/Lnax Sin 0))/R = (4,30 Sin to)/2,30

(4)

= (9/5) Sin w (90:9). The instrument described here is the same as the parallactic instrument described by Ptolemy (Almagest V, 12). The length of the alidade is taken as twice the radius so that the graduated alidade will read sines directly rather than chords. One observes the moon in the meridian when it is 90° from the node and notes the zenith distance. The latitudinal arc lies in the meridian only when the node is near the equinox, and the equinox is at the ascendant, an event that takes place about every ten years. At such a time one need only subtract the lunar zenith distance from the zenith distance of midheaven, and the difference is the maximum latitude of the moon. To avoid parallax, one should make the observation when the lunar latitude is to the north and the culminating point is north of the equator (see Fig. C-9). (91:20). This description of a second instrument, missing in the Latin version, is not at all clear. The text may refer to an instrument composed of only one alidade in addition to the gnomon where the angle between

zenith

Fig. C-9

212

NOTES

them can be measured directly. In that case the rest of the procedure would be the same.

2.

Latitude of the Planets

(92:11)- The first step is to find the distance p from the observer to the planet, called here the radix of latitude (see Fig. C-io): _ Sin P

(Jo

Sin a

Sin a

r Sin a ~

Sin )/p where i is the inclination of the deferent from the ecliptic, and argument of latitude. In Figure C-ir (for outer planets only): sin co = AC/R where angle AOC is w, and

213

(4) is the

(5)

IBN AL-MUTHANNa’s COMMENTARY

sin

i = CC'/AC

(6)

Therefore:

CC' — R sin to sin i

(7)

If the planet is at P, sin /3 where /? is the angle

— PP'/p

(8)

POP' and PP' = CC' (for an outer planet the epicycle

is parallel to the ecliptic). Since both /3 and i are small angles, we have:

P = (i Sin c))/p

(9)

as stated in the text.3 For inner planets this formula will not work.4

The values for the inclinations of the planetary orbits are given in the text as:

3. Cf. Neugebauer,

Khw.,

Saturn Jupiter

2;o° i;o

Mars Venus Mercury

V3° 2;o 2)30

p. 34 ff.; Aryabhatiya,

vi, 1-3; S-S, i, 68-70. 4. Neugebauer, Khw., p. 37 ff.

214

i,

6-7, trans. Clark, p. 16;

K-K,

NOTES

These all agree with the Hindu values except for Mercury, which is usually given as 2;o°. However, the commentary of Amaraja gives 2530, and Brah¬ magupta gives 2;32.6 (94:2). The values quoted as Ptolemaic are: Saturn Jupiter Mars Venus Mercury

2;3° i?3° i;5° i;o i;o

Of these parameters, the first two are attested in the Almagest, the Handy Fables, and the Planetary Hypotheses. The parameter for Mars is attested only in the Planetary Hypotheses, while the parameters for Venus and Mer¬ cury are nowhere found in the works of Ptolemy. Our author shows no knowledge of the complex Ptolemaic theory of planetary latitudes.6 (94:6). The few lines to the end of the chapter refer to the tables for solar and lunar velocity in the Majrifi text.7 They more properly belong to the beginning of Section VIII where the diameters of the sun, moon, and shadow are related to solar and lunar velocity.

5. K-K, Sengupta, p. 116. 6. Almagest, XIII. 7. Suter, Khw., Tables 61-66, pp. 175-80.

215

VI. Conjunction and Opposition

i. Mean Syzygy Q58 (94:10). The first step toward an eclipse (or new moon) computa¬ tion is to find the mean syzygy. For units of 30 Arabic years, the first mean opposition is on the 14th day of the month. For single years one must pay attention to the leap years of 355 days instead of 354 days. Moreover, the months vary between 29 and 30 days. Thus the mean syzygies move from the 14th and 29th days of the month. The Arabic calendaric month is 29;3i,5od while the astronomical mean synodic month of the Khwarizmi tables is 29; 31,50,5, ... A1 and so the syzygies move forward with respect to the calendar. The mean syzygies are tabulated in the Majriti version of the Khwarizmi tables (Suter, Khw., Tables 69-72), and discussed by Neugebauer in his commentary to the Khwarizmi tables, pp. 108-15.

2. True Syzygy O59 (94:23). If the mean syzygy is known, it is then necessary to com¬ pute the true syzygy which either preceded or followed the mean syzygy. One takes the differences between the true positions of the sun and moon at the time of mean syzygy and divides by the equated velocity to find the time to be added to or subtracted from the mean syzygy:

t — (Am —

— v.)

(1)

The denominator is the equated velocity, vm the velocity of the moon, and v, is the velocity of the sun. For opposition one uses Am + 180 instead of the longitude of the moon. Since the velocity is given in degrees per day, the text has us find 24f, because t is expressed in days and we wish to know the corresponding number of hours. The procedure is iterative because we have not considered the motion of the sun or the moon during time t. Hence one redetermines the positions of the sun and the moon: A'» = A, ±v,t

(2)

A m — A m —!— Vmt With the new longitudes one recomputes with equation (1). The iteration process is described in our text as correcting mistakes, rather than as a pro1. Cf. Neugebauer, Khw., pp. 109, 114.

216

NOTES

cedure of successive approximations.2 The mean syzygy precedes the true syzygy if the sun is ahead of the moon at mean syzygy. On the other hand, the mean syzygy comes after the true syzygy if the moon is ahead of the sun at mean syzygy (owing to the greater velocity of the moon). See Figure C-i2 for the case when the degrees “belong to the sun” in the terminology of our text.

2. Cf. Ms Par. Gr. 2425, section 59.

2i7

VII. First Visibility of the Lunar Crescent i. Day of New Moon 06i (96:20). The lunar crescent is never visible on the 29th night, is sometimes visible on the 30th night after the previous new moon, but is always visible on the 31st night.1 The length of the month is determined by whether the crescent is seen on the 30th night. If the moon was ripe at its previous appearance, indicating that it just missed appearing on the previous night, then the following month will be short, i.e. 29 days. Also, if the velocity of the moon is great, the month will certainly be short. The moon must be approximately a day’s motion beyond conjunction to be visible.

2. Polar Longitude Computed Q62 (97:17). The rule stated in the text is exactly the same as the one found in the Khandakhddyaka (K-K, vi, 4) 2 except that here we divide by 375 instead of 371. The text tells us to find A': A' = A±AA

(1)

AA = /3 Sin (90 + A)/375

(2)

where

We shall see that /3 in this formula is the polar latitude which we shall later designate /?'. In Figure C-13, P is the pole of the equator and O the pole of the ecliptic, and we wish to find AA where VA = A is known. We apply the law of sines to triangle POS: Sin Q _

Sin x

(^ ^

ShTPS ~ Sin PQ Arc PQ = E, angle Q = 90 — A, and arc PS = 90 — 8 where 8 declination of point S. Since Sin (90 — A) = Sin (90 + A):

is the

1. For a chart of conjunctions and the corresponding new moons, see Neugebauer, Astronomical Cuneiform Texts, III, 140-50. 2- K~K, Sengupta, p. 120 ff. Cf. Millas Vallicrosa, Fund., pp. 162 ff.

218

NOTES

Sin x = [(Sin e Sin (90 + A)]/Sin (90 - 8)

(4)

In right triangle BAS Sin x «=* R(AA//T)

(5)

or A A. = f3'[ Sin e Sin (90 + A)]/R Sin (90 — 8)

(6)

For equation (6) to be equivalent to equation (2) R Sin (90 — 8)/Sin e = 375

(7)

If 8 = 0, then e = 23;350. The Hindu denominator, 371, implies that e = 240 for S=o. Rather than introduce a new value for obliquity, we assume that our text rounded the Hindu denominator to a multiple of 15, which is easier to use with the Hindu sine tables (computed for 150 in¬ tervals). The error introduced from the assumption that 8 = o is equal to l/Sin (90 —8) where 8 «=> 240 (since S is never very far from the ecliptic).

219

IBN AL-MUTHANNA’s COMMENTARY

Thus the computed AA will be, at the most, *% of the correct value. Note that the coordinates of S, /T and VB = A', are polar coordinates.3 (97:27). The answer is hopelessly confused between the change of co¬ ordinates described above and the parallax procedures.

zeni th

(98:24). This example is incomplete (the longitude of the sun and the latitude of the moon are omitted), and so we cannot check the result against the procedures in the text. (98:31). The meaning of this paragraph is not clear. The figure I have drawn (Fig. C-14) is reconstructed under the assumption that the author wished to show that the closer an object comes to the center of the earth,

3. K-K, Chap, x, Sengupta, pp. 148—53, where the star catalogue is given with polar longitude and latitude. Cf. S-S, p. 205, for a similar catalogue with polar longitude and latitude.

220

NOTES

the lower it is; i.e. parallax increases as the object comes closer to the earth. The figure in the manuscript (M-27) suggests that the author wished to indicate that, as the altitude of a given line segment increases from the base of a right triangle, the angle under which it is observed from the opposite vertex decreases. This remark is correct for a plane triangle; on a zenith circle, however, equal arcs subtend equal angles regardless of their altitude.

3. Right Ascensional Arc Corresponding to Lunar Latitude Q63 (100:9). The text gives us the formula: 4 Aa — (/S/ Sin 95)/Sin (90 — cp)

(1)

where Aa is the right ascensional arc between the moon and the parallel to the horizon through A' (Fig. C-15). This correction on account of latitude is a right ascensional arc rather than a longitudinal arc as stated in the text, since the angle between the ecliptic and the horizon is not, in general, equal to (p. (100:19). Again the answer has confused parallax with the problem of first visibility.

4. K-K, vi, 5.

221

IBN AL-MUTHANNA’s COMMENTARY

4. Oblique Ascension Between Sun and Moon; Digits of Light Q64 (101:20). The criterion for first visibility of the lunar crescent is given here as 120 in oblique ascension between the sun and the moon.5 The ascension of the moon has been corrected for latitude in the previous para¬ graph. So we have: s(As — A'm) + Aa = 12°

(l)

Stereographic Projection

where Aa is defined by equation (1) of 3, above, and s(A) is the setting time of A. (In the next paragraph, q.v., we learn that the setting time equals the rising time between the opposite points.)

5. Cf. S-S, x, 1, Burgess, p. 262; al-Battam, I, 87; Millas Vallicrosa, Azarchid,

P- 134; K-K, vi, 6, Sengupta, p. 124; Pancha-siddhdntikd, Thibaut, p. 38. In all these places the 120 of ascension between the sun and the moon are mentioned.

222

NOTES

(101:23). The answer does not deal with the question, but introduces a new relationship:

d = E/15

(2)

where d represents the digits of the illuminated portion of the moon, and E is the elongation between the sun and the moon.6 Equation (2) indicates the use of linear interpolation where the maximum number of digits of light is the size of the entire lunar disk which is illuminated at full moon, i.e. at an elongation of 180°. Thus equation (2) is equivalent to:

d _ E 12

180

l3j

For an elongation of 120, we find, from the formula, % digits of light.

5. Rising Time and Setting Time; Projection of First Visibility Q65 (102:12). For the proof that the setting time equals the rising time between the opposite points, see Figure C-16 which is a stereographic pro¬ jection of the celestial sphere onto the equatorial plane. In the figure arc LM = arc AB, and s(LM) = p(AB), since angle LPM' = angle APB' (cf. S-S, x, 2, Burgess, p. 262). (102:13). The visibility of the moon in the east would refer to the sym¬ metric problem of finding the lunar visibility of the night of opposition. (102:16). The diagram described (Fig. 28) is a projection of the celes¬ tial sphere onto the tangent plane at the setting point, D. On the same plane we also draw the horizon through D, though the horizon lies in a plane per¬ pendicular to the projection plane. The purpose here is to find the azimuth of the first lunar crescent. Figure C-17 shows the positions of the sun and the moon on the celestial sphere. Let E be the highest point of the ecliptic, and project arc GE onto the projection plane (tangent to the sphere at D). Since the projection plane is parallel to plane G'GE, arc GE is unchanged by the projection. Arc HDE projects into a straight line, since it lies in a plane perpendicular to the pro¬ jection plane. We consider the neighborhood of D on the sphere as a plane. Thus ZT is perpendicular to HDE. We wish to find the image of M, where M is the foot of the altitude circle through T. Arc GTM projects onto the plane as an ellipse (arc GTM') and arc ADM projects as straight line DM'. Now rotate the horizon plane about axis DM' into the projection

6. Cf. K-K, yii, 4, Sengupta, p. 131.

223

IBN AL-MUTHANNA’s COMMENTARY

plane. The horizon circle is tangent to its image at D. In the text, line GT is extended to M on the horizon circle. The assumption that M is the inverse image of M' introduces a small error, since MM' should be perpendicular to DM' according to the rules for this projection. Thus arc DM has been graph¬ ically constructed. Since arc DW, the ortive amplitude (more properly, the setting amplitude), is considered as known, one can find arc arc

BM = go + arc DW — arc DM

BM: (1)

Note that the azimuth is measured from the south point of the horizon. (104:11). The rule given here for finding the digits of light differs from

the rule of the K-K (cf. 4, above, equation 2), by using oblique ascension rather than elongation, a change but not an improvement. The 900 replaces the 15 as a divisor in order to put the digits of light into minutes [15 • 60 = 900], which should now be called the minutes of light. A digit of light is the amount of the lunar diameter in the illuminated crescent compared to the entire diameter, which is considered to be 12 digits. The diameter that bisects the visible crescent lies on a line from the center of the sun through the center of the moon. Tire two arcs of the crescent should both go through the same diametrically opposite points of the lunar disk, such that the di¬ ameter through these two points is perpendicular to the sun-moon line. The

224

NOTES

arc of the crescent would then be an elliptical arc through these two dia¬ metrically opposite points and the point on the sun—moon line where one lays off the digits of light. As an approximation, one can substitute the circular arc through these three points (cf. K-K, Sengupta, pp. 132-33). The inaccuracy of the procedure in the text is not too great for the first crescent, though later in the month the result would be grotesque.

225

VIII. Diameters of Sun, Moon, and Shadow i. Sun and Moon Q66 (104:21). For the sun we are given the rule: (11/20) v,0/d =

d.

(1)

The Majriti text has an equivalent formula: 1 2 0533 v,°/d =

d.

(2)

The tabular values for the diameter of the sun in the Majriti text range from 0531,20° to 0533,48° with a mean value 0532,34.2 The source for this formula, as well as the formulas for the moon and the shadow is the K-K, i, 31, repeated as iv, 2.3 For the sun and the moon the formula is based on the assumption that the distance from the sun to the observer varies inversely with its angular velocity, i.e., f _ v (3)

r — v

where f and v are the mean radius of the solar orbit, and the mean angular velocity of the sun respectively.4 If D is the true diameter of the sun in linear measure, d the observed angular diameter of the sun, then: sin (d/2) =

D/zr

(4)

Hence, if we substitute d/2 for sin (d/2), as we may in dealing with small angles, then: d

D/r

f

v

d “ D/f “ r “ v

(5)

or: d = (d/v) • v

(6)

In the text d/v = 0533. If we take the mean solar velocity per day as 0559,8, then 3.

= 0532,31,24°. However, the Majriti tables are based on a mean

1. Neugebauer, Khw., p. 57. 2. Suter, Khw., Tables 61-66, col. 5. 3. K-K, Sengupta, pp. 32, 83. Cf. Ms Par. Gr. 2425, sections 60-62, 6. 4. K-K, Sengupta, pp. 32, 33.

226

NOTES

solar velocity of o;5q,i2° per day,5 which would require a mean solar di¬ ameter of 0^2,, close to what we actually find in the Majrifi text. In the case of the moon, the text produces the formula: 6

dm — (10/247) ’v"‘°/d

(7)

Since the theory is exactly the same as for the sun, we may conclude that

d/v = 10/247

(8)

The mean velocity of the moon is 13;io,35°/d, and this would lead to a mean lunar diameter (from equation 8) of o;32,o°. The Majriti text takes the mean lunar diameter to be o;31,56° which would correspond to a mean lunar velocity of i3;8,45°/ 121-41-

248

REFERENCES

Levy, Jacob, Neuhebraisch.es und chaldaisches Worterbuch, 4 vols., Leipzig, 1876. Levy, R., The Astrological M^orks of Abraham Ibn Ezra, Baltimore, 1927. Millas

Vallicrosa,

J.M.,

“'Abodato

shel

R.

Abraham

ibn

'Ezra

behokmat hatekuna,” Tarbiz, 9 (1938), 306-22. La Autenticidad del commentario a las Tablas Astronomicas de al-Jwarizmi por Ahmad ibn al-Muthanna3,” Isis, 54 (1963), 11419. -— Estudios sobre Azarchiel, Madrid-Granada, 1943-50. -El libro de los fundamentos de las Tablas astronomicas, MadridBarcelona, 1947 [Lund.]. -Las Tablas del Rey Don Pedro el Ceremonioso, Barcelona, 1962. -- Las traducciones orientates en los manuscritos de la Biblioteca Catedral de Toledo, Madrid, 1942. Millas Vendrell, E., El commentario de Ibn al-MutannT a las Tablas Astronomicas de al-Jwdrizmi, Estudio y edicion critica del texto latino, en la yersion de Hugo Sanctallensis, Madrid-Barcelona, 1963. Ms Par. Gr. 2425. Commentary by O. Neugebauer (unpublished). Sections are numbered in the manuscript. Nadir, Nadi, “Abu al-Wefa3 on the Solar Altitude,” Mathematics Teacher, 53, 6 (Oct. i960). Nallino, C. A., Al-Battdni sive albategnii Opus Astronomicum, 3 vols., Milan, 1899-1907. --Raccolta di scritti, vol. V, Astrologia, Astronomia, Geographia, Rome, 1944. Neugebauer O., Astronomical Cuneiform Texts, 3 vols., London, 1955. -“The Astronomical Tables P. London 1278,” Osiris, 13 (1958), 93-H3. -The Astronomical Tables of Al-Khwarizmi, Copenhagen, 1962 [Neugebauer, Khw.]. -“The Transmission of Planetary Theories in Ancient and Medie¬ val Astronomy,” Scripta Mathematica, 22 (1956), 165-92. -“Thabit ben Ourra, On the Solar Year and On the Motion of

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Khwarizmi in der Bearbeitung des Maslama ibn Ahmed al-Majriti und der lateinischen Uebersetzung des Athelard von Bath, based on the preliminary work of A. Bjprnbo and R. Besthorn, Copenhagen, 1014 [Suter, Khw.]. -Die Mathematiker und Astronomen der Araber und ihre Werke, vol. io in Abhand. zur Gesch. der Math. Wiss., Leipzig, 1900. [ZDMG. Zeitschrift der deutschen morgenlandischen Gesellschaft.]

251

Index References are made only to the Michael version where the Michael and Parma versions are parallel. The initial Ibn and al- are omitted from personal names. Aban-meh, 187 c Abbas, 9, 49, 243 Abu Macshar, 243 n Adelard of Bath, 5 Ahargana, 190 Akhfash, 15, 151, 185 Aclam, 150, 246 Alexander, era of, 6, 186. See also Dhuff-Qamain cAli b. Muhammad, 185 Alidade, 212 Almagest, 6, 8, r49, r86n, igr, 193, 203 f, 206 n, 2r 5, 233, 237, 245. See also Ptolemy Altitude, solar, 7, 82 ff, 202 ff, 207 ff Amaraja, 2r5 Anomaly: correction of, 32 ff; maxi¬ mum equation of, 194 Apogee, 26, 27, 28, 31, 190 Arin, 4, 49, 147, 191 n. See also Naru Aryabhatiya, 2140 Asah, 152 Ascendant: 84, 85, 209, 236; of the radix, 143; portion of, 126 ff Ascension: oblique, 7, 76 ff, lor, 222 f, 242; right, 7, 69 ff, 202 ff, 221. See also Examples Astrolabe, 149 Astrological houses, 209 f Avicenna, 234 n Azarchiel, 9 n, 10 n, 196 n. See also Millas Vallicrosa, Zarqal Azimuth, 223 f

Bahman, 19 Battani, 150, 222 n, 244, 246. See also Nallino Biruni, 4, 5, 9 n, 183, 186, 190, r99, 205, 235 n, 243 n, 244 Blachere, R., 9 n, 2440, 245. See also Sa'id al-Andalusi Bonds, of the planets to the sun, 46 Brahmagupta, 4, 7, 190, 215. See also Khandakhadyaka, Sengupta Brahmasiddhanta, 4, 8 Brockelmann, C., 185 n Bullialdus, I., ipr n Burgess, E., 189 n, i9r n, 196 n, 198 n, 199 n, 200 n, 222 n, 228 n, 234 n. See also Suryasiddhanta

Calendar. See Chronology, Month, Year Carmody, F. J., 245 n Center, correction for, 30 ff, r92 f. See also Apogee Centrum, 194 Chaitra, 189 Chronology, 6, 16 ff, 186 ff. See also Month, Year Co-latitude, 202 Computation. See Examples Conjunction: 94 ff, r2off, 216 ff; of the planets, 35, 41, 189 Correction. See Anomaly, Center

INDEX

Daily arc, 81, 207 Darius the Lesser, 24, 188 Declination, 7, 49, 64 ft, 200 ft Deferent, 191 DhuTQarnain, 18-22, 186 Diameter: values for, 51, 91, 240. See also Moon, Shadow, Sun Digits: gnomon, 87; of eclipse, 130 ff, 238 ff; of light, 104, 222 ff Dome of the earth, 49 Dragon, 29. See also Node, Shadow East, men of the, 27 Eclipses, lunar: 109 ff, 231 ff; colors, 119, 234; half-duration, 110, 113, 118, 233; limits, 110, 232; partial, 232 solar: 120 ff, 236 ft; digits of, 132 ft, 238 ff; duration, 138®, 241; projection, 140 ff, 241; syzygy, 216. See also Examples Elephant, era of, 24, 188 Elongation, 223 Encyclopedia of Islam, 185 n, 243 Epicycle, 31 ff, 43, 191. See also Anomaly Equal hours, 81, 83, 208 Equant, 244 Equation. See Anomaly, Center Eras. See Chronology Euclid, 34, 37, 98, 194 Euctemon, 150, 245 Examples: correction for Mercury, 43; diminution of rising time, 79 f; lunar eclipse, 116; solar eclipse, 123, 135; visibility of the lunar crescent, 98 Excess of revolution, 143 f, 242 Ezra, 3, 7, 9-11, 147, 149, 155, 171, 179, 200, 244-46 Farghani, 5, 6, 15-16, 49, 78, 79, 148, 151, 185, 205, 244

Farwadin-meh, 20, 22, 186 Fazari, 199 First visibility, 188, 218 ff, 223 ff. See also Lunar crescent Galen, 234 n Gematria, 149, 244 Ginsburg, Y., 9 n Ginzel, F. K., 187, 188 n Gnomon, 7, 87, 207 ff, 210, 212. See also Shadow Goldstein, B. R., 209 n Golius, 229 Habash, 9, 109, 150, 230 n, 245 Hadid, 29 Hagin le Juif, 246 n Half-daylight, 120, 236 Halma, N., 204, 245 n. See also Handy Tables Handy Tables, 198 n, 204, 215, 245 Haskins, C. H., 12 n Highest point on the ecliptic, 238 Hijra, 6, 186, 191, 244. See also Chronology, Muharram Hindu: diameter, 51, 91, 240; era, 189; gnomon, 87; lunar eclipses, 233; lunar latitude, 89, 231; mean motion, 190; month, 186; obliquity, 63, 200; observations, 49; parallax, 124; period, 26; planetary inclinations, 215; plan¬ etary latitude, 92, 213 ff; plane¬ tary theory, 193, 198; rule, 57; transmission of astronomy to Islam, 3-6; trigonometry, 196; yojanas, 228 Hipparchus, 150, 245 Horoscope, 144 Hours; equal, 81, 83, 208; seasonal, 81,83, 2°7 ff, 210 House of Enemies, 86, 210 House of Hope, 86, 209

2 54

INDEX

Houses, astrological, 209 f Hugo Sanctallensis, 12 Hunain ibn Ishaq, 234 n

K-K. See Khandakhadyaka Koheleth, 11 n Koran, 147

Ikhtilaf al-Manazir, 194 Ikhwan al-Safa3, 210 n Immersion, 238 Inclination, 29; of the planetary or¬ bits, 214. See also Declination, Latitude India, scholars of, 26, 39. See also Hindu Instruments, 63, 200, 212 Interpolation, 198, 201, 206 3Isti (5Istiji), 50, 245 Jacob, 148. See also Kindi, Tariq Jaib (sine), 176 Jawzahar (node), 29 Jupiter, 42, 43, 44, 94, 197, 214 f Kacba, 24, 188 Kalila wa-Dimna, 147, 243, 245 Kaliyuga, 189 Kalpa, 190, 242 n, 243 Kanka, 4, 148, 243 n, 244-46 Kanon. See Zij Kardaja, 42, 196 f, 200. See also Sec¬ tion Kennedy, E. S., 3 n, 4 n, 5 n, 191 n, 200 n, 230 n, 235 n, 2360, 237 n, 242 n, 243 n, 244 n, 245 n, 246 n Kennedy-Muruwwa, 243 Kennedy-Sharkas, 200 n Khandakhadyaka [K-K], 190, 208 n, 211 n, 214 n, 218, 233 n. See also Sengupta Khwarizmi, vii, viii, 5, 8, 15, 17, 19, 22, 24, 30, 32, 43, 47, 63, 64, 74, 75> 79’ 93> l86> 1(A 2°S> 216, 243 f, 246 Kindi, 150, 245-46

Latin version, 11. See also Hugo Sanctallensis Latitude: argument of, 191; celestial, 29, 32; geographical, 66 ff, 201 f, 204; lunar, 7, 89 ff, 211 ff, 221, 231 ff; planetary, 7, 92 ff, 213 ff Lesley, M., 206 n Levy, }., 244 n Levy, R., 246 n Longitude, 6, 26 ff, 191 ff; polar, 97 n, 218 ff Lunar crescent, 188, 218 ff, 222 f. See also Examples, First visibil¬ ity

M. See Michael version Mahayuga, 190, 193, 243 Majriti, 5, 186, 188, 191, 197^ 205, 215, 216, 226, 229, 231, 243, 245, 246 Ma5mun, 63 Manitius, K. See Almagest, Proclus, Ptolemy Mansur, 4 Mars, 42, 43, 44, 94, 197, 214 f Marwadhi, 150, 245, 246 Mashallah, 11 n Macshar, abu, 243 n Mean motion, 26 ff, 190 f Menelaus, theorem of, 7, 71, 76, 203 Mercury, 42, 43, 44, 94, 197, 214 f Meridian: prime, 191 n; projection onto, 207. See also Arin Meru, 199 Meton, 150, 245 Michael version, 8-11, 12, 15 ff, 198 f, 242, 243

255

INDEX

Midheaven, 84, 85, 91, 124, 209, 236; lower, 84, 85, 209; of the moon, 97 Millas Vallicrosa, J. M., 9 n, 196 n, 200, 202 n, 203 n, 204 n, 205 n, 207, 208 n, 218 n, 222 n, 230 n, 23m, 234 n, 235 n, 243 n, 245 n, 246 n Millas Vendrell, E., 12 Minorsky, V., 199 n Month(s): Arab, 16, 22, 188; Per¬ sian, 22, 187; Roman, 21; syn¬ odic, 17-18, 86. See also Chron¬ ology, Year Moon: correction for, 30 ff; diam¬ eter, 104 If, 116, 135, 226, 229 f; equation of center, 197; latitude, 89 ff, 97, 116, 211 ff, 221; midheaven of, 97 ff; na¬ ture of, 101; observation of, 212; shape, 102; sphere of apogee, 28; velocity, 94, 104 ff, 216. See al¬ so Conjunction, Eclipse, Exam¬ ples, First visibility. Opposition Motion, mean, 26 ff, 190 f Ms Leiden Or. 199.3, 2°9 n Ms M. See Michael version Ms P. See Parma version Ms Par. Gr. 2425, 204 n, 208 n, 211 n, 217 n, 226 n, 230 n, 236 n, 237 n, 239 n Muhammad b. cAli, 185 Muharram, 17, 22, 23, 186, 188 Mumtahan zij, 9, 63, 200, 244, 245 Muqaffac, 150, 243, 245, 246 Mural quadrant, 64 Muthanna, vii, viii, 5, 8, 9, 11, 15, 150, 185, 188, 197, 199, 206, 243, 244, 246 Nadir, N., 208 n Nallino, C. A., 196 n, 246 n. See also Battani

Naru, 175, 199 Nativity, radix of, 143 Neugebauer, O., vii, viii, 5 n, 186 n, 189 n, 191 n, 193, 195, 197 n, 198 n, 204 n, 205 n, 206 n, 210 n, 214 n, 216, 218 n, 226 n, 227 n, 230 n, 231 n, 2360, 237 n, 238 n, 243 n, 245n Neugebauer-Schmidt, 205 n Newminster, 206 New moon, 23 n, 216; day of, 97, 218. See also First visibility, Moon Node, 26, 27, 29, 106, 109, 190, 191. See also Latitude, Shadow Nota, 17, 186 Oblique ascension. See Ascension, oblique Obliquity, 7, 63, 200, 206 Opposition, 94 ff, 216 ff Optics, 37, 98, 194 Ormazd, 19 Ortive amplitude, 102, 224 P. See Parma version Pancha Tantra, 243 Pancha-siddhantika, 222 n, 234 n Parallax, 212, 220 f; displacement, 98; in latitude, 126 ff, 237 f; in longitude, 121, 236 f; solar, 120 Parma version, 8-11, 12, 147 ff, 198 f, 243 ff Pehlevi, 243 Perigee, 29. See also Apogee Persian calendar. See Chronology, Yazdegird Persians, 19, 39, 188. See also Yazde¬ gird Planetary Hypotheses, 215 Planets: argument of motion, 43; bonds, 46; computation of long¬ itude, 32 ff, 190 ff; inclinations,

256

INDEX

214; latitude, 92 ff, 213 ff; max¬ imum equations, 44, 197 Plato, 234 n Precession, 244 Proclus, 229 Ptolemy, 6, 18, 19, 49, 51, 63, 65, 75> 87* 93, 101, 123, 149, 186, 187, 191, 200, 201, 210 n, 231, 239, 244. See also Almagest Qifti, 191, 244 n Quadrant, mural, 64, 200 Qurra, Thabit ibn, 245, 246 Rabic, 23, 188 Rabin, C., 11 n Radius, unit, 7. See also Diameter Radix, 17, 26, 27, 186, 191; for plan¬ ets, 49, 198; of latitude, 93, 213; of nativity, 143; of the sun, 28 Raqqa, 246 Revolution, excess of, 143-44, 242 Right ascension. See Ascension, right Rising time, 223. See also Ascension Roman calendar, 19, 25, 187, 189. See also Chronology Sachs, A., viii Safah, 147, 243 Safar, 23, 188 Sage, the Hindu, 245. See also Kanka Sagitta, 135, 137, 239 f Sa'id al-Andalusi, 244, 245. See also Blachere Sanctallensis. See Hugo Saturn, 42, 43, 44, 94, 197, 214 f Sayili, A., 200 n, 245 n, 246 n Seasonal hours. See Hours, seasonal Section, 44, 197 Sector, 136 Seleucid era, 6, 186 f. See also Chronology

Sengupta, P. C., 190 n, 193, 19711, 198 n, 200, 215 n, 218 n, 220 n, 222 n, 223 n, 223, 226 n, 227 n, 234 n. See also Khandakhayaka Setting time, 223 Sexagesimal fractions, 7 Shadow, size of, 104 ff, 109, 111, 116, 227, 229 f. See also Gno¬ mon Sharah, Ya'qub ibn, 148, 243 Shiraz, 245 Shvat, 19, 21 Sindhind, 4, 25, 152, 189, 243 Sine, 31, 42, 49 ff, 200; of midheav¬ en, 97 Sizes, of sun, moon, and shadow, 104 ff Smith, D. E., 9 n Soghdian calendar, 19, 187 S-S. See Surya-siddhanta Station, 45, 198 Steinschneider, M., vii, 9, 11 n,

*94 n> 243> 244 Sufi, 150, 245 Sun: apogee of, 245 f; correction for, 30 ff, 197; diameter, 104 ff, 135, 226, 229 f, 238; radix of, 28, 191 n; sphere of apogee, 28; velocity, 94, 104 ff, 216. See also Conjunction, Eclipses, Exam¬ ples, Opposition Surya-siddhanta [S-S], 190, 198 f, 205 n, 214 n, 233 n. See also Burgess Suter, H., 5 n, 185 n, 186 n, 188 n, 189 n, 190, 191 n, 197 n, 215 n, 216, 226 n, 227 n, 231 n, 232 n, 244 n, 245 n Syzygy, 216 ff. See also Conjunction, Opposition Tables: ascensions, 73, 75; begin¬ nings of months, 23; diminution

INDEX

Tables (continued) of rising times, 78; mean mo¬ tions, 26; planets, 42; sines, 49 ft; stations, 45 Janah, 49, 199 Tara, 199 Tariq, Ya'qub ibn, 199, 243 Thabit b. Qurra, 150, 245, 246 Theon of Alexandria, 150, 245 Thibaut, G. See Pancha-siddhantika Tishrin, 21 Transcription of Hebrew letters, 14 Transmission, Hindu astronomy to Islam, 3-6 Trigonometry, 7, 49 ff, 200 ff Twersky, I., viii Ujjain, 191. See also Arin Usfuwana, 163 Varaha Mihira, 190 Velocity, 94, 03, r39n, 216 Venus, 42, 43, 44, 94, 197, 214 f Versine, 66, 82, 83, 201, 208

Waerden, B. L. van der, 3 n Wolfson, H. A., viii

Yahya ibn Abi Man$ur, 150, 245, 246 Yamakofi, 199 Yazdegird, 6, 18, 25, 27, 49, 148, 186, 187 ff, 198, 244 Year: Arab, 22; Egyptian, 6; inter¬ calary, 19; lunar, 16; of Ele¬ phant, 24; of nativity, 242; Per¬ sian, 22, 191; Roman, 19; sidereal, 187 n, 242; solar, 19, 143; world, 242. See also Chronology, Month Yojana, 228 Yunis, 245, 246

Zarqal, 150, 244-46 Zarqallu. See Azarchiel, Zarqal Zij, 4, 5, 9, 109, 150, 186; -i Shah,

3n Zodiac, sphere of, 30

258

Hebrew Texts

mats? neon

451|

J

pea

UDpDVK p« •’TnttVx MnaVK

mr» p« ,’2N3nD!?«

•’D'l^N nKssbK

•’KmaaoKn ]«n mp p nan 'aVn

Meton, 2r:i9 Ibn al-Muqaffac, 2V:3 Al-Marwadhi, 2V:2 See: ^nabK |a ~tnm See: mrs? pK amaN Al-Farghani, it:9, 2r:27 2V:22, 3r:i,3, nT:24 Al-Sufi, 2V:3 Al-Safah, ir:4 Theon of Alexandria, 2 :4 Thabit ibn Qurra, 2V:3 See: ovaVaa

260

Index of Personal Names in the Parma Version

srx amax □rnax

Abraham Ibn Ezra, ir:i, 2rp Abraham (Ibn Ezra), 4V: 11, ioT:i Hipparchus, 2r: 18,20

patJp^x •’•'nOXbX pX

Euctemon, 2r:ig Ibn al-’Istiji, 2Y4

DT^plX

Euclid, 6r:22, 7r:8

WSDXbX noxVx □*?yxbx pX OVaboa

Al-Akhfash, 2y:i8

"l2n31?X •’T1DD bxp"ITbX D’mns •’3“is?n wan •arxn a’anbx nxnrx Dan nw

•man 'ax p •nrabx aipsr (pnxto ms^ p aipsr pTivan aipsr

aipsr* H33D ,,amabx ’’Oia p Tana •’inabx p 7ana oxaybx p nana

Al-Asah, [al-rishi, the Hindu sage (?)], 3r:n Ibn al-Adam, 2V:5 Ptolemy, iv:n,i8 et passim Al-Battam, 2Y4 Ibrahim al-Zarqal, 2V:4 Habash, 2V:2 The Hindu sage, 2Y3 The Hindu sage, 2r: 5 Yazdegird, 1Y4, 3V:23, 4r:3, nYiq Yahya ibn Abi Mansur, 2Y2 Ya'qub al-Kindi, 2V:3 Ya'qub ibn Tariq, ir:23 Ya'qub the translator, iv:2 Yacqub, 1Y17 Kanka, ir:2i, 2r:5 Muhammad b. Musa al-Khwarizmi, iT:5> passim Muhammad ben al-Muthanna, iv:22, 2Y:6 Muhammad b. al-'Abbas, the computer, I2r:6

261

Index of Personal Names in the Michael Version

DT’VpiK p nanx ora^wa ot’&Vm 'bun

patwin •’bya naNban 'Vya ]DpH XINT p pnpbin larfrs ^an mir 'anKia^K p»K» p ns^Va oxay p ■^y p tana pnaaVit nann •’ID&bN ’MUnfiVK a'Bnpn n'annnn

Euclid, 5or:n,32, 63^:25, 6-jt:22 Ahmad b. al-Muthanna, 45r:i Al-Akhfash, 45r:y Ptolemy, 53r:2o Ptolemy, 46r:2, et passim Masters of (astrological) judgments, 52r:28, 30, 6ir.-4 Masters of computation, 53r:i8, 65r:i8,23 Masters of the discipline, 69^2 Darius the Lesser, 47r:i8 Dhu’l-Qarnain, 45v: 19, et passim The Hindus, 47r:i9> et passim Habash, 66r:2o Yazdegird, 45^: 15, et passim Al-Khwarizmi, 45r:i, et passim Al-Ma’mun, 55r:22 Muhammad b. 'Abbas, 52v:28, 55r:i3 Muhammad b. 'All b. Ismadl, 45r:i The scholar of the Mumtahan zij, 55r:2i See: nnttbN ]3 nans Al-Farghani, 45^9,15, 52^:25, 59r:n,i4 The ancients, 66r:6 The ancients, 65*: 12

262

PARMA

D,l?Ewn

unit? D'&'lKn

nsoin ppna rrhan 'Vn maian nsnan naian nsnan avian ppn pnan ppn man ppn

GLOSSARY

II 142

Inner planets, 6V:5, et passim Radix, 3v:i, et passim; square root, iv:6 Gemini, 2r:2i, et passim Increment, nr:n, et passim Maximum equation, 9^23, et passim Node, 2r:7, et passim Phenomena, 7r:26 Motion, ioT:i4, et passim True Motion, 4v:2o Arrangement of the Houses, ir:24 Equation of center, 5r:9, et passim Equation of anomaly, 5r:9, et passim

263

nirttn Jupiter, gr:g, et passim

mis mis »*?s ninp pan naip V nman ip ip aaion ip nbuyn ip mt^n ip 31")p manp 21*17 "P"72 -nt^p iwp Pnn mi pn trm a^yaa nT>sra*i

»n ppn an arm a^ain

Constellation, iT:i5, 2r:io,i2 Figure, 4v:i7> et passim Side (of a triangle), 6v:2t> Eclipse, ir:25 Dome of the earth, nv:2i Line, 4r:i5, et passim Line of sight, 5r.‘3 Ray, 6r:2o, et passim Circumference, i2r:i6 Circumference, i2v:20 Equator, 1r: 17, et passim Close, 7v:i, et passim Closeness, nv:io Approximately, 6r:3 Bond, iot:i8, et passim Arc, 5r:6, et passim Ascending node, 2T:rj, et passim Ascending node, 4v:io Fourths, 3r:24, et passim Quadratures, 5r:23 Moment, 7V:20 Moment of correction, 9v:i7 Latitude, 4^7, et passim Arrange an astronomical table, 9r: 2 2, et passim

nns ati> Tia^

natr

Retrograde, ioT:i5,25 Saturn, 9r:9, et passim Area, 13^: 5 Thirds, 3r:23, et passim

nman •’latr-

Optics, 7r:8 Seconds, 3r:23, et passim

1XV

bxau> wm ni’istr-

Chapter, 3T24 North, 4v:i4 Sun, 4r:io, et passim Perigee, 3r: 13, et passim

264

PARMA

GLOSSARY

unw »r*2na iysa

Take the square root, iv:6

nanan mpa mana mpa pyn mpa myiarpa savia

Place of Observation, nT:22

Rising time, ir:2q, et passim Node, 3r: 13, et passim Place of the observer, 7r:i6 Angles, i2r:i3 Square, 13^2

ama

Latitude, 4^:8, et passim

pma aana

Distance, 6V:6, et passim Attached, qv:i6 Triangle, 13V: 1

D'tasiya ]rby niM pina rm nna wna n^aa man

o^naa rmpa 113 naaobK ■’a*' naaoVn nbiay nay T™ maiyn *]na •pyna m'Tpy ]ipnn ny hnd hrd nine bsa naa prana

Astrological judgments, iov:24, nr:z Outer planet, 6^4, et passim Corrected, equated, 3r:2i, et passim Venus, 9r:6, et passim Node, 4v:n Declination, i2r:i, et passim Inclination, ir:24, et passim Parallel, i2v:i9 Point, 4v:i, et passim Meru, nT:25 Days of the Sindhind, 3T:2,3 Sindhind, 3r:i2 Circle, 4r:i2, et passim Perpendicular, i2v:i4, et passim Ratio, 5r:io, et passim Form a proportion, ior:i3 Form a proportion, ior:iy, et passim Tenths, 3r:24, et passim Time of correction, 6V:6, et passim Maximum latitude of deferent, 4v:i4 Direction, nr:22, et passim Less,

6T:'j, et passim

Divide, i2v:i,2o, et passim Separated, iov:n, et passim Perigee, 4r: 3

265

o'Va na'tzn

II *39

in' ptwnn tid:)

D'31'Vy

First magnitude (of stars), iv:i4

D31D

Planet, ioy:6, et passim

3D1D

Mercury, 9r:6, et passim

D'DDID

Fixed stars, ir:24, iy:i2

•vna

Sphere, 7L9

m

Power, i3v:i

]naan 'Vd naan hV'Vd

Astrolabe, nr:8 Kalila wa-Dimma, ir:io

mni

Kardaja, 8y:20, i2r:n

n“D naaV mmV

Intersect, 4r:i4 Tables, zij, 3r:io, et passim

D'lKa

Mars, 9r:6, et passim

Moon, 3v:i7

ninaa

Luminaries, 5L5, et passim

D'2TKa

Libra, ir:i7, et passim

t^piDE ]rna

nrvna 'i?22EN

Chord, sine, i2r: 1,7,10, et passim

Unknown, ior:i6 Astrolabe, 2r 17,8,17,20, nr:8 Speed, 9r:i5

iVna

Mean motion, iy:i, et passim

pma tzna

Center, 4r:8, et passim

m*?Ta

roVran *pn Vy mVran yn *]snV n-ira “>aina Tima atrnan

“in'a nVya nVya nnaiy nVya naya n'aaiya D“iya

Tangent, 6r:2o, et passim Zodiacal signs, 3V:6, et passim In the direction of increasing signs, ioy:7 Opposite the direction of signs, ioy:9 East, ny:24 Conjunction, 5r: 23, et passim Cycle, period, 3y: 3,4,5, et passim The computer, i2r:6 Chord, sine, 5r:6, et passim Magnitude of a fixed star, iy: 13 Degree, 3r:23, et passim Ascendant, 1L24 Station, ior:22, et passim Nebulae, iy:i6 West, ny:24

266

PARMA

-inmVtf pp ViVa nspnn mmin rnhr&n ntn: bw&sn 'yfry,

GLOSSARY

Node, 4v:4,io Epicycle, 4^: 15, et passim Epicycle, 5v:2o, et passim Circle of Apogee, 4r:9, et passim Zodiac, 4L9, et passim Inclined sphere, 4A5, et passim Parecliptic, 4V:6, et passim Cylinder,

rjr\c)

Geometry, iv:2i ]V?2T

marm pt^n nn-'bn rma rv’iT 33T

T’lnVx nnx mm rv’mnx -inn m’pran •nnbN •’asn nrmsbR 'aan mmn naan own nann a-'pVn pnon na-rin na^nn *nt> nba

Figure, 4r:i2, et passim Phenomena, nv:9, et passim Tempus, iy:2 Motion, 9r: 15 Right angle, 7r:i2, et passim Descending node, 2L7 Perigee, 4V:3 Retrograde, nr:6,i3 Retrograde, iov:2 Scholars of the constellations, iov:i7 Scholars of India, i3r:2,6, et passim Sabean scholars, 4L7 Geometry, iv:20, 2r:i Chronology, iv:3 Minutes, 3r:23, et passim Difference, diminution, ioT:4, et passim Section, kardaja, 8V:20, et passim Arc, 6r:i7, et passim Row, 4r:i Aries, ir:i7, et passim

ds?d

Reason, cause, ir:25, et passim

Ka^

Sine, i2v:6,io, et passim

nNn&av pa** *w

I'pnaa ir ■w^nn

Geometry, iv:20 South, 4D14 Equal, 11^25, et passim Direct motion, xor:23 Direct motion, nr:6,i4

267

o^Va nmun

^ll nw n»» nirmsa nnsixn •pxn room nVmnn

Equal hour, 59r:25, et passim Unequal hours, 6oy:23, et passim Gemini, 48r:2, et passim Era, 45v:i4, et passim Increment, 5ir:io, et passim Trick, computational device, 57r:i2, et pas sim

namn m’mn *'1?n namn nsnan mns1? nsnan ppn D’nwn mmwn taVwn pwn

Astronomy, 52r:26 Maximum,

^oT\r], et passim

Node, 47t:6, et passim Phase, 52r:26, 28 Motion, 48r:3o, et passim Retrograde motion, 52':21, et passim Correction, equation, 48r:4, et passim Geometrical, 52T:28 Geometry, 45r: 13, 19, 57r:i2, 581*: 15 Complement, 6T:25, et passim Tishri, 45U29, et passim

Glossary to the Parma Version

m qVrwa mnx nxna-’x

Apogee, 4V:2, et passim Retrograde motion, ior:23, et passim India, Hindu, ir: 5,8, iv:i Diameter, 4r:i3, et passim

•’ttOattbX

Almagest, iv: 18,25 Cylinder, 7r:io

•yrm ynx pK nrrD pwan mma

Longitude, 8r:25, nv:22 Long, 7r:i4, et passim Arin (Ujjain), ir:i7,i9, nv:25 Place of observation, nv:24,25 Corrected apogee, 7V: 11, et passim Apogee, 2r:2i, et passim

268

MICHAEL

*lDj? *l»p nwp n^p mn mxi pn mi srm

|| 136 | [

GLOSSARY

Diameter, 48r:n, et passim Node, 48r:24, et passim Arc, 48^5, et passim Sagittarius, 6ov:2, et passim Appearance of the moon, 6ir:6 Ascending node, 48r:24, et passim Rabh, 47r: 3 Fourths, 47T22, et passim

»n

an an am mn am wawn pw na am mx-ib mm

Moment, 52v:i7 Maximum, 5a laym naai mannn riaVzzm1? Ka^VK nmzr mV? annaK (2) DDX .maapn maVam mtz? naa na V? Vaa nazzan ntzwa mm (3) .mV nr nanmtzr mirVzz mum mainaa naV mm vana m a*>pVn 7aa *TX ntz>p pVmi ima^t? amioaVK 'zv mmai TQX nViav amp mmai xn riT n ■’pVn on mVva 'V ana p1?n Va mmtz? m? nme? (4) nw mm nvnm am jvfijVsn mm natz> mV tzmnrv’ mm DD HT DT DT (5) *rnK Vaai HDT wVwai DHT tzrVitzra am d'&Vwb amw nm nawai DD (6) DT »Van poaVx mn mntz> TH amnmai D D ivpit am nasa rmr ana (7) mn poaVnn nmmn DT »Vxi HD aVzz iaa mm'm nizwa mVsra 7o Ka” (8) mnw DT ww nsnV t2?paa do) ntz?Kai .DH iaa mm mVs?a ,!? Ka^ (9) naiV nsn (12) iaaa lanon mm DT ip iaa DH ip m (11) mV»a '0 xa^ X

Fig. P-9

nVa xar» mntzr mVaa 'V poaVan (15) mna mVaa 'V i^sm mm a-aiKn maa Vta Va pVm (17) mannn

|| '*3a Vi iidi1? nsn 12

(13) xa-* mm* ]ioaV*n rraran poaVxn Vaa Ka^ nom ]a nnm naaai (14) mrz> Ka^i naaai (ie) laanaai XD ip Kim Ka” Kin nKtrnrn Ka” nsnV tzrpaa ntzazai .nnzzn ita Ka^ 'z

"33 or 103 9 || "33 on mr s || "33 on na :ao nr 5 •’ 33 nisnnn iinn is

278

||

’’33 if3 is ||

’”33

tn’DiNn is

=13!

II 126 II

PARMA VERSION

ipi (22) nfrya 'V *77 ntyp p *?y nf?ya '*? n ntypi *71 ntyp R'm mVya (23) '0 “in*1 'an ana nnR *?a *?d ip iaa DT ipi m*?»» '0 Ra" .nbiaa (24) am 'an nVya ^a R3" 'a la'tyyty nRtn naiaa aaann nam naVi (25) a'pVn yy r3"Vr iaty naVi a'lpn nVR my'ty aya HQ a'amV mntypn pa pR d) 'a “1QTXT mRaa poaVRn my'ty inp*? r1? naa'ty ■•a nam ama (2) poaVRn la'ty'ty patynn 'Vya pm r*?i pay aitya *py a"p imaty (3) annRi nan'iR 'aan nyn ^y a'p^n niRa 'a poaVRn aty pT xb ^a (4) "im ir mns laity*? *?ia' naa'ty 'ai ovaVa nyn *?y mVya ma'ty' (5) a'amm laaym a'aioa'iRna anm Raia 'a ana r3"Vr a'ty'ty 'aam (6) naarn patyn nrRa poa*?Rn am' aR iap'r xb nam mntypn iVr Rasin’? (7) a'aa a'p^n iV tm aaoan nr 'a maya apVn niRa a imaty nan^R aaai (8) a~p poaVRn iaty orabaai aaya 'aam a'aVty amnRa a'aayn •o (9) nny nr tyasai nnmty am iaa nViay Va poa*?R *»asn *»a laasoa ''a paabR ,,an iaa apaian mty pa tmty na nam atyRa nmy mVuy amty b'xb nainan naiaa nRan atyRa mty'Vtyn nmana “pasn pa nnR Va nam (10) maty m'nnty ny mViay amty poaVR (ii) 'an HD ip amity nt ‘YlX'O a'pVn a"'a nViiyn hrt pVnai m pmaa (12) nViay 2 paia *?y ntyyn anRi XO 02 10 O1? *72 22 On m (13) TH ITT 71 IX ybn am a'rn nViay laa rDl n’puy 07 pmaa (14) ntyya aai paia n mipa a^a *ja nnR Va Vaa^tya man' nam (15) nnR ip ampaia -a» pma 'a Pi IT “THX nViay n'lmVty PiU (16) ntypi naaa mnRa *?aan n^Ra nmana ana 'a my “IQ1X1 (17) n'VRatyn nViay mtm^ty nQH ntyp ]a *?y maam nn rn mp R'aiaty (18) m *riT3 mw an' laa ana nnR ’ra poaVR 'an mViay a'nty poa^R 'an (19) laa ana nnR ^ai ipa ana nnR ^a 'a nViayn 'pVna a'pbn (20) amu^ ip*7nn' a'ipna nnR *?a a'lty a'lpn nam 'an laa n’may ba (21) mtyty nm p 'py nbuyn mtw a'pVnn amtym naioaVR 'an (22) laa nViay Va mtyty nm 'a mannty nnRi naioaVR mVya ,5? (23) Ra" n'm n*?iDa am 'an laa nVya Va Ra" 'a aaanm mm (24) i^a Ra^R nan mVya 'a an' 'an Rin 'a nViayn iioaVR my'aa ]iDaVRn (25) n'y'aa Rim yin' nViayn |a m*?ya 'b R3"i iioa^R 'an laa ym'

.’"33 IDtP 10*?1 24 || ’*33

.’"33 N3”D :«3”1 21 || ’*33 n H 3

14 ||

3

Hllp: 13 || D’l71»3 3in3

br

n»pi 22

iV’pV . . . *1®«3 10

«13 D

naiKa noia

m ana nnK Va DNi n®p rrnn nam X nmpa *?» anran m®1? DX“J n®p (3) mipaa Kmia nam X“T n®p run1? ®paai n^iayn myan an® m*?ya (4) ■’ana? p®Va Knpan inn m nm mn Kim HX mar mm DP poaVK may (5) mayai nm mn mn xy anmm mn®pn nV?a msaan in'’ mn Ka^ (6) THk may DX n®pa amsrn amayna D"T pcaTm *?y Via-1 kV® (7) rrm nViayn psia Km n nmpa m m va" mm ks^Vk pio mn m PIN ]a (8) mVya D"p am D“7 poaVm nViayn poaVK mn DPI npn DPI ip laa PIN ip (9) nViaa am mn nVya by xa^ p by mVs?a a~p am mn Kin m xa” nam do) mnn nam a^pVn ra by IX n®p lapVn niVya 'b Ka^ nyf? ®paa n®xa (id T mipaa Kmia p anm XPi PIT T"7 •’pVn am mVya 'b ana n®p by (12) mVya a^®® Ka^ ®paa n®xai nV?ya 'V Ka^ DT ip mmi DT miTaa? (13) am nV?ya 7o Ka^ DJI ip mmi Dn mm may n mipaa Kmia (14) poa^K bx aaay nai amman nVxa Ka^ ^a aiyts? naa nyf? ia®pa (15) mn n*?ya by (17) xa” m laama aaai ana xa*’'’ by nx® Kmiai (16) Kmm DT ip mn® (18) mVya 'b Ka^ my® naa r\yib ia®pai nViaa nm mayai n®pa *? nmpa (19) by Vian “D n®p *?x n®* *]*vt ^y DT may laKKin n®xai (20) a^naa am m ip ^y amay a”a® DTi PIN np m n n®p laa (21) n n®pa l?a£D*’ nr nViayn ip yaa' n®x n®r *?y DT

.'‘32

Of 19

||

’’33

-lb

”73 18

||

’’33

N’XIJI

280

l’niflPp

yiy

313^3

’3 16 ||

’’33

3

PARMA VERSION

I 124 ||

vnim1? w aaK (20) aa?Ki 'm aa?Ka aamar maVaa pa?Ka nr Kin w k1? HDD aaK k1? (21) iVki [?dd in pK aiana am mnVK iso mm1? on mpa (22) Kin ayan pxn naip1? an m amrn aa?Ki ia BTaK» irm mVya naian p iaax m ay&m pis i1? pxa? aipan Kim mman Kin awn Va *>a pK1? myaK aipa Kin w pKn naip aytn'X aayan pi (23) rmasVK ^ Tim aipa iniKa mn ann’iK nrna p Vy aaya1? mraa (24)

.pax nnK pa^a xap' •’a rn xap1 nrnan aipa •’a (25) [(135]

mann1? insoa wna p aana anaa? mm1? (l) a*wn nVx Pitt K^xm1?

iaaoa

mit^n

aiy

my

na*?i mmam

(2)

amn

Ni2 a

aya nai mnjam amn

.anaa? mmVn pa *?y (3) mma? nanK pa Vy mmnm amn

■wami naan amaKa aaa

(4) ovaVaa nya Vy mmam amn •’a ya

nrxa am*?y mxam nbp paai nax (5) paa K-sm*? aaK Via' pxi aiwa oxayVx ]a aana Vx manaa? (6) naixa m Vai maan maixa maiaa ayam

oraVaa

nya

Vy

(7)

m'twm

amn

mwb

nxian

nam

.aa?nan

nnmn •»xm (8) amma paa?nn maa x'xm ap nViwn poaVx aw .manVx nya (9) *?y awan aata nrn aaoa naann nmxa 'm aa?xa na?y mann (10) a?a?V mm1? na?a? an aana anaa? mm1? ama?n i*?x **3 aaiKi aaoa (11) 'm aaai VjV’Wi manna mann a?a?V ay»m amn 'xn *po ana? amn 'xn (12) *po aa? na1? ayai naaaa ixaps naVi mannn aya na nr nya1? (13) a?paaa? mVyana xxim aaiyn Kin amn 'xn ’’a mann ana?

(14) myispan wa? rmi nVwyn poaVx py1? nVmyna

a?a? aw

aam mn

nVnyn (15) ipa a'xxrn amiaya -paxm amxa poaVxn *?y aiayn mxa iVm (16) amiayn f?x m nVuyn nryaa rnnpia xx'a? na Kin mioaVx Vx aaai (17) aaian ipna aiaaa Vis'a? npman rmpan xmi mVmm pyia *?y nVuy

(18) myaa mm nVya

era? ’py aViy1? mpVnnn nV?iw?n *o 7m

iyn rjio nan*’ (19) a?aa?n ppna na?y aa?Ka mVya r7a *?y ry p’pm aa?Kai 'y mm1? naiy (20) na?yn

]did *?k ymn anm mn m mann a?a; awa amn

mayn 3*7 poa^K Kmir n (22) naipi npyiai “THK nVwy (21) na?yn pma ama?*’ amyn awa?1? (25) nVwyn Vys’ n*?iw?n (24) poa^Ki n'piayn pyia Vy (23) n*?nyn nKT poaVK Vys'’ aa?Ka mm o/7a? *?y mpVina m*?myn •’a'm aaai (l) my pVnm .mVya s/7p ana na?p Va nmn nna? mna?p ama?a nnVixy

(2)

25 || ’’33 -1BD m]nV« 20 || ’’33 -INT’-irN 19

.’’33 V1«

.’•33 th-in3 :nn-i3 n || ’’33 3*r :3'p

281

7 || ’‘33

py1™

8

||

’’33

-W’?1

:_ln,n

1

^2 a

II123 II

naaxn non

Kip"1 (7) tx atm ip Vy nViy xw i^xn n»T D nnipa nxn *?x 3 nmpaa aaian nayan (8) aiytya aaian pai tyatyn pa mmiy na pnaa mm tx ntyn nayan

psian nxsn *7 nmpa (9) nxs1? n nmpa nxna yyiam atyxai nana warmy mtyn i:>Vna 'aa (io) manya maran *?y nmx mxia iana« nam nnx nxnn yaa (ii) tnptyr mr x1? aaiam imanp ix tyatyna lpnaai ppn VaVaa nf?yan mVy pVnmy aaian naaa amn atyxi nxrn naixa atyxn nnx X

-w xwty ix mamnx aaian nma** nynV napinan naam nayan pa nmy (12) nr i^naa laaonn atyxa 'ysaxn naiyan nr *f?na xinty tymsa mt (13) ■jVna anna xin n^Vstya pa marby nmy nvntyaa nn •’ysaxn tyatyn (14) nm1? nan mtyyV mm *,y^axn natyan nr "j^naa •’ysaxn tyatyn nr (15) amnn pa pnaan -a tyatyn pi aaian pa mpnaan -aw Vy piVnn (16) layna am Vy apaVn nrnV typn nam napinan naan xm (17) tyatym .mya laintyna (is) mm xVty ay 037Dn (19) .•’tm'rty nr ,,sn |a tymtyn nmatyan mmVa nty pjfr1?

282

1"D

122

PARMA VERSION

m^Tan taipaa (22) amp?: vm lanamw mavann nnvna mnnwan n*?R mm nan (23) nyim napina waw.n n*?ya naai nanaw ny Van yim *?y (24) nnvna n'nnwan nam mapina rvfryana yim nma'aw pman 'DDu;a 'Vya lansinw nnR nan1? nana 'iaw Dn1? mm mavann iVr nynV lanxim onr ir Tins n'aty dr nnwan naya nynV mVtan (25) mywa nma'm nmnswa nvnV mapinan nVryana nma'aw pman d) «n a nnv Va^an many mynv manVR 'aan 'a p'an1? p'ana nnwan 'law (2) vn r^i VRyatm maVaa naann hrt noaaa ntyRai mansn 'aan *?aa (3) pma mm nwo DinaV an 'a n'rfjR 'nan nnaR m*?Tan 'DDwa n'ynr (4) naiy aaian mm tr pi *p mapinan nV?yana watyn pi nna>an pa (5) n'niwp wiRnpi nVapa nR'na'n 'aana nr i*?api ntm'nn1? ir mnR nm) (6) naa mn iVri nmwpn iVr 'a lawn mn nnR mn Rai mam ianR nwRai (7) 'Vaa mRin vri mnonna ir msoia mVyan vn' rV iawn nu>Ra (8) aaian pana ppm wawn ppn m»wa nV?yan n*?R Vy nsoin paan (9) hrt (id tjyanm n'non nmaw vm nwRa ir msoia nmaw vm nu>R3 (10) fpoinV (12) 'awn yma1? ppnn nr mm nwRa mmwpn ^ pnonm nooinn laa (13) napinan naan na"nn no>Ra 'a 'amaVR imam rV nnR nan um nayan (14) iaa n'nn nwRai mnR mm1? naiy nn^an mm pwRnn nayan vVy mVtan Va'ia ip (15) no>ya .nr Vy nms no>yai nw"nn*? naiy mm 'awn mnaan nViay T pana (16) Vy ip nwyai n nmpa pRn pasia ipaiiai “T222K ip nwyai DW1? mby ppn (17) VaVa n'aiay n psia *?» ip nwyai DDPi m^y nn *7 nmpa Vy nVnna aaian a'trai (18) KD27 vVy wawn VaVa pana *?y my *?y tr n'nn wawm nVyaa nVya wawn (19) ay ppn I'aaVa mnaaa mvna mn' mVran baVa -jnn (21) *?y ]apn iVa’aa mnaa ]a (20) aaian nyiam nVaVaa y nmpa aaian yyian' (23) nam nnR nRaa am^aVaa amaw (22) lyyiam nam tri nVaVaa D (25) nmpa *?« t»at»n yiam a nmpa (24) *?r ]apn i^aVaa nmpaa na1?^ aaian Vn' tri pwRnn nayaa laa amaa> (l) pa pnnan mm 2n a ]a Vy ntm ip ^y nnv Rin V?R3 pnRn ^y nRn'i 2 (2) nmpa nRD *?r 73 iaVna i^Ra nRn' 2 nmpa nRD Vr y'a'i (3) nn' n^Rai naiy iV naR' inn ‘ry a'aVna 'awn rmw (4) ppn i^a mnaaa mn'wa iaVna pn m 'a maya naa aniR (5) n'Rin ianaR na1? nyom .nnR nRDa mVian I1?' nwRai (6) nRa naap R'n 'a pnRa p'pa Rin na nRa pp Rin VaVan

.’’33

non’

IN O’BDU :iK O’DDU 10 .’•33 on’J® |’3 :on’]» 1

283

ii121 ii

nooKD noin

nVKo 'yam mm ppnn p (16) mVyon nVKV wv no mranm amnyKo omnyno onty amsoo (17) yaoK Vo m typiaon Kim yim mam onaoon "yaon iV kit tk (is) ptyKnn Vy nViyn pVimi mrana puwnn Voa1’ ax mannn mpna (19) ioVty KVty mannn pVn VioaV ms p Vy yim in^Kty omotyoV (20) naa nm* nann onty niKO '» Vy ipVimty nisi aa onty amityKo1? (2D ammnty nnK nann mVyo an vVy pVimty mss? no Va ntyonn .VaV nnK *pm

mmV ay msoa (22) mio p nono atyty mmVn nVK nfa [Q34\ T'*’ ntm ax nynV ntypa ax ooki aaia VaV (23) ptyKnn noyoa onty ooki amipnn mitna rma Ksonty no npi ]pmon psioa (24) onan mnKaty ik laVnoa noyon mn oKtyam jiiVto a^o ion Vam mVyo (25) mVto ptyKnn noyon noyon Kin mtyn '••a annaK pftX .matyn man Vk (i) nKn p *inK mtyn 3ioa mtyn noyono nmnai ptyKnn noyon ]o (2) mv nmm aK mm mtyn *iaa T]2WD .nty Kin naa mm kV (3) bki mmnK mm Kin nntyon •’ysoKn tyotyn iVno pnon (4) omrVyn omntyoa •’a Km no mon 'nm Warn rnsoKn tyotyn *]Vno moo (5) noimty ihk 'ysoKn aaian pnoo my /u on *piK3 tyotym (6) aaian pa w no pnno Kin hok Kino? mnann p ppn iVnVna (7) VnVnno nntyon mi nopnn ■’VaVa amniyoV m nKT ppn iVaVao (8) p'Vy.n mna iVaVanna nntyon mm mVron *pn Vy inyian nKnn Vatyn (9) mna wvnai mVron yn Vy tyotyn nKD Vk iaVno pnnon mm *0 do) mKi Kin p Vy niVron yn pnV pno Kin m •pnV nntyom (ii) nntyo VaV ptyKnn noyon nsn»a tyotyn pai aaian ya tmty iVko (12) nKT> tk tyotyno nnaaty yano ]Opn iVaVa myan myty *]Vn naa o'Ksim (13) ompn 'a maya noiy iV noyi mty.n ipa mnn inyian m noiy Kin VaVa (14) maa nVKO nVK ommp rm mVron VnVn Vk psiono aaian Vk mVton (15) *]m inyian mnn Votyn mnn Vk laVnoa yyiam ntyKai mVron ppn (16) iVaVao Vetyn •’snn laVnoa a^ty ntyKai mnK ao> Kin iVkb nKTi pnno (17) iV mm an noiy nViy iV noK" tk nnK ip Vy nViy Kin iVkb nKn' pKty (18) mVron man iao>n naai iVty mmVa nono nminty Kim tyotyno ntyKa (19) amityp anV vr mi omityp amntyon m a'VnVnn noana nyn onV ipna’ (20) tk ano tyotyn pnnn ntyKai nVnyn Vk monam omion intyp' m npym (21) pynty man an ]a noiKm mVK mtypa oaotyn moo an

.’’aa pmon noyon 24 || ’*00 o’anyno 17

284

PARMA VERSION

Kxrn apn •wro PPn aman nvna mm Vanin ]ipn nVR rrr mm naRan mwa aaaan paaa Vy naay ppn V1V1V waa^w pxaana •>xnn mm anw na^Rw ap^n *,at2?a ppn V1V1 nVaiy panna a’ap •wn aVaVaa aaaan pma *0 nay 'wma naaa Vawn 'anna Vani p'Vyn naVran nVRa aamp pna aaaa wawn pmaa ananai aapaa ppn •*a a^nin ampnn anaaaw naVyan an aaaa VaV mamw naVyana amarnw naan iw Ram ppn aVaVia p’Vyn 'anna Vawn ^nn nsnw naVaiy Va ^a ppnn yraa aaaa wawn pn*aa aaa mm nt ai aaaa VaV .naVya o"w Vy napVnna naVanam naaapn

II 120 II (io> (id (12) (13) (14) (15) (16) (17) (18)

amwV namV nwyw a*»mpnn R^aanV amatnw (19) amsoan nVR HQ [Q32\ D"*1 “lonn*’^ mxw naoan Rim aaaa Va naamn namVa a^mam (20) a,|’’w,’,?w'?a nanaan aapaa nRtmw na *aoam p nnRa ppn Vaa paVna nVy^w (21) na aaaa mVR pax Ranw panan nanaan aapa “iRX2?an mma a^Vy (22) laa'cr’ ir aVayV *0 ppna mVan aaaa VaV 'a mn nsoa *»n«nD naa (23) PDIttfr) nomw Vm tr aw amana mVann nnaRV yrnw ay aaaan ppn fT,ov (24) naa ppn *po nam .ppn aaaaV mm rV tr naVaa 0 ay aaoan man? ay (25) aaaa ppn *poa anawRin apVn nnRa B’wVwa naRa naawa d) a^aVR «io D a^T^Vt^a naRa ya-iRa a-^VR a^aRa ppn *poa apVn (2) a'ywna amRaa «f?R ppn *poa apVn a^wam a^awa naRa ww pax (3) ppn *poa apVn nnRa traina am ppn Va *po an ansoan nVRa (4) apVn 'aa naRa a maty mm® apn ay aaaan nrna wawn i^aa (5) aaaan pa ww pnaan aaaa aaayaa ama'ama aaaa aaana naannn (6) paVna r:tw na mmi ppn ViVaV □VayV )apnn ^ anaoan n^Ra )apnn (7) ]a RR'tf na aaomty nax nT .nomiy Vm aw manaa (8) naoann ^ao *?r mrw ny *pcm naann ppn r-w "iwRa (9) naxw naRa ywnn n*?R na [Q33\ 'm n^R 'a ayan mi^n .naanan (10) amVy pVmw maVna wawn apbn naRa 7a ama anman mna na*?ya a7/a (li) anw nnR nann an naRa Rbw mnnR anw n^yan pya naapn Va ^r nann Va (12) py ^a yan' ’a nnRn a^anyn p7a a^nsoa 'i n^na aiapn *?r (13) na^^w nann1? ay^in nnonn nannn Rnp'a naapn “wm apVn (14) p^nn1? na^wnw nannn apVnV aar’-ia^w naxw ana na^w (15) nann1? ay^n rVw na^yan ^w'Vwna

,i«33 i«ni :«im 20 || >’33'd »r 13 || ’’33 d’:p3 :»W3 12

285

X''

II 1X9

nanKD non

iann tmmtn (7) orntyan 'n pn mnaan pnnaa nVya m naaa Kin ana p-^yn mna (8) ontya nan aaiai DmKa ppna ’a mn» *pn *?» ^ran iann ppn (9) tofaa *?Btyn mnai mann naiatya ’ran iann ppn VaVaa Vs Vra do) Va iann pnm maty *jk nryan 'on mann nV Via Va iatyi VDtyn (id mna ik ppn ’r’rana p'Vyn mna rmty pa mann yanK ppn iVa^a (12) nty mmi nVyan naaa laaa ’mo, nnr pKty Vmn ppnn mm1? *poa (13) nr lanai ppn Voton n’nay *?k mnan lpn taiary Dipaa mmty no? ppna (14) n’mai nVya Va*? aw na nyn*? ion ntyKai .mannn maya naa'ia (15) nam .nans 's’? namn ba iann myi latypaty n^ya *?a ppn mnty an1? ppnna mrty mm mamn yanKa namn *?a iann nna-’Vn mmna lanai n^ya myty anV Kom nfma Vy nr ipVn D^aityKn n"ai mVya a ppn (16) -]Vnan lapnty •pin Vy maipnn lamnm nnK n*?ya may ppnn (17) nt pm Vy kV mia p nana aann tyatya pn m*?Ta 'in (18) *po ny msaKn Kin ^a pna iami Kin aya pn omam (19) map tm ••a nam man imty manna ityy ntyKa mann (20) n *?y nmnann 'tyaK lannty namn *?a ■pn mbyana n*?yan (2D ^aa tmty imty na iVanon ntyann omnwaai naa’m naaa (22) lana^ morni manna mVya *?y apVnn aniK ip’im ]ipnna namn (23) yVn nVnn m nan aaiaa pmm amnn’? iVm Dtyai nnK n’rya mnty mm 7t *?y nooan nt ipVn ntyKai apVn a"p rmamna naityKn V?nn Dtyai nnK nVya ppn myty inn apVn pf?na mm nnK namn nooa (24) nr Vk ym*1 ntyKai nV?ya a"ai m’ra 'a naaa ]aipn pa yanty ny amnn5? (25) kVi mVta nty^*? y•arty ny nnK piVna K^^ty na laaa non1? I’m1’ ^lon d) .mVa ppnna (2) ia nKty 'a naan nmm dk mniKan mtya mamnn ]ipna naK n^1? [Q3l] naKi (4) iKtyaa oaam nty^ya nanon mVra 'ia mr nn^n dki na mVra (3) mbra 'ia nanon nnr dki na oaan m*?ya rmi mVra n ny naan nmn dk naaa oaan mm mVya a"ai mVta n ny naan nmn dk aaiaa naKi .nKtyaa oaam (5) nmm dk nmKaa nam nKtyaa oaam niVta na nanon nm*1 dki na (6) niVta na nanon nnr nmn dki na oaan mm mVya m mbta n ny naan (7) mm nViya m mbra n ny naan nnm dk nn)atyi pnaoa nam .nKtyaa oaam (8) m rntyno naa nmt2?ri .na oaam na nanon nnr nmm dki na oaan (9)

.,'33

Nsisn ltn pidit 12 || ’*33 Vro VdV :Vro

9

.’’33 'n '10 :'id 9

286

39 3

X"*’

118

PARMA VERSION

dot (22) nuns ■’ms? p^Va (21) mxnpan mannn n^x nft [Q30\ (20) “dd nDlt^D (24) .mpm ppnn rawa amx ipn1? (23) mu naVi mamba amx nba xin mi 'a my (l) n»»n amrasn trmpnn (25) nr naoa tibtib pw*?a /,di pwba (2) ampn nmx •natw n mpaa 'ta obux xm pn irons WT’tt? manna (3) baba “pnnb itppa nt^xa aanum mm nana namn ana manna bian (4) aann |a nnxa mbtab nbnna imann mm nann ba ppn mm an5? appear (5) mann a*wa bran aann nanaxa mt»a mm nabya nnx bab aaaa (6) nnr pxty banan ]apnn "a mbta 'ab vrv mann vn

.’■33 nmmD 21

.’■33

287

ni3nnn

3

*9 3

II “7 II

naiKD noia

amp mm tk jtapn Va^an psiaa naiyn rnnaan VaVa (id psiaa Ksim ipn mm ny^n non’? psw p *?» japn Va'ran psiaa naiyn pKn (12) psiaa Ksim ipn *?k pKn nmpaa Ksim ipn naa?aB,,a? nmpan *?k la^n1? (13) amama? Ksim ipn mm nf?Ta 'ia *?ma pman m mm na?Kai (14) Vman VaVan rnnaan ^aVa psiaa Ksim ipn amp ppn Va'ian psiaa naiyn (15) pKn nmpaa pman *?y mya?n nT pom1? psw p Vy japn VaVa psiaa naiyn (16) nmpa *?k pKn rmpaa Ksim ipn naa?aD,,a? nmpan *?k Kin aa la^n*? (17) lamin'? (19) psiana ppnn mom*’ na?Ka ms naV aytam .'man Va^an (18) naan nm *»a Kin (20) ayan .naana lanom psian *?y pia nm bki naan 'ay nam aaian psiaa naiyn (21) rnnaan W?a psiaa amsian a-npa nniK )pn naa psia*? nVu? Kina? pKn psiaa Ksian (22) ipn *?k jinonn ik nsoinn nma la^wn nnm amaw psiam naan ]pn naa (23) nan naa nu?y na?K3i mtyan nta?p mytra laaa “ism ik pman (24) pman nr Vy ^•*0T*ts? psw p mm1? imnaa aipaa ]apn I'aa'aaa (25) nnu?an *im naan |ipn ma? nnr maa?n naan mma Kin ]inonm d) laaa jinon ik pwa riDDin it naan w ayam mVra 'ia runs naan (2) nnm bk Km nsoinm nV?Ta 'ia *>sna mntt? nna?an mm tk rn'iia ria (3) nmns naan nnm bk kb maa?n mm tk mVia 7ia toy* naan nnm bki (4) nfisam ny rnnaan 7a |apn i^aba nam japn I'laVaa rnnaan ^k nf?Da?n (5) 7a Kina? nnmr *»sna mwan X nmpa Vy aaian a^ai nx japn (6) 'raVan *?y ainaai mis na?ya n^ya naina a?aa?n ay nna?an (7) mmV D nmpa Vy n'la'iaa a?aa?m 7m DD nap mna? psian 7ipn (8) Kim nnK 7ipn pn i1? mm kV nVyaa amaa? lyyiami a?aa?n ay nna?an (9) nanm na?Kai non'? ik pom5? Kina? nmpa ^k a?aa?m 2 nmpa 'ik nna?an (10) aipa 7ipn nya yam am'ia'iaa nna?an aipa 71pm Dtt na?p Kina? (11) pman 7pn amipn ama? rm tk *? maipn ama? f?K3 nna?an (12) jpim *ia?Kai 17D nap Kim jtapn I'la'iaa mya?a laaa non'? (13) ik 7apn Va^an psia Vy ^min1? psim pmo' mon1? ik Y’Vy (14) lamin'? ms 7a *?y i^aVaa ma?an •jVna? na in'’ na?p mm 7a (15) *?y 7ia?Knn Va^aa mya?i nKm nmsa DK nap Kim naa mm (16) naan 7pn nya a?aam pmaa amp japn iVaVaa ma?an pma jtpn i^aVaa aaian ^Vna nm nap Kina? lanama? 7pnn iKmra? nm m'?Da?n ny nmaaa Kina? mna 7apn iVa'iaa ma?an nm bk psian Vy (17) 7a Kina? mna 7apn iVaVaa ma?an nm bk laaa noimi nx na?p mm (18) .nmsn Km nmi xn na?p mm rnnaan Vk m'?Da?n (19) .’*ra 3D n || ’’33 3D »

288

38 3

II 116

PARMA VERSION

nnbty ••ssasn ibnan pn tyatyn *|bnan ^ban nnbty ppn baba p2aa *»d (4) nans 'yyaRn tyatyn ^na p by tyatyn bty 'yyaRn ibnan by yoaa (5) bty •'yyaRn nonb m2 p by O’batyn n^aty ^yyaRn nntyan -jbnaa (6) p2aa (8) tyatyn bty ^BRn *]bnan mmty ay aaaan bty •’yyaRna tyatyn (7) .p2aan ]a aaaatyn Rmty ^ra aaaa atyma nmatyb 'aa nans naan

nrrn

DRty

o^r^yn

Dmntyan ppna naR ntt1?

[Q29]

'D

(9)

panan nanaan (id nRtyan mma nanaana ppnn ^n non1? tm nabta (10) m2 n^bstyn ppnaa *,y2aRn (12) nntyan ibnaa panan nanaan nonn p nnRa banonb na2a p2aan Ran nRtyana tyaty (13) bty ^BRna panan mnaan nonb pans nanaa ab mm tr nanaa by ppnn •T»oa,» (14) nabta 'aa

nm*1 naan dr

aas'cam p2aana p2aan ppn non nabTB 'aa nans mn (15) dr nRn naR naya •pan nabTB 'aa nans mn DRa 'atyn naan nra oaan

p nnRa (16) ^BRn by

by amoan aaaa men aR a^by nooanty nnR ]panan nanaan by aba (17)

ppnn

bR namb ntyyty ■’a baa •’anaabR nyn by ppnn •’a yn rOW) (18) .p2aan .nna’ Rb (20) mnpana maty an nnra •,aana ons raanaba na nRnrn (19) tyatyn ay anannn (21) yana ppn ababaa ^ban nntyanty Rana naan ppn nnRn aapa aa pnb

*p2

Ranty (22) nanaba aanan ]apnn Rana apbn aabnab np^a

Ranty pa ppn ababa nanaaa (23) nntyan pnna nm ntyp ^n Ran 'a nntyan nnRa a^by aas'Da’’ aR nanaana nomty m2 nab (24) nytaa yanab Ranty pa *poanb tyatyn

^bnaa

nanaan

baba

D'bstyaa p2aaa

maa^bya

Ryam

apn

mtyan pnna

“]bnan

nynb

]a

nr

■pyan •’a

nom Ran

(25)

p

nytana (26)

nanaan by *poanb aapnb na ppn baban p2aaa nntyan p2aaa naayn d) ananaa tyRna nmntyan •'babab nRtmty na non’’ ntyRaa aaaa yanab aR (2) nan naa ntyy ntyRaa nntyan p2aaa naayn nanaan baba p2aaa apn bR (3) naa

nnannb typa^ty

Rana

p2aan

]apnb

*p2an

p

nnRa naan ppn (4)

p2aan (6) nt ]apna aaaa yna^ty aR a^by poa'ty nnR panan nntyan ^bna (5) nnR nanaan baba (7) p2aaa pRn p2aa sp2aan -aty pa tmty pnnana nnam yanab na2 nab nynb p tma (8) nntya bab nanaba aanan anna '•»*> ntyRa ntyy-'ty nyan .nabra na nans p2aan mn or (9) naan by aas-’oanba p2aana ppnn m tr nabta na nans panan nanaana ppn baban (10) p2aa pnna nanna *>3 Ran

.’•33 rural 23 || ’*33 a to :on 19 || ’’33 non ns’Din 17 || ’*33 ’yxottn inion i« :psin pmo 9 || ’'33 i’3 pxiDi :psim e || '’33

3Dn

289

:mt>on 3 || ’"33 psioo pxwn :pxioa 3 .’’aa pmo psin

K8 a

II

nanxs non

n5

"j mp; Vs ]upn iVjVja

mcaai (26) mean as manna nsra nViVj m

bib* naVaas? tma^ yn V x>in1 (1) i^aana xxaan apn mma? mpan Kin 'iq no>xa nan y nmpa *?x (2) yaw nr mwi pxaaa naaar’a napnn ntrxaa napnn W?ia (3) bafam mwan 'a aVa japnn xm 2217 na>p mnn paaana xxaan (4) apn mm Dp na>pa p rmpaa pn nnnx nmpa by mm D Vx nrvp (5) naanpa » rmpaa nana mVran 'aa’aaa nmpa *?x a?'*’ rVx nmpa (6) *?» aaaan mn nx naaa Vanan ppnn po xanu? pao px p by nntran (7) pxaa Vx pxaana n-’xxaan tnpn arra naVya s?"na mna nxra 73 D rmpa ns*? (8) 2 rmpaa mna D2 ntr'p *?» Vaan “Tfap ntypa yyaann n?i>xa na?xa psrn **a (9) nxnan •naty au? nartt? anaoa omVpax aana1? naaa nxaaaax ana? do) jau?Va xnpan nPPan Vxa D'Vaayn Vxa mnanan Vx tran ^anna mna pan nxnn (ii) a’rya’? nxna nanna npann nmpa nmm nnx rmpaa rmpa ^a? anna’ti? nmoaVx n'W (12) napnn ‘aa'aa nPaa?1? arxiaw nr by mxnm nVa nm xanta? mVa?a ,J? nm npa nam (13) 22X Q’aaoa'ax am maxa nvar by n naia?n anaty -janxn mm oner* a-np aVa^ (14) nVaaa? Vaa pfa ap ma? x'saaa aaa\a> ax *0 nVaaa? •’xna nans xam Tip1? nta?p nxnan (15) p by n'aaaarn pxaa by napnn VaVa atma1’ p?na xxam paran ap mm p&n aapa (16) n mapa nra nVaaa? rsna nnana nta>p napnn VaVaa nxnn nam pa fa nampa by (17)

*7 3

.atrn nxaaa (18)

a^aa^arn (19) trnnrcan ppna nonV ma p nana nax HfaP [Q28] a^Va^n amntraa naxa (20) u?awn Vw •’arxaxn fanaa a'atr ^xax ^na matyn nr naVa 'arxaxn mwan (2D ^Vnaa u>aa?na *>yxaxn ^na nonV •’arxax -fanaa nana a^aa^Varna nnx byb (22) •’arxaxn ^Vnan "a rnitt7D xan ppn Va'aia aaVna pn ppn Viban pxaa ^na (23) xam a?au> *?w a>ati>n *]*?na mna> naaa?aa aaaa nomte ax a^a? (24) f]-’Da,’a? na naa?*1^ mnx aa?a anannn na? a>aa>n aa? anannn na?a ma?ana nn^an (25) “fanaa *aana ax a>aa>na pnnan *»aD naaaann na^nna naan naaaan namn a^a' (26) nnta?an bv 'a?ajax ■['ana nonV inxan mVx aanp nna>an mn^ *»Ba (l) fl'oan’a ^axn aapaa poa’’ tx an-’a^a^ pnnan na?n*a ta>a^n bv ^a?xaxna (2) naaa?a a^Dwn a^mwan ^a^a pn .nonV ”axnn aapaa nam ax (3)

□m

]’5?no :]’yna

is

||

’’33

nxum poVn^ :nmin is || (?) nr’nn d’d’ 26

.ehit invn1?

’"33 di 3357 :3~iy 10 || '*33 non || ’"33

by

mipa

7

17 || ’*33 yon ?n

.’’33 t£)D»na :»a»n Vtt> 2 n ’'33 ’33 in :’S3 1

290

'U

=7 3

PARMA VERSION

ntyya .pm rmnb mis (id -|b was mm inaab bnan ppnn ntr p by H pRn psia npsia byi (12) IDS mbs? mbran babab nbwaa mis w'w na (14) nsr®D KH ip p npai n by immty (13) maioabR R-'Siai pmaa (16) ntpyai psia T rrnpa tnwi TH (15) ip Rim n^psian nty p psia by my (18) nbiay ntryai mbyi (17) mpm baba nbiay Dn by nasrw 1D02 (20) mp Rrsiai 7D171 mby tyawn 09) nbiay smi PI S7 mipa xmi (22) mbran baba mipa bR isrrw ny (21) mwan psia

mn'i D mipa by tawm D mipa by mtpan (23) naicRua m lavana Q^^ai o-’yyiana omaty om ryaiyn ny “Dina (24) nspnn baba mniaaa nntyan “lnv1 riRTi D mipa bR nbabaa yan (25) watwr ’a *iaRa nam ombabaa

.1*33 nn’Vyi n || ’’33 niViaj) ie || ’’33 bjn 133 R HUN 11

2Q1

X14 ||

IIX13 II

nanas noia

irowi msaan nnwan -fma msaxn watfn *|Vnaa nona (6) tors nim ppnn mna npVn mm» ma npa nam .mVra ntroa nmns ami (7) 217 n&P mnaaa m man XT w mVran VaVaa mm ppn VaVana itt (8) w mm psian mm TTT nRt&n Tt7 nt&pa nr lanon T1 ntyp nswn X) niyp (9) Rinty Btoiai 170 nwp Nina? psian )ipn laaa nona nam mVia 'ia mns do) mn aa ntrpa oaaa nam 20 ntyp naan nmm nam 2$7 ntyp ama? naan Vy (li) ppnn nr ■roswi nspn VaVaa l1? n^p ami ppnn Vaa nnaaV v'v na a 2) npai 170

-iKm lawpaty inn 17 nmpa by n*wan btmi S70 (13) n®p ^aVana .nmsa

13

nt^anV p na?y rVi mVya '* naaVm awn -aipn aty HO1? [Q27] (14) nn nra mns nr mawa mVya naaV nna Va ppn aro na-w pn (15) n-mwan •-ysasn ppnn m OS7OH .niVya a"pV ana nna y^anty na? nra Vina (16) poaVK nann ay mVran VaVaa mann wt-d ntyaa la^aya mn a 7) permit nyna pn mannn iVx naian nynV aana nVia^ pm ppn VaVan (18) mwan mma mVran VaVa nanna namty Vmam psian ay nwnnna anty (19) ppn VaVan ipn m w-a- -ms-a ip pan psiaa «srw aipaa ppn VaVaa (20) anpna nwnnnan nmm Van nVma imir mnn mi mwan psiaa iy^an ny (2i> mma nVman mm 'a msoa pwn ny^ya orVpR lanaV naai .a-Rman (22) nmtyp Vaa nVma mm nr myp nrnV misn naa nannw inm nVnan yVsn (23) pn nntyan psiaa y^am ny psian p asian ipn wia** rVi a-aipnn (24) nya imnaa aipaa ppn VaVana laVnaa pn naa nntyan (25) nw ny yyan n^Vtya mnsi ppn ibaVa mmana nm*1 tya kV 'aVri! mnxn *pn anaits? asm a* pan (12) nr n^Vuma atrn iKa mnKaK nann nn maa amianpn Vs? miVn (13) Va mipn am aass?a •’aVn mipn on •’Vw naoa ana> amipnm naan (14) *paai mVna mm ipnpm ipnai mm mmV nana? VKS?atm man (15) maa mns Kin n mVn ppna mr« naV trjotrn ppn pn nr as?a "iK3K (i6> paa? paan Vs? *]ao n mVn Vs? ms?an Kai amia?m apVn a~aa anp (17) Vs? “jnmoV Vam kV •’a naa? nanmtr-n arniKa mVn msV mna? a>anaK (18) *nas?a ama? *]Vk laa mVna amianp mna? paapmKi poa nan (19) a?anaK naK na?Kai •»aVn ]pn *ia?K3 paan ’Va ]pnV is?r KVa? (20) amp aa amiKna mVs?a a>ana mara nn a?aa?n ninas aipa n (21) s?s?mna imK a?aa?n mna* n *»aVn aam p Vs? mVn iKaa aipan nta (22) aman iniKsa mm amna?a nwna nnK Va mna* s?s?mnn na>Ka (23) laipa mm amna?a na?ann mna* ns?mna ms?mn *>a nKa aai (24) a?aa?n mna* aipa nas? a,’Vns?n paa?nV n'wi nKai pjVk ma? aim (25) kVi mna*n aipa mnaa? VNs?a^-’ nan an nVKi trauma mVs?a ra d) TnaVtn *nsaa nK p mm nns?n a?an ]ia>Kin nnK mna aVa rn (2) ninn annVKi nnp p nam nmaVK =nps?-*n mnsVKi s?apaVK pm (3) pm •’tiokVk pm mnaVm mao VKpnVK amain nrinaaoKn pm (4) .aVs?KVK (5) mnaVK p nana (6) nao nVnn .•)VKs?aa?’’ aann nao annV nnS71 .aa?mrnai •wtdVk mmV msj&a Vm?aoK (7) ’Vs? p pnpVK maVK nas? ]a npiVnai a>iaa? amna>an mmVa mmia? na aam (8) “f“772^7■, mDT lamaV kVi mm a^awan ]a 11:2a? na Va Vs? mns kV annana **ai (9) maa?V mamnn nam a,,as?an mV ia?na kV n naa ma?s?V ims naV cio) m^an *pna anon an anaon naa mm na?Kai mm KVa aVipa cii) nts>Ka onaon n ananaa pan pnonm Kmpn naian annual (12) mas?a naa ia?s? ananan m naiV Vma?V ik ps?»V ]*?it?V tm naa mm (13) na i2is?*,at2?m a*»as?ta is?m kVi ]a aana iVapa? anan ma?a nnK (u) mm mas?a mV*V isn kVi ams?a is?ma? ik anmrKa is?aw (15) I'nw nmnK amana irm naai nKa maas mVtan naan hkt (16)

.’’33 |lD’t3’BK1 JIO’D IJlDQp’lNI ]1BB 19 .’’33

DlV’yVx 4 ||

300

'"33

’03nV['33 ’SipV«l :'sisV«i 3

32 s

PARMA VERSION

II 104 ||

misaRn *]Vnan a^m □•’nun naam paamn naan mo snv» VR»a«na Vna (3) maVa Va a;aa mapn 'a oaa mVa *pa amir’ Twr patimV amaamn bv (4) man Vai miaRiaVR mia ]a aana mn aann tri imV aa^im did (5) aaoa ainaa mwn ammiai a'pVim paumn a^Vaia nVRn amm amasn (6) nVp nanR paa mmVn nws?a Va Rmin Rim a*mR pns?ia Nina? aann (7) ]na rV pa nRmamna aann naaa namaV naianRa nw rvhp a'TaVnn by (8) man VaV aa?a ]nai ■maaaVR iaa; VRaa^'a aan vanR api man*? aaa (9) aana; mVtaa aaaa aao pvmnta; Vim aan ap vanRi 'tan mraiaVR (io> aVw Rin naan nn ana; pVR lV tn *»3T» mn Rin amsa f?a ovaVaa (id amaian mai pRna aaVnai amoi a^VaVan naana laaa nVyaV pR (12) nai^Ran nVaam mVs?a w by apVn Rim mVtan VaVaa anty (13) anrVsn »®na nnR Vaa ana; a-aaian VaV isoa pm ptwnn naan naRap' (14) anwi antm >f?R VaVan Va mna ana; mam nma ana; aVVai mVaa (15) a-’Raaan VaV Warn amaumn mpn VaV asm jna Rim anaman aaV (16) R^ana; (18) mman Vai nRmamn aan ma aipsr pmana> mmVn mraaa (17) mmaa (19) mma an moaaVR Rapa Rin Vnan laaoa mVn Rapan Rin oraVaa asmm (20) aisnem naana an mma m amVs? piVnV aan aia; Via’ pR amaip (2D amnpn VRatm mam nRmaar ]V ptrVa paiptr nnan naan aan anai (22) VRsmtma ama aman inmnm ason nn niRmaan iniR raiapa anRV aam (23) ^ana ]a aana law mVraai nnaa aViaa RVaia a-ainan mmVn (24) mraa Va aa?ai amawan mpna aRa aaaa aao amapsn nap pan (25) mama maraam mman amrm mraiaVR aaoa *]Van mVna anm d) aRaV tpoim; maipa urn monaVR aaoa amipV naanV naia man aiaai (2) aisntzm naana miRa laVa points; maipa ur an paV naitpm nVRts; pa by (3) iaao na;yi amva maipaa pa 'ran •>aiaasVR aaoa aminan amatran ^aipn pa npiVna pR (4) anaaR dr3 .aam aRiaa inrn amtR •’aRi amaa maipaa pa nR^aam nan aaoa aminan pai mVn (5) 'ib ^ssaR ^Vnaa aao maan ••aRi amam naum Ra naV nRan aipaa (6) maipa m ]naan 'Vaa naRa an aa;Ra aatm ••Vnn u>Rm amatran (7) mVaa m laa ]naan ma aaipaa naRa aim anon mniaVR mmVa (8) amaa>an anVa; amatran bw msaR ”f?na m nRmam man aaa m aaiR ^ri mm (9) nama naana rV mamn na;ya naana naR am VaVan nam ma Rin do) nmaaV mnn^a an m nVvin ana pR •iao,’aaVR aaoa mmVn aa mVran (li)

.’■03

uan :uoD i2 || ’"oo VtntP’o 9 || ’"03 oinot? no «npV nu>p rcroiyn e || ’"oo i”i»’

3°i

*

s

nttnKD non DW3 ■.‘neon y"x annax axa nmy5? 'V mm xim xmam "naan atya d) n^nxa amiatyn ^xyacn maa m x5?! naan x5? nrrn x5? a'aianpn (2) namy nna a*? man rnnx nasi wm m la^a on1? pai xnpn apty ny (3) nxax^x men ^xyaena Vm ^a nayty ny amaatyai amama amn ansa (4) pwVi nrirwi piy5? ynr aan typa5? npai man maan nxma'xa vr 'a yaen (5) ona' ax pox iaxnp' p nax m anaan msaa nnx i5? pw 'any (6) lrim iaaa maan V?apty nai pnpn nae pn Vxyatym m'ais'n (7) maan xim nai*?an msyiaa nxa naaa naaa iso nxma'xa (8) ty'ty amaixn iV naaa naan mm5? nann min in em am5?** ansia (9) 'a *?y a'Vem *pn an,5?y 'a nwm mnxn aenmaty naam nV'Va cio) naan aen xmpn 'ama nxm '*?ix aim naran "jVan naynn nam (li) naona ptyxnn nytya nam Vxyaty' ptyVa naan i5? p'nyn5? aV?nn (12) Vya i5? mrny ai^nn *?ya ynr nna mnty 'mm maya nVty tx (13) inmyna ma miVna ]a nxm '^xyaty' lapmy' ax nna 'a naan (14) nT p'nyn5? imxi many^n 'my myea x’raia xinty naan nxn (15) myxai ma' ,,5?ix maie^n 'nty ynr pn ]na rx nxnan 'a5? maana (16) nyn5? ityaa naaaa tx naxa x?n pi mem ip nnn x'nty pnx n'y5? (17) -jV'e? naran naan p'nyne> 'mm5? an mxp' x5? p a'w m5?'5?? (is) am' naem 'a' *?ai a'atxai nVa e>xn nnn 'mmn -f?m "i^an ai anty naaan (22) me' Vxyaer5? na5? xim naaa men ^Van Vx aann nt aan (23) ^xyaen anna 'mmn m Vy aann 'aa pmyn tx rmmx m a'nysam (24) pixn myya Vai nyaem arnnemn mm5? naa nnxiy ]a aipy’’ men mnnrpi (25) a'ai'Vyn a'aaian ny'mi a'nan ppm nnaixn nVyam n'aam *]m (i) na5? nema pn B'a'ayn nVx Va5? aya naT maaa pxi mmxan annna ampty nxma'xn patyn Vy avnemn *?iy mxaxn "jVnam nVap aan ap p'nyan apy' nnxi a'ae; ^Vx a,/5?i pVx mxa h ximy ]aTxn (2)

,’”33 )”-|N 17 || ’’33

14 || '*32 nynr ;n:oil 10 || ’*33 ’313»n 2

302

«i 3

3i 3

nft-iKD non 212 •’OI-T'H T 3DD *>3 Vl7

II

bxam nom

99

matya aayty na py manx a^asa nr naan ax aax [Q86] rpoim o^aty rmi apa mtyi arsa omx nam ma'btyn nnbian (12) aiaoa anv yanty na rrn axi (13) abian apm naixa nbyty na omby aipaai (14) omyxan mbyaa namn p imon axtyaty nai mao laaa non 70m abiyn maty mibn nyab nxan axi .abian naty nam xm yanty natyn nmxb naixn xm yanty aipaai nyr* aaaty (15) natyn naix by nr an mitfnn .rmtyn (16) mbyab ambnna amn nmx myxa mbyaa am ia max aaa omann p anx *70 nan tyatyn natya apbm o^tyaxn m a'm dv mam am1’ mmty obxx natyn •’a*’ nan aanbx man naax (17) o^aty mm 07a a"mi av moan otyi (18) .amanx ~[tyaaa o^aty mi 07a mbyaa (19) mytyn ibxa nVmty na mm a^aty mi 07a a,,,i my "yn (25) Kim "IT ip nipVn yyaKV mm n’nay n (26) tana ^y mpai amp laiKaa? na w1? D (27) nmpa Vy an aaoai ia*iKaa> laa ]pman annn n"mi "in ip K"yiai mp'm nVnnna nn"n inanma? na DT ip n"n"i mVann ma"’?®1? n’may ma^a? ny mp'm yyaya mm manma? na (28) HD ipi mVann ma^a? Vk nVnmn mViayn ]a (29) ]pinan mm ann Kin -ia?K "in na?p "a nVmnn a^iama n n"iT mnn maya nrir "na? Vy nn"n (30) tanaa *]Vn K"m nKan laa a"yaipa (31) iayya xk *73 "TH TH "ip nKan mmi naya "IHT mm mp1? imaaa "IT nKana noma imaaa "in nKan nan imaaa "IT "in nKan nonai awiy"am "aa? "yn Kin "IT ip "a ianKa naai TH Kin (32) nVnnna mm ianma> na Kim TH ip mn"i d) -iKa?aa? na na npai imaaa 373 a nan (2) -jVnan naa Vy mpVm *raa nr nan naV aaaKi .iyyaK Vk mpVn ny*rV *pyim iyyaK Vk mp’w nVnnna mm *]inm a"pn V? ikk" na?K Ipinan Vy mpVm *raa a"pnn nan nam (3) myam ]a nr ]aT mm naa mp'm (4) yyaK mya?a imam a"pnn aniK1? m"n “jinn mya? i1? iKy"i mya? imi mp'm yyaK mya> 'ay inswim mp'm nVnnn mya? riKaan .mmy nKTi (5) m'mnn ma"*?a>

|| ’'33 jr :ar 27 || ’”33 jr nr 24 || ’"33 n :~in nN3n

3Dn:i

24 p '*33 f

2s || ^33 na :jx 22

32 || ’'33 trays 3133 tmVnyn iy ruyp ’3 ia 29-28 j| ’’33 nr na 27 .’•33 nr Kim .•■'33 ins’Dini nmn nVnnn 4 || ^33 my® imam 3

310

MICHAEL VERSION

nDTin

“dip V? Ran r\"b

Rin

ntPR

'a

II 94 ||

bx wpVm •wan nw ntPR

n"aa

(26)

.mrm nP mVn 7a m»asan iVr 7pnV (27) nann dr naR nfa'pi [(182] o?fm (28) rosaaran 7a rraaatp na npi Vnan mVa wo^am tpatpn rfaiaa? yn inpVnm a"m mam nnm n*?ia» yn np nnR .jitPRnn jipnn Rin mn mnatpi npi Vman mVa imraan nRtpatp nai }pman nnm VaVa (29) msrtp bx inp’rnm imaaa nnm Va’aa m»tp (30) nan amt -wn ppnn Rim naaatp na ]ipnn (3D bx inomin Rartp nai mtpn ppvia man Rartp nai map bx ©•’lan maR aaaR nmi^nn .fpman nip^n msmR an Ras^tp nai fitPRnn poaVR pan m»aaiR bx iVtp nV?n ma Rin mn m*?a tpatpn (32) n*?ia» yn msasR mi mapina myaatR iaaa Ram (d m»aatR a"-1 an ntPR tpatpn *73 a poa*?Ra (2) msasR paa Rin patpnn ib Rar ntPR tpatpn nViaa? yn on ntPR nViar yn o^aantp -pan jpman mm bibi paa mm nViaa? ym tpatpn mRnp’ ntPR Rim ompman apim m»axRn 7a rraaatp na (3) npm nV?a tpatpn ma tpatpn nap pa1? mm n*?ia» (4) yn matpV poaia Rin mn 'Rn ppmn nnR aio r^r ia oma'tp *>ir-i pm nnR nfm m a~a mVn nr 7a maa^tp na Vr a"\n oma mm n’rnay yn bx ]pman mm nwtp am iVatR nan (5)

Ram Vaoia mmnn arnmna amooa (6) m i*?r mm nwaatRn 7a atartp npVi (7) mVn incaam a"*1 an ntPR tpatpn nViay nap aiaa msasR ib ]pinan mpVn msastR on paipn mtpn f?Ri mtpn ]ipnn mRnpi rraaatp na paa myaatRV imatp*? •panm tpatpn nViaa? (8) poaVR aioa an m rVr *jnann mm (9) mm 7a namn Rm mpVn 7a nRm ntPR m nnm poaVR Rin m -pnR *jma nra 7iatpnn nmi 7pinan nnm Va’aa aioa nmaatRV laonV mtrtpa bx pVmi nr poabRa nr ntppi nr poa*?Ra nr ntpp (10) na'tp “jmat poa’aR aaina Va? nnm (id 7ioa1?R yama pVmtp nisi 7iatpnn nspi .nnR Va a^a pVn *7S7 looinn laa vbx f]om .nnm 7ioa'?Ra a^a pVn i1? Rar1! tpatpn Vr nnm 7a pVn a^a pVn ia 0> Ratim pVnn mmi tpatpn 7ioai?Ra (12) iatapi 'itp.n ppnn 7a iV Rat'tp (13) naa pVnn nr nRan mm tpatpn 7a a^a p’an pVn iaaa msaatR bx m»aatR aioa f?a ppnn mmtp ma 7Rn ppnn ’tr ma1? nnm 7a namn Rm mpVn 7a nRm ntPR m nnm 7ioa'?Ra a^a (14) mtp

mn

nan

mpVn ma?tp

(15)

nxib

.’’on pm :pn

natnn

dr

e || ’’33 fD^i

311

*

naR

|| ’'33

[Q,83]

poo pnn m’n 2

II93 II

to naaa

nym run xan nan n"aa s?pan nan anax max rmtPfln .piVn Kan •»w» s?pa amamVa nw xV xan m n^pn m ns’r mm s?pan nT ntz>p (12) •o n"aa mam 07*1 '«? poaVxa (13) s?pa Vx anawi a"*’ poaVx Vs? ntz?p '«? poaVx paa xan no>x s?pan nT a^n ms? 'v xan n"aa a"m nxan nsn ntz?pn s?r»w mas?a xan m nnx pVn o"w nVaas? paa ntz?p aV xa-n (14) paa ntz>p V"*i ntz>pn nt tut® •pass (15) mm DJHK namn ma ns?nV nt^x (16) nt2?pa ppnan nan nnx mmatzn nu>Vo>a a"m nxana nVaas? paa nwpn aV aa^ nana a"o Vs? s?*>antz> na pVnm pVn o"w nVaas? paa ^na rnnan nxra DK xana no>pn nr (17) nan nnx a"' naaoaVx nVaas?

Fig. M-40

nVaas? (is) Va *»a inDX namn oma® mm nw xm *wx nVaas?n poaVx oma nVaas?n nmx nitres *aau? xm na^pa •’ana naaoaVx 'an nxan nan .amnantz?p Vx naVaas?n •’oawa namn oma nmmpa (19) Vx nVaas?n mew ■wp na na'tz? na xm (20) noau?o mm nnap ax nVaas?n na na*1 *wxa uwen xm poaVx ^ana nVaas?n 'am namn natz? mma nnap ax nmannn mVa naamnn map am naamnn npp •’ana poaVxn 'an nan rpian (21) nn PDDK namn nap xm poaVxn •’ana (22) DN npp nxan m lanax Vx DIHX (23) namn nap am aVax mm *pnx aa ianaT npx papnn aV xaam nppn ppxnn o'omna amsaa n aVxa 'ta 'an Vx 'nn oma anpp mamtm 'an nm nponn 'am 01HN namn ma (24) •wm papnn p □Vaa nrmatz? n’wapa (25) npVpn atzn n**s?**aw paa nap •’Sjmna mm nppn nan nnx n"V mm 'in 'nn nam rrpyap a"a man d^pV paama a"a ama

•’’=3 max 221| ’’33 max 171| ’'33 tannx 14

312

MICHAEL VERSION

II 92 ||

p rr' nwR nnm VaVa mmw oma myaRR (21) a"*’ Vm mwn hth poaVR (22) rr^w amm nnm poaVR myny Rm nwR myassn nina mm jpwan mm ^aVa ■’anRia'pR laRnp** nwR mm a^a mm ^atyn poaVRa nRwaw na mn DT ip mpVn myaRR mi nt (23) ip nniRn DT ip ana nom DDi PIT ip (24) pap nwRai .DPI mm poaVRa n ip ~\m p^nn mn m mar n m mn amaipa on TD sip nRW n^saia Rin m inaT nwR mn amaipa mip *»awn Y?r npV aa»m vVy pVm (25) on^ -a amaipa a»ip -awn iVr (26) npV aaaRi mVy pVm nm pVnn mm (27) poaVRa nRwaw na np’?*’ rVw naa wawn poa’pRa nRwaw na rVr ammna amsoa n iVm mm n'piay pn *?r ®a©n n’piay pn oma amnn -aw paipa nmi amnn ••aw am a^aoia ana (28) amsoa ■’aw m nwR amaipa mmi wawn (29) poa’pRa nRwaw na om nmi laamm yrr nVias pn ^r mp*?n myaRR oma wawn poaVRa nRwaw na Vr ipVn Rin Rim Vaoia ana 'r ammna amsoa n n’piannn riRia H irr^i nnm (30) nwR mn i1? rr^i 'Rn *?y mp^m >aa mn (31) nan nam mm n’piay pn ba (32) nmi DT ip nwR mm nViay pn yp nwRai mm nViay pn Rin Rim DD ip "iRwai DT ip mpVn niyaRR Rin nwR DTa non yir t?T ip laaa nom a^ pinan mm ^a*?a *?y *poin naR (1) mVi wawn n’piay pn 272 a .nr (2) miR in pVnn Rin nRwaw nai a^saia mpVn myaRR nViay pna nRwaw na nam a^a wawn n’piay pn non naR HD1?! [Q.80] .mnawn nwpn ypa Rin man p rr^w nai (3) ima np rr^w nai wawn n’maya lannm anp *»aw *73 m nnm mpVa (4) -jiasnin naa PDWDPI aminn 'awn nRan laa nRwaa ana nnR Va mamna nnR (5) riRan nan Vy DPI ip JX ip wawn n’piaya inn naa naiann nRra mm nRwaa (6) amnR.n DK nwp -a JD laa DNi IDs DN nRan laa DDa DD nRan nan D nmpa (7) nr ma nnp^i DDa DD man nwRai DN ip laa D3 ip nan 3D nwp laa (8) .nwpn ypa Rin m naR ntbi 3Di DN mwpV ypa Rim DK ip ma mn (9) nai nwp wanm yanw nai n^aa ypan nr nan naR HD1?! [Q81] (10) nam innawn rr^w nai n"b (id bv mpVm a^aa man nwpn ]a rr^w

.•'33 II

’'33

33 3 as

n’3n

a ||

’'33

Dr ip

tan mr241| ’'33 tan :taai

njn 3a

rwp a ||

’’33

**

dr :3R « ||

|| ’'33 pnf23 ’'33

ip :np

2

.’’33 id« nrVi rn ip ’nt^p1? 9 || *'33 31a :or 9

3X3

ViO's noaa

II91 II

mravni pawn jaoaVN xm na'aan nVaayn tint [i] •pn rrxnaa wawn bx i'ov* naaa nvn nVaay Fig. M-39 n"pn poaVN m»as» do) nutw na>5N ^a^n VaVa Vn mm VaVa naanw onm omasa n a*?N nana omr* on^x Nan 'a nano in a^a nDa*1 warm rnpVa

bx anpVna 'aa 'an nam ,l?aoaa nnx trorrma (id

nn 'Nn i? x:r 'nn

*pan nan a"' vbx *poan anax onrnian (13) mm poa^N

naax

xaro> naaya

(12)

.]panan

nnm ’paVa

nsp»

Nan 'a nn pawn jaoa’px a'1?*?

mpVn nayaxx aaaa nonn oWa'P ma pawn poaVx

vbx ^•’onn pawn noa'pxV

nNPap na nam nnma papn •’aaoa'pxa np'n nVp na aas? nxpaa n'poaaa (14) na nna nsnw naasra Nan m p’mn axnpa raapVn nnN (15) onaoaVxa nnma pawn mppa nnN Va yn (16) nayp nan5? -psan papna npV'P rfraaa? (17) napna n

tana bx JQX

nbnaa?

aaoaa .araxpa anxp np’aaan

n Tana bx PIT nVaay a"a aaoaa papn bx “in lanm JX ap xmaaa n&X raamn (18) papna np^aan nnapa bx in (19) laiarr ap ma raap^n naarasx DT ap m naaNa D 'a aapnaa nnm nVaaa? napna

rvnapa

npx (20) papn n^aay fm DD apa nnm n'paaa? pn dt ap m naaNa D on1’ ma n'pn y'b nnm babaa cpn 'V papn ^a^a nasnp "a aapnaa anat

"ID1N1 19

II

’'33

HDN

18

|| ’*33

r

n 17 || ’'03 rt3TIK

:in» 11

.’*33 ar

3*4

ip

’3

|| ’’'33 m’TI

mpVn

ttfOPn 10

myaxN tar

ip

’3

MICHAEL VERSION

II 9° ||

.npVti? na tishp Kim mm mmon *wk ini D] Fig. M-38 my® oma wawn p np^a? na *?k myasK a,/*’ Kin upk

(d

mm laaa ]pman annn non (2) nnK 'aa’aa nyrca nKtyaty na *?k bwh k^'i (3) an *?y mp*?m 'na 'K»n nam ’aaoia 'awn ammna amaoa

n V?k

.anpina mba mp*?n myasK am •wn

[Q?9\

mam nnm Va^a (4) ny» np myasKn |pnV mnn dk naK

imawm fpinan mm ’aaVa nria Kin Kytp nai urnam *?aVa *?y mpVm a^a nai 9iDa mp'm myasK iaaa non kw nai a"' vVy ^oin nnK (5) m»ajtK «t.t

pn1?

nxnp

na»a

mW)n

(6)

.mnawm

pVnn

Kin

iKwarc

vmm (7) naVnuo mm m nn np’?^ na nna |pnV na>y mpVn

msasK m nnm mp1? fpVna nn nnr ik 3"**a runs poaVK myaxK nru ampVa m is’mrv’ tpatrn mp*?ai man niyaxK a/,a a®nai pa® na pnna Nin (14) n®N )pman nmn anm apn N"V nn Dn 'ip nnNa ana nnN nbiaya oaa' nan (15) prioa'iN "sna runs aranan n® nViaya npViam amioaVN m® "sna aranan (16) 'a® pa nsp® na my'® a®nai nmsn an mrnai m ip my'® mpVn myasNi DnX (17) n®p ®a®n nnaian 's’? nmsa ®'® na nN®i V?a ]pma ann i1? n'n' nV nmn 'a (18) npVn ia'a'y m'Nna nnN nmpa Vyi nnN ip *?y (19) mniNa 'a®n 'tana 'a pina ann nn'1? n'n' n1? Nin 'a nnN nmpa ^y an'rana (20) i'nr ®a®n N'Sin Nin 'ai (22) .inms nNTi nnNn ja pnna D'tana 'a®n nnN1? nan (21) n'nn ®a®n (24) my®a nnr nn'n ^aVa my'® (23) n®yan nr iaV npVn nt^i ®a®n (26) n^iaya nVna nnr |'yn niNna (25) nn'n n^iay D'sn inai mniNan (27) '^a'pa my® ma ian"s® naa nNana naai .nVia nan npV' n'n dn naN na1?! (28) mpVn mn' -pNi pinan nn'n ann inai

.’’33 nn’n am n’n dn i”y nnN ns ynm ’Vnm nnVn DipaV nn’n 28

3X7

ll87ll

Vxam mm

ViVn mbtan VjM paw na pnna 27’w na1?! nnm rm^m m» (5) sn-w aim 72a axn pnnan m 72 n27’? *p2m (6) amwn 22ax V2X nmm .mm mwa Vrtoa amn (7) nr pnna aipan nra srp© ns pB2 723 ix xm ppnaw na rrrri mnonn ix mVran Va^a rmm V» X2im amn T*onm mmw an ma2,,i aim mwaa amn aipa pnonm naoinn mix nnx (8) msmomw “|‘,ns nm am 723 nm axi aim ix (9) *?xaw 723 pipan nr ms 2TW na nnn do) anna mnonn pD2 72a nm axi V2n ann *?2 ixnp (li) nrbi piapn aipaV wxnn nana n*?2a 'o xm namn pnonm naoinn anna nnon nwxo nnx ’a pnonm naoina pina T2n ann maxi pinan annn room ix pax 7sa nm nwxa amn mwai aaian paw na pnna m2n (12) naa paw na pnna pnonm naoinn nnx yan (13) mm 72a mn nwxo pi mVi?a '2 paxn 72 wxnn (14) naaa "a T2n nmxa aaian 72 wxnn ]a xxmn (15) annn n^m nmn m*?2a 72 avn mwa5? amn nwa aopa amapn maipa pVma pVnm n^xn*? nnmi wawn paw na ann xnpan pawnn maennnV mi >f?nm an manan ann *p*?nai noian W?n nfnan ^aVaa (16) naxw na aaaxi .amapn aipaai mvn anna (17) iom npV m1?! nnan i*72x m'aan mVan xm (18) nwx 'a on-1 nan 'a *?» inp*?nm jr*a man paw na ann amnpm iV anramn apnn Vx arm oma ]pman annn 2pa*7 •’sV xm **aim xm m nax aaax .Vaoia •wanm mmn1? nnmi wawn (19) xm m an naixi nni2a mnxaw laa (20) ame2n mamaa norw na maVnnn (21) mmw a^nna wawn ]a pxn *?x amp nnm nnm mmw na1? .aima nmn anna irmxn 27nw (22) mxm mpVn 22ax nmwV nnm ann nw2 nax nfr1?!

[Q77]

ni2W paw na p^nw m nymm mpVn 22ax nmwV ^nm nnm aipa amx nam apnm mown ]a mpVn 22ax maw (23) pai pwxnn piapn n2w1? nnm aipaa (24) mnonn X2,tw nai nn ^2 apbnni nnm *]Vnaa amn mn nnx nm ax mV2 msmim amn *»2n amp piapn nm axi piapn pi mpVn 22ax*? nnm aipa xm pnonm naoinn nnx nxwaw nai (25) piapn nm ox ^nn I?2 *f? X2,,w na (26) pmm nnxw x’ix 'Vna nw2n ann nw2ni aim mn pnna nm ax (27) naa mnonm amn 'xn amp ami nnm (28) ann nm ax p2 nnx m2 27m ■’Vnm naaVn aipaV nnm

npsnnnV innDnnn1? ie || >”33 tpi-ino :^iVn3 is || ’'33 Vab’jn :i’yn 9 || ’”33 insp nmonn 7 .’’33 ’Vnn Vy ’Vnn p -jV

n» n'010 nnKt£!

’Vm 20 || ’’33

II86II

MICHAEL VERSION

.wawn roaV ton ma naxnn yap^ na [1] .nrn

ton ma naxnn xjpp na [2]

.^xn^

maa

nw x1? im wawn ’to

[3]

Fig. M-36

nxa (32) naaxn naa anna naaan naa rv^w an ax nax nfr1?! [Q76] naatrna annawn nxttw naa nannn ]a aaan non anatoa vn axa axapn mx axa T>»n am *?» ansroan aama ans nna nxa toan nxa xana anx (1) «7i a annawn ]panan amn xan nx»a» naa (2) Tan ama amon jassa nn xart? na 7a (3) *?» anp'anna a'^a anana a?pa ana^n ran •’aam xana rm$nn .annawa T’an •’aam xana n^xn1? mrna tra^n p» na am xan na naan ntppn aiaa nan1? na naasn w aaax xan n faasnan naa (4)

3X9

ii 8511

Vkom non

nan mytyn nnwa tyotyn ^Vno omo nVnl7i nr myty nn (3) nyK T'oV hot mo wv'VyV tyown (4) mpo nr Vkp naom .momno annoo m iVk no not s'? nr vv nyKoi pSKn p nn^Vy mpo an ntyy’i pDKn Vy mit py oan’’ kVi on nyin non mm nniKO Karty no m .latyi mrow (5) .rnyty (6) oanw no mynn p nro norm p non nn« mpVn ysoK mytyV norm moyn iok HQ1?! [Q75] nmxn mo mua np *inK .irnotym norm nan -iKtyan (7) mmi mVyo n iKtyny nm papa nytyV •’Vnn iaoo mm (8) norm nao np rnK .irnotym norm nao am (9) Kin KJPty noi m Vy inpVm no mom ypo mo^n m on amn nma yvw moyn Kin m mwnn .ns mown irnotym nViayn 000a mis mV mmai .mpVn mam iaoo n a^tyyn npm (10) ny nniKO •wa toioo *jVm tymn noa nmpao Vmnn (id mn?K mmoan p nViayn hkti n tom Vy (12) TDK nViay K\m wan noa1? Timmy oooai (13) Von Wio rnyK Vnan anon notyn 10V' im mVnan mVaVan D*7 rVy mm VaVa om nn nViay Kmi n toio Vy tyotyn VaVa nViay nVaVao n mipa Vy tyotyn m aitynai (14) mVy pKn nViay om toio p (15) kkv moy iDfrn ip "0 aitynai iVaVao *? mipa Vy mmi mpo fri psa noip xi amn ooip 2 mipa *»o aitynai tymn noaV pKn Ipinon papn myty lampm otyK lpn Kin OH ip m aitynai lammn (16) nip-1 nam *piKa mpVn nyo mmi tyotyn (17) Tonon •jV** m ntyK Kim nmoan nViayn (18) p 17 mipa mm lammn ny nma iaoo mm VaVa mmi tyotyn pty no om Kipan Kim pmoo mm mma 170 ntyp mmi noKai .now ioo omasa moipoa Ton mimn nso Kim mmiV (19) ntypn nynV mo typai motyn ysoK noiun |o yT»ty (20) mo norm yr1 Kin m iwk (22) mman mo yvv mo ppn mpo Vk omm noao *ityK (21) mmi iyotyn pw (23) no om mpan ]iok ik nm mm ns ama mp' pw no pmo iaoo yT’ty mo •’Vnn (24) iaoo *iom noisn npVi mmiV nnK (26) p-’ay nn n mm ama ntyyao? ioo nman VaVai (25) mVron VaVa p •’Vnn non mVi mm (27) om nKKino ntyyon ioo m ntyyon nam am iK*ip om iV KKm m Vy inpVm no mom (28) ypo iKtyan ntyi nomn nnpV iamm (30) nVyon nmo mm ook Vtyo nm oki .nomn (29) nao mnK ntyyom )^ayn m ianoK (31) mVi amn m laV kk^ mo lam

.’*33

3

'a :yp

27 || ’*33 an

:DH

« || ’-33 nn

320

13

II

’'33

nans

11

||

’"33

mm

rnoiacn 9

MICHAEL VERSION

II 84

asm )pn ^y V?aa rvmxan ^® vm® an mp*’ my® am jpman (29) Kin 'a mpVn yxax my® on nax nPi Tran (30) ana 'x np!7‘’i irmxna .mnnan mpVn yxax1? a^aca ^aa>n *|Vnaa amx nam p®x-in papa my® np nax (3D nft1?! [Q74) w»®n pp-na anonn a”3®n pi (32) mpnn ]a xr® nai *ra «?y opVnm

mwnn .tnawm (i) irmVy1? ®a®n aipa xm nx®2® nai papa mpaV ^7oa nV’Vi nr my® an n®x T'a ^y inp^m ®a®n -jVnaa pa**pn my® nan naax papa my® am 'a nr n®y nnaxi my®n nmxa ®a®n -jVna x:r® na (2)

.’’33 np^’l 29

321

ll83ll

n013

mpnpa n®s mpnpn n®p nintn amsa ss*1 “inn pasm a,,a®n (32) Ennnn mit •’maan n®pn ma ss^ nns mmsan (l) m® aipa *?s tfmn *70 irr Vy mman n®p (2) mitn imsa smr nns irmsn aipaa pnsn Tana ^ss mapinan amm 77 msnpi nns p®na as-sin 071 npV mVm .rns-a V*' D"?2\yn ysas pa® na pnna nV?ya V? issn n®®a a'pin ams nan nns (3) ■’ansiabs aaasi mmana i1? iss*1 B'®ya nT (4) nns n®y ams piapn Vs nn nasn Vs anp’ (5) insmin nan sin is nra nanVs mns i®m as sin nan na mam ypa ma®i piapn nVya Vs awn ysas nVya pa® na npV sin m nn mmnn my® asnpi mman my® iV issn mw Vy nr pVn -ins (6) oma ny® nrtman n®V®i ny® n®p (7) ypa Vs iVia ypan on'1 iVss m sim pa®nn p iV (8) ss-1 n®s mman ypa*? awn ysasa ny®n n®p ypa Nan® no ny® nwan 'ai ny®n n®p ypa pVn a®i .nyn 711s ia pa®n nwan 'am ny®n amV nan sin nns mw a"a pa®nn ma iV (9) ny® nsan nn my® n iaaa paipan (10) mm n®yaa imVpV ma,1?tr m®»V 7am p (id nwn iVsa n®y n®sai nsnai an®a nwmn 'ai innsmin naai .mw rm mm an®a (12) p aa ypan nam 7 amaoan ]a .Vaoia ana rnmnn ammna amaoa n Vs (13) nViannn nst mn my® pi ]pman piapn my® pa® na np nas (14) Hfa1?! [QJ3] mmnn nw (16) an ss*® na /sn ny®a ^ima nta n®ym aim (15) as inamim aim mn amp (17) mn as piapn nwa innonn mn®n nmtynn .awa 'a ]pman piapn my® an ssm (18) nai am mn nns mn s'?® ma an ®a®n (20) mma n®y1’® 7ns mn nmn mma yr® maya (19) non n®sa my1’® nmn mmia (21) 70^’® na mm nwn vby aia'1 ®a®n n-^aia amp n®yaa my® m®n pa ib s^^® (22) na mm3 i:aa -ny®n imsai ny®n )a 371 ib isr1 wn n®yaa ms® naa ms (23) mman p i1? ssm na ]pn^ 7ns mn sin nns (24) ayia nan ®a®n ‘i'?*’ p*?n sin ]ipnn nn (25) yean ppm nwn f? ssm ma my®n pm ss' asi n,,®,’'?®n ayaa nra m®yl? nisi nta amp is iiais pVn a^a nan n®n ama1’ sV asi pa®na -\b nasna naai ,,a®n bs amo-’® na (26) 7ns m®',’?®a -]b nasn-- s!?®a sin m nn y’r® (27) ny amya n n®y ppn iVa nn ppma (28) n®y® na aioa ayia nsoina nwa ni®y5? piapn my® isnp’ n®s my®n n®yan nra m®1’® ma piapn ia pn-1

.’*33 riD^n: nsn 3331 26 ]| ''33 idd’

322

:iV’ 23

II 82 II

MICHAEL VERSION

naaa nnx np’m xVi nnx nVaaa »a»n aa nrn nr [1] .nnrnaaa nixnn nma Fig. M-34

‘worn warn (25) bx mmxna xarn mam ip *?a nn (24) an papn »t niaian xt naxa xin "a (26) max nnai .mp^n frcn irmxna ^pi 170 mt^p am nman xin anVna nmi nmxa (27) mm -pna ana p aaVna ana amta mn niatan amx nan1? -pain aaaxi .nman nxra (28) nr nonm na ipnna ix nipVn mn papn amp (29) nipVn pi papn a-’ntanana 'x Vai nipbn nan laipa (30) ami insnoT* ix nmm ®a®n aipa p xin nan (3D ovaVaa aaax .naxn *?x inxmnV minima n*?innn ib ty aaax ^ax nimnnan nvnrn np^ xin •o nn nvirn p mntapn ibx xm*’

.’•33 iVk n:n «’si» 31 || ’’33

323

27

b’KDJi noaa n^aan 71 nra axara naatan anaxa nanaxan aa’?' naa arta na niatan (29) nan nT.n n^aa oaaxa aaaa nan1? ‘i't xan nan (30) taatan rv^taa naax npoxn ja apnna VaVaa (3i) anaaapa pVna vim1? nn nxan^ anaata n an .anaana (32) nnran nxa amaaapaa naa rpnn*’ naa nanaa cratan nanaaxa ]QK nVaaa xna nVanan taatan n'aaaa natana n’aaaa aaoa mis nr5? T-saa

n

nPa nn'1? naVtan M^a*? nan nVaaa an nVa aaoaa Vsx pxV px nan (2) POD

(l) Tana Va

n^a pxn n’aaaa an n^a aaoaa TITT

taatan toa nnaa^ta n nax*?an ^aa aanon naa nan naa^ta taatan Va^a taatan n^ea nann xaa xan n oraV&a nar nr1?! aaaa amp ax nnapaa (3) nnaa'ta

aaaxa

a^ata no nnx

pn

pnna

xVx

(4)

npnnaa nnn

ntaxa

taanrta na maun p rVa oaaa (5) ana aata ax Vna naa^ta nnn Va’aa ^ax taatan nnpVn (6) oraVaa anar ntax nn rpnnaa aoaoa aan nrVa anaa'ta aaoaxna xsan oatran apn naa Va papn naa nnna ntaxa xan 0 nnn ]a ]tanpV xan nan ntannnan moan aaaxa .aaoaxna nas!?an taatan *?x (7) pxn ranaa naxsan anp ntan aaVnnn pxn (8) p ananpa nn\n VaVa Da nnta X nnapa o (9) atanaa nnn rana’? aanra naxn aapaa nnsaa papn o naataxn atanaa n’aa mapa o aa atanaa notan asax la anaa nnapa ana nataa naVaa aoa asapna an na anaan nsn nan nn nnx (10) papn naa taatan aawxn aa nxna 0

nnapa Va

cii) nnn VaVaa

nn,no*' aaaxa taatan (12) noo*’ nnn .nxo x’aa nnn VaVaa

D

17 nnapa

nnx (13) npVn aaaxa taatan npVn xVa papn nn naaa Po ap *?a n\Ttao naf? *psam nap1? rrrr tx aVabaa 170 ntap naa^ta nan *f?n ntaxa papn p a1? axs** ntax naatan p nr ara 170 ntap xn ntax (14) n^nan nxr aaapna nnna taatan na nan nn (15) anap papn n no atanaa pntann apna 0 nnapa (16) ba pxn ns Va aax nan 22 nnapa ana nnxa aoa xana pa nnn nna X nnpaa (17) an** taatan aaoaxn aa aaoaxna xsan nnta nsa napVn n\ra aaoaxna (18) taatan nnn aa nno1’ ntax aapan nT*?a moan xn ntax 22p ntap naa^ta nan5? psana (19) naxn papn anap aan (21) nn anap papn nn ax a5? axar ntax n^xnn naataa nax (20) papn naata ba ananan (22) aan nn nnx nn axa papn naataa annonn naata an n ]apna nsn aaaxa .panan (23) papn naata an a^anta naa

•’’33

n’m 17 || ’'33 na

is y

’"33

na nx

32

|| ’’33 333

my an :niyt2>n 14

y

.’’33 ps«n rpnpn

nvon

’"33 23

y ’'33

31

y

ns

’'33

:a’D»n

31

di a nmp: am 10 19

y '*33 mpon Kim

'v

369 a

P°ll

MICHAEL VERSION

pna pawn nipV m in rnwin /jar ran nii?2?i pr ni2?2? inn anx (5) rnpVm nsa nnx rv,*,t33 nma mm impVa nam mma m nn (6) mm mpV Dm2 anxm pixa ana 7x 2>a2?V nrtaa •w (7) nrtaa n i2?anm 2?a2?n 032*’ xV inn? **2 oraVtan (8) 22?m ni?ia 227 nmma nan 2?a2?n naaxi nxiVi (9) mma mm2? iV ur nan nam mm maxi .laimtj? V27 2?im ip rVa? “pa? Van ma2?n nmsxaxi mpsxn ]a laipa nma? Va? mnVnnn mman pman rnVan Va? maann mam xV nrVi rmipVn imsVnnna do) inVnmi awe? Vi? p|Vnrr nt '2 mpV mmr (id xV mae?p 'am 2?a2?n p nm a2?x m2?annan (12) nan itawa awe? Va?i manV ix px axV nrn msVnnn a2?x fiapn nan nan nrVi na-’V nman VaVan •’212*1 VaVaa nam mma iaa naa ianp nVa?a nat xVi aatn ]a ix trsm ]a amp (13) 2?a2?m mm .nKT’ xV nipVn nan nr ci4) fiapn mm xV a2?xai nam mpVa aai2> na?aV pm inaim*’1? ynsn (15) nan nr •’an ma?2?i lar ma?2? nv'v naaxi n2?annan nrrn (16) laaa na?aV mae?n a?xax laaa a?T,2? ma mpVn na?a naixn •jinn nrr laaa a?ra? ma nrn 'an ipi mVtan VaVa pna ma2?n a?xaxn pa? na pma a?T’2? a~a *pmn nam Vx 2?xan naa Vmaan 'naan na?pn (17) niVa?ai nmvxan mVa?aa n^at^n a?xax pi fiapn na?a (18) nami a?aa?n iaaa a?ar nrn •’an ma?2?i nrn (19) ma?a?a ia?ar nVia iVxi mVran VaVa .man nmnx mam fiapn ma?2? p na np (20) anx fiapn nia?2? a?a aax [Q72] n'wm ipnaa ix (21) fiapn mm nrn ,,sn naip nrna?an ma?2?n ]a nrn •’sn pi nmaaa mVa?an p np nVm iaVa? nai nVa?a apa n Vai mVa?a rts nna na?2? Va nn xm2> nai n"a?2? Va? mpVm na mam maapa niVsan iVx m2? ami (22) naip fiapn nm nx fiapn ma?2?a inaon (23) nai2?xan mman ma?2? m:i72; nn aR2?32> (24) nai rVa? insmin nrn mn anx nm nxi nrn mn mm •’xn ni2?2? f,a2? na m a?a PDI^nn .maa2?i nnpinan mxiapn msw nvaV nxa «m mi mmaV ni2?annan nnta 2?ar fiapn ni2?2?i (25) iaa nam n^m nr anx ya'2? ma ]xa "pixa n2>annan mman nxr (26) fiapn D273 maixan n2?i n,,a2?n a?xax p2? (27) na pnaa 2?m iaaai2? nxt m pixa (28) nirma 7a mm2? nna i?ar ni2?2? nta ixarw ma nmx Vx2?i mm ni2?2?a maixan ianm2? na ai2?2?i VaVan ]a n^pa n^nan

:ynpn 25 ||

’*33 ni^pD

21 ||

’’33 own

ny )’3i is || paV a p ip msia nnK fan (5) mpV iasya 7213 riKanV ppa iasya 7372 nKan m m (6) laKSin aaaK npa “inK ,nn®n mVann ma-’1?®'? (7) n*?man 71 an n®K 7313 Kin mna “inKm (8) D Vy nmsp **axya nnK a’wai mp*m n'mnn'? m®n rfraan pn mm anyaa oam n®K nmpan mm D nVy a'®ai *?fa pa yy® apaa namn Kin pm1? mpVn ysaK Vsk mm rana p (9) Ksim ipn m mp’ia nVman 71 msn apnn V?k my®i (10) mm ia yy n®K naian VaVan p pa 73 ’py Vaai (id m*?ann ma*’*?®'? n'nsan pna many ]ai mp'm n'mnn'? ba'iaa namn mp’m mm1? mm Tana1? 73d Ksrn ipn nmn 73\3 Kim 7372 “inK .mVann ma-’’?®’? n’raan pn imy®i mm ia pm n®K ntaian (12) ipn m nmn mp’m mm1? mm tanaa I'm (13) $70 ip D nmpaa msia nnK .imVan ma'’1?®'? (14) mp'm n'mnna mm i'?'1 iai naian VaVan ]?a namn Va Vy aaoai Di 73 mipa *?y nmsp n®a 7k mtyai mm Va^a my® mn npa nViay nan D nmpa1? dk mm (16) nVnay am nViay ana mipa (15) mipa1? aKi ia oaaa Van mma immai mp'm (17) n'mnn'? mm Van (19) mma yy immai rn’iann ma,,V®'? mm nViay nan 73 (18) annan nan mar n®K (20) nmV nmp'm mna aaaKi .nt mis in iaaa Kan Kim nnn®n “inK manaV Vyian Kim (21) pan amyaa npmi amama Kim *]®nn lasm niK pfm m pna )an'?n p Vysm (22) n®R 'manan nan lay annna®a pVm nan nann □■’ana ma aanm nnK (23) mnn®n niK mpim nt 'yy nmnm mpim amam n'?m mnn®n ]a ayia nnK (24) Kin m nmn mom V? n’n'm maianna innpai (25) imaiaa1? nnK ®a®n DKi .nipim (26) anyaa ®nnnn n'n’m nmmn®a nan nmn ay anym nmn® iay anyna am marnnKn laaa n*?n may niK mpmm ma anyna ®a®n nyp® ay mm® n^sm mVi pna pya mpin*’ (27) laaa n'y pa m mm® n®a,,Ki ®a®n niK (28) am’iy nK®an a^nan Vyi apDKa mamnK mnn®n ik )ai'm ]a n^ian mpmm iniK aTa'® na my® ’m mpmm my’® Vy nan®ai mnn®n ik pi*?n p mpmm Vy ks*1 aaaK msn (29) bsn tana 'ly Kim mpVa 'isia nmn (30) mm n®Kai lay aTanan iniK lana nmn apan iniK Vk na ®a®n niK*? mm kVi *]®nn pna nm

dj?s mnj imhjy ny

33D31

id 16-15 || ’*33 moo

:n~\yoo s

|| ’*ra 3

.’”03 ns: 29

327

:d

» ||

’'33 am \'im

4

|| 15-143 3133® 1B3 n’3W

bKa'a'noia

77

ip npbn •j'by mm 'bnm mm nbiayi bmn baban (22) nbiay nnsn pbnn® *jis ntVi (23) nKiaa “inn naK nni' niK'an mm® ms now ik 'pn nsoan npai jiaa “inr Kin nan ipa pbman ipn ja npn nnK mpbn nmsa ip baba my®a npbi binan baban nbiay 330m 13 ms® na mb 'bnn (24) nbma nmm nbiayn hkt mnn® (25) nnK nsn my® nr» npbi 'bnn TDK (26) nbiay Km nbiayn nKn m'®yn *jnn jpnb nra ms bam mbiaya® n rana by aaoai 'bnn baba nr® 'sn ms® na by npn nnK n tanai

nbiay

jpm

bsn nbiay jioabK mn Kin nr (27) *»sn m *>bnn baba nbiay

“T2 'ip (28) am nasa mir by ranan by mbiay 'a®n pbna my nt npa nnK .'jibs “ji aim 3i mr»n 3 nmpa byi anya K nmpa by ainaai ■wa nnK m®ai mpbn ysaKa mm am my®a (29) pbnnan ipn p nmm bKa® nsa mm (30) am m in ipa mum n nmpa by imisp npa nnK mpbn ysaKb mm ann DPI ip nmm rby jam n®yai D nmpa m nnm nbiay D nmpa by aaoai mm baba my'® mn notn®an ja (3D nbiaya ks'® na j"yn nnK .ibaba my® (32) mn Kin nnm jiaabK mn n'®m mpbn d) nbnnnb nnm ann npn nnK npb' nn bsn nbiaya nnm

“in

ip by nnKn nspm n nmpa msn

mbiayn

nbnan nbiayn *ppab ip D nmpaa K'sia nnK

3

nmpa

.nbman 'pn vm ip tib Da K'sia® -na *? nmpab (3)

“in

.’'03

328

imisp 'a®a nnK Kmi annn nsa (2)

m® tana by

ipa nasa mir by

n :ni

28

||

'’33

“I :j 2S

=>68 8

MICHAEL VERSION

II 76 ||

.mpVn nbnnnb nantyn 7*1 mnna napbn nbnnnb natyn nb^san pn paan dh K2Ptp naa “jbnan naa by anpbnna T'aa (26) ana 'r ba nan nnR Nyw naa napbn (27) nbnnb nb'san nayty art nsoann 7a Ryty naa nayty »*»■’•» nb^san pn vvv naaya mwnn .napbn nbnnnb nayty an norm 7a maapa ntyn abR ann aynn nabann naaptya napbn nbnnnb nTn sapa (28) aPaaan nra yyty •’na nb'san pn yRnpan apnn am» (29) natyaty aa^un “anx RbR a'nany nyy a1? (30) "itysR *>Ra .napbn anap aama *atyR 'nn npb ntyRa nTn *>3 napbn nanya nTn T'taab ntyR apnn pbn nyy ]aa nnta nun npbaa Rana an^taa maan arv^aa nn\n aman nua (31) pDRn p ays baa aanar -iwr ^a*?nn nr (32) a~a ^bnma .a1? naann *jabn ba? ^bnna d) anaban pn anPsa jar rrm anaan 'atya a^atyn yuaRb *68 a ap *»a aama naa aanaRa naaanpn ntyR maun anpaa nasbnnnn nr nayty ban nbaay ]aoa'?R ^una nr non ntyRaa napbn (2) nbnnnb nTn am ftn ntyRaa 17T aaaa (3) noanana 17 D ap apy *]oaan yn *anRn 'unn by pana ntyR 17H nRan by pan ntyRaa anaaaa 017 nRan aaa yn 177a 170 nan ■ana npba napbn yuaRb aama napbn nbnnnb nTn am pty (4) na Ran pa naua 17 namaT ’a ]pana DH ap (5) man p Ryty na n\n ppnaty na naaPwb my am Ran *atyR ap ’run nbaay jaoabR 'una non yyRa .pT aaaa noanana pD apy paan yn nnRn 'unn by pana nabann (6) pH nRan ppna ntyRaa aauya ip (7) nRan aaa pTa pD nan ntyRaa nabann (8) naaPtyb aama napbn yuaRb nTn am paty na Ran ntyR aauya yan nowaa ]pana in ap man p Ryty na yn ppyty na ma npba nbPa uv nayty an *atyR n^aa ]aatynn 7a “7V ru*> *i»r (9) a^apanan apnn nayty npa’rnn 7a ^ ru' do) 7panan ^nan Ran ntyR ‘I’rnan naa by anpbnna *7bna naa by anpbnna n/raa man 7a a^apanan apnn 7a na^ty nau nrba naa-’btyb n“rn am npaty aanaR 73a .amau ara nb^aan nyanR nayty Run (11) •7b (13) Run napbn nbnnnb nn-’n ama 02) aa^tyyty aa ntyyaa nabann

btyaa (15) .maun 73a *mR ntyyana nabann (14) naa^btyb nb^san pn nar nasu apn (17) rb nTn ama nmn napb *0 (16) atyna aa^ay m napbb aanpba npn (19) 7u 'hnn baba naytya apn V'b (18) n“rn baba *aaytya ra “iRtyaa nmn am nta aanon apn a"o yna (20) amaywn nty •'un aRa .anup (21) npb'ty a^nna n^na n"rn babaa naytya nans n^na .apn nabaayn naoabR mayty aaaa npa .npbana naty 'nbaa ntysR 'R amu aa^un

.,’33

Dp :]p 7 ||

’’33

o’Dys p-\ :pr s || ’*33 nn :dh

329

6

||

’'33

ys :ya 3

II

75

||

Vxa'a noaa

ppoana on v ssam *vn '* aanxa "l23 wux "3 nTn ^ ,:jn x:svn inm nn ap paapan mna rrrn VaVa poaVx 'sn Nan -awx (4) no ap napVn i?sax n? imp1? (5) n'nnnna mm “jnnn-* nwx noun VaVaa “pnnn n? napVa anVsa m?a mm nanrr iwx 37m nn '3 nVman 71 axip1’ nr1?! .nip1?! ysax (6) n jpanan *?s? anpVnna anw tfraan 71 an? lax [Q69] ikww na pi aa^wna a^awn an xs-w naa (7) panan Nan “jVnan naa nwna mx *?» anemana (8) naaapa nwa napVn ssax1? panan mm man tjoiam napVn n'nnnnV nTn napa ionn nan nnxn *71? amonna maipa mm am1? 7111 nrn Nan ’a i?i rnitWin .mVann ma’52?V aaapa (9) nVman 71 axip nwx 371m mVann naa1?^ cio) ni?aa np1?'*^? na nVnnna aaaa ioana nTn i^na 71a (id nasi panan 71 paa a1? axs^ 1 wx an1' (12) mna nTn i^na aaoa iwx 37m *?x aa-’wn1? -prim wawn i^na Vaaa axs^ iwx 37m ^x nVman 71 oma mm ^Vna Vx panan ^Vnan B’Vaoaa axs^ iwx 37m am ommna amsoa (13) n aVx nam .paVna span iwx mma n71?n (14) Dsax1? nrn pana 37m aVa nonnw 72*1 mma n71?n ysax1? panan nrn *?» ansmana nap’an nVnn1? nrn napa nap’an rsax1? panan nrn nn? lax nrVa na’aann maV^V nrn 37a (15) anonana naaapa nwa mxa npaVnn ]a f? xs^w na (16) nonm ma7a nwa naaxa aa'1? •’Vnn i^naa a~a nr (17) aaa nwi? nax mx .nnxn aapan Vi? 37m (18) aVx nrn ^'w na ias?wn ^nn pV-’ir- na namw ymw nrn nsi 37a! mm 37a i?ma 7nn 37a nia s?mw na npa*?nn |a a1? axs' iwx am nws? nax ntVa mVann naa^wa nap^n n’ann1? nrn (19) anm ^nn naVann naa^w1? aam nwi? mx (20) amawa 1112 aim nap'an nVnn1? nrn .amawna ns am 70am maapa (21) mtaa "Vnn VaVa naata nn a'w nax Htt1?! [Q70] fioaan (22) nan laa mxn p amonna nap’an rfann1? nan am ana 'x ^a aama napVn asax1? nan am paw na pin' np nnx .amatana nona 11a np nnx .maw nwx Va ansnana anaaaa (23) anana nap*?n n'annV Va ^oan nnx naaapa (24) mwa ann^an nan |a xs^w naa papnaw na nan (25) anawna nnxn p amonna •’Vnn VaVa naaw nn naaapa ^awa mx

nioipD ’3@a 21 || ’”33 lmonm

20 || ’’33 n5’s:n n’pi 10 || ’*33 ms'Dirn nmonni s

.’•33

330

MICHAEL VERSION

II 74

n2tym nViw n tdiq Vy 2"2 22021 DT nVi2y ym Vyn nVuy nniR aisma nVny n2tym nVuy IN ip2 n nnp2 Vy 2202 pi mpVn yy2R2 hth (12) nVuy am m ip traai Vyn nVuy rpp2 (13) rvwViy ntonn n^VtyV mm nVuy nmn 'Vnn V2V2 nyty Rin im Vyn }102Vr TD ipi mpVn yy2R2 mm nVayn n22R (15) .urns Rm in mV2nn nwVty*? mm am Dn ip (14) nVnnn R*ip2n mpVa inVmi VyR m’n (16) nVuy Rm n 1222 Vy ntyR *wk nVuyn Q22ri (is) .m 0222 Vyn nVi2y y22 222 mVi mpVn (17) nVuyV y22 *122 mVi mVinn moVtyV nrn (19) nVuy Rm 0 12*12 Vy mnRmVRn mi (21) mpVn nyy Dn ip2 1? mmi .n222 Ryv Vyn (20) yy2R ny imp1? nVnnn2 mm 12 (22) -ray nn rwp nynV my mn (23) nViayy mnnm onp my V2 m an “12R22 orVpR nynn 1221 imp1? nniRi nnn»»2 mnRn nymnn nR2n 122 mnRtyy ipn rvomn tor nR2n nrB Rin 12102Vr *>2 12*121 im Vyn 2120 TD (24) nVuy i22ty niyR2 T"T ip (25) •’Vnn V2V2 nyty mn Kin *iiyR 12102Vr mn 122211 *>Vnn V2V2 121R2 *i^n m ip Vy 1200m ntyRoi nVuyn hri tiooVr Rin TD ip Vo m V2V2 iot ran Rim “TD ip mn mpVn yy2R2 (26) mm onn Rin m •’yn Rim TH ip (27) nVuyn pooVR2 nRom mm 2m rVy tpim •’Vnn TH ipa HD ip mon *io>R2 m mm 2m 122a *ioin2 'Vnn V2V2 nyty riR2H Ttt 12npV ItyROI 2PI2 OH JIRDH 122 (28) 121R22? n2 mV fpm nViiyn m (29) nn in ixd OH m OH ip 122 12V Ryty n2 mm THo HD mm 2m Rin ntyR “TH romn n222 *i2>r mVran V2V2) mm 12*12 Vy Rm nyR T“T 122 D“7 ip n2n ni2y2 mm mty Vy mm 12*122 *]’?•> mpVn yy2R2 (30) ihr Vy umim ni2ip2 myy 'Vnn V2V2 (3D ny» mn a^nty *riy mVi n2 T12 npm (32) non2 »]0i2n n2n *mR .mRn p imonm mm 2m an2 an n2n Ry-’ty n2i mm V2V2 *nyty mn Vy insmin “mn ]2 Rys? n2i ppn'ty onp myn2 7r Rim HD ip nyty nyi Rin m Rin nn nVmin pi (1) ipn ]2 ni2mn myn vt ntyR2i n mp2 (2) Vy ID ipV *|mnn nViiyy amyim m2 Vy ^min (3) n2V D22ri .m2 OH riR2n 122 *mR2 nn2 'r nR2n mn

:-ny’K> Kin 24 || ’"33 on

:nn

2* || ’*33 2(1

:t2n

2 || ’’33 -id

:jd

13

|| ’"33 mp» ipa 12

[| ’’33 n’n’ (2 :nm’ m 27 || ’"33 n :rn 27 || ’'33 n*1 Ip V2 Ip 25 || ’”33 ton || ’'33 rn nn 29 || ’"33 n^tt mpn nVuyn 29 y ’*33 m nn 28 || ’'33 on :dh 28 .’’33 ntVi rn 103 on ip n:n 30 .’’33

an :dh 3 || ’’33 rm cm

331

2

|| ’'33 o’pn 1

^67

2

tomj noia

II 73II

nama naru> 'iKa mamra natoa k1? im nViasn rnnw nKT [i] ^ aa>K nViayn Taaai T mp3 n1? y^an naaa nat?p am' mnx

•inyai n aina m*?iay -n^n -jinn Vyi T rrVv manwa nabtra

.m*?ann ma^ttV m^n [2] rtrb yaian mpVn nVnnn1? man [3] Fig. M-31 Vya inyap’ aa>K DipaV iann (4) ]ata mvi *]i*?n» nt y*n orr’aan a^Viaan nT'n ami ■’Vnn ^aVai man VaVa awt? ya'tya Kin *0 iVsk miysK nr n\Ti ** nri .aaiKty na ^sV araan n^iaan iVk aisra wr iV aa>D'K (5) mma pan «?s -ai pan Vsa (6) nrn mp1? -a mpan nVira

-f?

-naKa aaa

nTn *»ai tz?at£?V maa a^an man k1? nasa n-ir *?» mVtan Va^a mroV piaa at^Kai anK ip Vy naan tyatyn Taaa n^m Q’atyp •wn anKa n\T at^Ka (7) nniK aitynai

DK mp mpai aa*? (8) anK ip ^y Vsm tyatyn nan lpnan**

3^ ip nn^n ia pa** at^K naian ^aVam mVtan Va.Va.a mamn *w trKan dk a-awp mt^n anK K naipa aurnai ntaian VaVan JX ipi mVran (9) Taaai rnpVn yyaK nya *n naipa ^sk (10) tyatyn taaa 'a aitynai aam ik H pnaaai (11) 1 taaa ^>y aaoai mpVn ysaKa

ipa n naipa «?xk na’n

•’"-,a sf :3f || ’’33 ax ax

332

10

|| '-za ~n :jn 8

MICHAEL VERSION

iniRm rVi trairn iaaa “iirV Vapan ran mm papn mm nrRai (23) irairna ^an mm pai irairn pa D'lraRn rm (24) naan V'V rr’rr nrRai pRn 'traR •wa ms pai iam (25) mi: mmir mp** dri ia'VR raiar iVsr man mm ">mo' Van 'a npV'ir wnRiaVR VaR nrwR nra (26) mns ir mV»a a"' amirp hr mVa n*rn iriti (27) rn'1? wana irairn *iir war irairna mm nrrty iy pipin'* me* rVi ay pRn naira Vmm utr pRn Va mna maya mpaa raiyi 'pan nr nyan nsw d^ir iym naai .nVar pi iRa iy (28) aipaa raw nwra ianir na mrm inor inyipr (29) nrn ia •jV'* iitr pl’?nrv* mpan nr ■’a 'Vnn VaVa (30) iniRip*’ utr Rim nrn ia np*?** utr aya *inr (3D nnapai 071 'a avian napa mmi nripVn mnya mwir D3aR pRna apmai mmi irairn *fma mytr Vy nn nam nnr anapai mm pmaa aaaRi .iam impn Van m miy mm nan trairn pmaa (32) nnpVn mytr iaaa lyntr ma nr nwmV laiaim .mpVn d) nya Van nrna Rin inR amtrp nr Vy (2) mannnan mViay mum raiai .imVan mytri mwiri mVran VaVa naira na*1 naian VaVan nair m naV pRn Taia nairV T*an pan pRn naira -pain Vam .mm am Rin mman nniR (3) nm*1 rV nmirp rirn 'Ra mm nrRa mmi (4) iaaa Via** rV mVran VaVa VaVan mma Dipan (5) iniRa mm mVya a"** mwira nirpn ]a no** dri .am iV Ripan mm poaVR mmtr na mVam mr aya 071 a~o mVran VaVaa naian riai VaVan nr mn ppna itrRai 071 mV iy 071 a"V mm VaVa (6) 071 a"oa runs nT mmtr na am 'Vnn (7) VaVa nwir mn Vr mm VaVa mmtr mp nrRai (8) .Van nViay mm nViay wan rVi mm am Rin utr mns aya lam nm' mVya a"'a nine 'Vnn iy mm pair na nVapnn nya mm aaaa nam 'Vnn VaVai mm VaVa nan omytr 'aim iVr 'ana (9) mna nRn aya ir (10) mVya a"' iRana *iaR nrVi Vaa nVyn nr Va non

a66

a

*67

a

]Raa

mm VaVa nwir 'an Vy mmoim ppnair na na npi nona [Q68] d) mpVV nmir naya (2) Rin 'a yi rDWfln .nVman 'pi on RR'tr nai mpVn nVnnn nan 'tr'Vtra aaaR D'Viaa 'a m'atrai D'Viaa 'a naiirRi npiVna iVr nwir nyiV pain mVann ^101 mVann nVnm mpVn yaaRi (3)

urn nm nmm mp’ dki i:’Vn vxuc’j iPb’i -i’nopi mm :im 25 || ’"33 rsn pai ira nua 24 .’*aa nra 26 || ’"3a

333

II 72 ||

ii 7111

ton non

im mma?y 'mi man aa? nam naVnan ^y psno*> omn nr nxsin **3 nnra a~a nr nma? ns inma?ya mm d) -['maa -mxn p'mn aa?i ran *66 z inpVnm (2) nima?y 3"a nr nma? ns inma?ya mm *f?n» nana? *]ixi .nn»u m *raa p*?n (3) nnp*7 nan ran ^y inpVn *ia?xa m ran ^y naa m-iKa 333 ianax nan **!?nn VaVa *nsr» xin na?x *?xn nViay poaVx aaaxi apnnaV amoa nr nnma? ns mm a?aa?n pma ]a sir m m anpa? (4) amoa mm nm na?x aipaa ^xn (5) poaVxa xs^az na nww -a mxn ammpn (6) noam anoana *?xn poaVx nma? ixxan mm a?aa?n •’pnnaV non nrn *fma mmazy ma?a? nma? Vna VaVaa na?p xnv mnpVn nya poaVxV amDoaV nxn nmm nn a?aa>!? am nmaa?i (7) d'S’ a *]*?na 1333 nan paa?nm nooan (8) nt nmi nnx nxa aa^na mm a?aa?n m Vxn inVya nnx aioa iVia nmaa?n nmaa?i na?*?a?n a^a?1? *pxim jpinan -jVnaa na Va mma? na nm® an mu “^na a^V *pxim nn nr nm (9) na na nrn *j^nai nna a?aa?n (10) i^na nana? *pxi nnx 3io iaya? n"a (id mai nmaa?i na’Va’a nnxn nxan m mmaa?n 3iaa Vm nvra? my nmaa?i nn’’ 'a *]*?na nmpVa nna a?aa?n -[Vna nxana H pmom nax my /na nrn “iSnai n"aa a?aa?n ^na (12) nan naan nxrVi 'na omxn ]a nann iaa?m (13) a'pn ma^n nxa?*1® nai nannn ]a ayan nan nnpV “ptm aaaxi myan emVy Vian apn aa^a?1? •pox*1 ana? nr 'a iV a?^ na ninan xin n a’m mpV nm^y n^a?! nr rvma?y rra?a? (14) a?4?a? xin mean ja iV xx^ na?x m (15) naix -tax axi .nma?y ma?a? ma?na (16) nnx *?y *pnx xm inmnpy ma?a? nra nnp1? *]nx xin n nya xV a^a?3X ip’pn naai .rnmVa nr naT x1? •’anxia1?xi a^a? ^y mpVn’? rti3ian nya? -jVna na-1 anxp nax amy^a?n f?x nm (17) a^aVan iVx nxxina xx^a? nai nannn (18) p ayan nann n“ta?ya a?aa?n nya? *jVnBi ana nrn ]iaa?na xx^a? na laa xin ]iaa?nn nra xx^a? nai 'Vnn ^aba nya? xin xx^a? nai 'a Vy inpVnm na a?aa?n ^na nan nnx naxi (19) nnxiabx aaaxi na5? mnaa? nnn (20) *f?naB np' nn ^nn Vaba nsw xin xVi .nr Vy (21) la aai anam *»anxia*?x naxa? na rnimVa nat a?an

mnna npiV ana nr^x myT* a?aa?m (22) nm na?x mann py nnn mp1? rn^ya -tax HQ1?! [Q_67] mn mx *»a PDl^nn .mns ix mVya a''•> my^a aarn ]a ix a?xm ]a nanp

.’’33 n::n mran 21 || ’*33 D’oyD aina :ni’Tts>y d'j ny -]i^i ]d 1

334

MICHAEL VERSION

II 7° ||

Vtiio mR*aa aaaVn "aaan (8) napVn ny VaR anmanV nmsa mam many •’D

aaVna

p

maan

nyn

nnaRa

aaVna

mynya

nr

i»ri

.papaa

•mpVn nya do) amyty aym “iVnan aym -warn aaVna *pVna apmaa (9) rf?nnn myty Vy a'm ^nrv’ Ran (id nan 'Vnn ViVa aRip ntyR Va nap aaaRa amatymn *>a tyatyna pp inr pun m nn 02) amVaVia nanasan "pma nVa^y naty Vy pan Va Vmma pRn aaa yam ma a"op (13) nasaaa 'an ma a"a (15) Van ^Vnma nann pn rrrr’tr? ny pn namV mo" rV my pRn (14) tyatyn npm (16) myRaa Vnr aty Van m pRna tyatyn 'pma ^Vnnna nn pi nap Van mm (17) aaaa nap ntyRaa ay -pnR Van mm pRna nya Van mV nrn “[inn (is) am, f|Vnm nam niRay VaVaV •pa1’ Van patynn "Vya aanaan nam ara pawn pmaa *pVnaa apnna *pVna nanpVn anaR

-[mnn

apn

(20)

ny®

naan

nanpVn

nya

ammy*y

nynV

(19)

nmna tyatyn "pm ymty "a rVr nr ym rV (21) mRin nya Van p nrn maoaVR naytya patynn (23) "Vya apVn naaa .anma an apnna yma (22) aaaR .m (25) nRaana anaRa asVnna pa Van poaVRa (24) nanaRan nty .amanty aaa amp nt nanny ma nanpVn (26) naana ana ynr nan nr nnaR anaana (27) nr aym an "a aatyn an nan aanaa anaty ma mraRaaVR aaaR mmtyyi "ann nya nVana nViaya Vna" tyatyn poaVR m ammpn naam Ran nan antyy Vy pVma R^a nan naaa Vaa (28) nanpVn mnya mnn np naVnaV naRny (29) ran mmtyya nsoan nr "an myty npaVnn p Ran -inr nt "a naoaa anaanV anaaa nt a^Vy ntypn van mmtyy tyaty.n ^Vna ’an an "a aatyn an nan nmn nan *paR aa ty aRa nnR pmy Vana Vp (30) Vna1 Vaa aVaa nya nam poaVR "a amnapn (3D naana anaana aRaa rVr (32) naVnaV nt naR" "a atyn na'mtyy m n"aa pVn nVana nVaaya

335

6911

’par?: non

.mxm on anpn f?x [i] .mxm 'mma np [2] Fig. M-29

IN

no “loixi n mp3 x'm pm p (3D pmn ^nna xim in 2N mm xm (32) *wx Tl msrwn Vm inr nxr pan *?x mop mm xm mx n mip3*?xx ttnnnai TPI nn in 3n nn Nn np x^xia i3n:x ht nma .pirn *wxi nnx rvnt anaw Vnan mnr u>Vu> pn?an mm aipa xin d) rnx 365 a nxm nxim m naap nma (2) nxm mxa Vna inv nVna mm nxm mn Vm mn 3N msnw nxmw nxm 13 nxpi 3N nxp T1 mixo ay man ipnmn pxn ]a imp-' nmxam amaiani .ixa5? wx-iw na (3) mm pxn p naip mxai .anpty naa nr (4) noxa moi mxn nafm npnnm mxai (5) laoxnw na tfr na*?na ma’ nn aim VaVaa nniVowa mm o>ao>a nT laV nxom na mm naVna ma' an mm ^aVaa nnm pxna laVnm an nm 'nn amaian aaaxi nanxia Vxx napnn *?a*?a anV mx (6) min’? parnn -Vaa mxm nam .amp nt laixa (7) oaai *qpn Vaba naian