ANNALS OF THE NEW YORK ACADEMY OF SCIENCES
~dil" r!o
DAV ID A. KING 00
ANNALS OF THE NEW YORK ACADEMY OF SCIENCES VOLUME
500
PR(OM DEFEREÏ\TT TO EQUAN-T : A VO)LtJ~AE Q)F STUDIES IN THE HISTORY OF SCIENCE Il\ï TH.E A_NCIENT AND MEDIEVAL .."1.'.JEA ·r.;~ E .i", A. S'T E·O·~TOR OF ""-'_~, '-' .IN . ~. J~I ._ E "S " KEl~NEDY
THE NEW YORK ACADEMY OF SCIENCES NEW YORX, N"EW 'fORIC
1987
ANNALS OF THE NEW YORK ACADEMY OF SCIENCES
Volume 500 June 10, 1987
FROM DEFERENT TO EQUANT: A VOLUME OF STUDIES IN THE HISTORY OF SCIENCE IN THE ANCIENT AND MEDIEVAL NEAR EAST IN HONOR OF E.S. KENNEDY Editors DAVID
A.
KING AND GEORGE SALIBA
CONTENTS Editors' Preface ....... ... .. .. .. .. .... .. ..... .... .. .. ..... . .. ~. . . . . .
ix
E.S. Kennedy: A Brief Biography. By the Editors . . . . . . . . . . . . . . . . . . . . . . . .
xiii
List of Publications of E.S. Kennedy . ... .. ....... .. ...... .. .... ... ... .
xvii
The Collected Papers of E.S. Kennedy: A Brief Review. By J. Lennart Berggren
xxv
A Late-Babylonian Procedure Text for Mars, and Sorne Remarks on Retrograde Arcs. By Asger Aaboe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Spherical Trigonometry in Kiishyar ibn Labban's Jami' Zlj. By J. Lennart Berggren . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
J5
The Zlj of J:Iabash al-J:Iasib: A Survey of MS Istanbul Yeni Cami 784/2. By Marie-Thérèse Debarnot ........................................
35
Developments in the Solution to the Equ~tion cx + bx = a from al-Khwarizmi to Fibonacci. By Yvonne Dold-Samplonius . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
2
Zoomorphic Astrolabes and the Introduction of Arabie Star Names into Europe. By Owen Gingerieh ........... . ................... '. . . . . . . . . . . . . . . .
89
Descriptions of Astronomical Instruments in Hebrew. By Bernard R. Goldstein
105
A Survey of Medieval Islamic Interpolation Schemes. By Javad Hamadanizadeh
143
Chemical Technology in Arabic Military Treatises. By Ahmad Yousif al-Hassan
153
An Unknown Treatise by Sanad ibn 'Alï on the Relative Magnitudes of the Sun, Earth and Moon. By Anton M. Heinen . . . . . . . . . . . . . . . . . . . . . . . . .
167
Abu'l-Jiid's Answer to a Question of al-Bïriinï Concerning the Regular Heptagon. By Jan P. Hogendijk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
175
Sorne Early Islamie Tables for Determining Lunar Crescent Visibility. By David A . King ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
185
AI-Khwarizmï as a Source for the Sententie astrolabii. By Paul Kunitzsch ..
227
AI-~aghanï's Treatise on Projecting the Sphere. By Richard Lorch . . . . . . . . .
237
The Theory of Quadratic Irrationals in Medieval Oriental Mathematics. By'Calina Matvievskaya. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
253
The Chronological System of Abu Shaker (A.H. 654). By O. Neugebauer . " A Few Notes on Sundials. By Olaf Pedersen ..... . . ...... . . . . . .. . . ... . " .
279 295
Indian and Islamic Astronomy at Jayasi~ha's Court. By David Pingree . . . . The Two Versions of the Tüsï Couple. By F. Jamil Ragep . . . . . . . . . . . . . . . . . Kennedy's Geographical Tables of Medieval Islam: An Exploratory Statistical Analysis. By Mary H. Regier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313 329
~a'id,
373 403
the Toledan Tables, and Andalusï Science. By Lutz Richter-Bernburg On al-Bïrunï's Densimetry. By Mariam Rozhanskaya and B.A. Rosenfeld . . . Tables of Decimal Trigonometric Functions from ca. 1450 to ca. 1550. By Grazyna Rosiftska ... . . .. .. ... . . .. . ............ .. .. . ......... . .. ..
On a Mathematical Problem in al-Khazinï's Book of the Balance of Wisdom. By Mariam Rozhanskaya ....... .... .. .. .. .. . .. .... .. .. . . .... .... .. The Takmila fi'l-Ifisab of al-Baghdadï. By Ahmed Saidan . . . . . . . . . . . . . . . . The Height of the Atmosphere According to Mu~ayyad al-Dïn al-'Urçfi, Qutb al-Dïn al-ShïiaZï, and Ibn Mu~ adh. By George Saliba. . . . . . . . . . . . . . . . . . AI-Zarqal, Alfonso X and Peter of Aragon on the Solar Equation. By Julio Sams6... . ...... . .. . .. ..... ......... . ..... .. .. .. .. . . . ... . ........ Ibn Sïna and Buridan on the Motion of the Projectile. By Aydm Saylh . . .. A Treatise by al-Qabï~ï (A1chabitius) on Arithmetical Series. By Jacques Sesiano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jabir ibn Aflaljs Interesting Method for Finding the Eccentricities and Direction of the Apsidal Line of a Superior Planet. By Noel M . Swerdlow . . . . . . . . The Solar Theory of Az-Zarqal: An Epilogue. By G.J. Toomer. . . . . . . . . . . . The Dominican, the Benedictine, and the Moon. By J. Vernet. . . . . . . . . . . . The Heliocentric System in Greek, Persian and Hindu Astronomy. By B.L. van der Waerden ... .. .. . ....... . . . ... . ... . ...... . ..... . .. . ....... The Influence of Islamic Astronomy in China. By K. Yabuuti . ........... On the ~-Lines and I}-Circles of a Triangle. By Peter YH . . . . . . . . . . . . . . . . . /
357
419 427 437 445 467 477 483 501 513 521 525 547 561
.r
E. S.
KENNEDY
Editors' Preface plotting this volume for many years. We missed Ted's seventyfifth birthday, butwe now compute that publication will be about the time of his seventy-eighth lunar birthday, and this is the best that we can do. Shortly after he had assumed the presidency of the American University of Beirut, Malcolm Kerr agreed with us to write a preface to this volume. We are certain that Malcolm intended to write the same sentiments that he had expressed in his inaugural speech, namely, that he was proud to assume the presidency of a university which had Ted Kennedy on its faculty. But alas, Malcolm's tragic death has deprived us of this preface. Q As an alternative we, the editors, would both like to pa)Vt'ribute to Ted Kennedy in the same spirit, and since we came to know him from such different backgrounds, we present our own individu al reflections.
W
E HAVE BEEN
GS 1 have known Kindi on two levels: First, as an undergraduate student in the Mathematics Department at the American University of Beirut (before 1 had any interest in the history of the subject), and second, as his graduate student in Islamic science, and hence as a friend and later a colleague. J. In order to understand the impact that Kindi had on AUB students one must review the historic role of AUB within its natural surroundings, namely Lebanon and the Arab world. It was puzzling, for example, why one did not find there a de part ment of Islamic Studies, or courses on Islam, and this in an institution that was so naturally positioned to conduct such studies. Was the only reason lack of interest7 The University seemed to be saying to its hosts and neighbors that it had chosen a route of deliberate political indifference. The reasoning, 1 suspect, was that by remaining aloof, you could continue to operate in an environment that would not allow you the same privileges had you become "partisan, " or in any way counted amongst one or another of the numerous forces op erating in this period and region. One could say that when, after ten years of civil war, the US presence in Lebanon was well-nigh effaced, the only American presence that could still be tolerated was the AUB. In this university which had represented Western culture for over a hun~ dred years, Kindi was a refreshing maverick. He not only cared for the cula Malcolm Kerr President of the American University of Beirut, was assassinaled on Janu~ry 18, 1984. '
ix
x
ANNALS NEW YORK ACADEMY OF SCIENCES
ture around him, but indulged in studying it with enthusiasm. He dared even to tamper with that aloofness of AUB that had become a long-honored tradition. One gets the feeling that everything he did, the people he met, the books he procured and the trips he took, were all aspects of his involvement with his beloved Lebanon and his neighbors who became his own people. To an AUB student, Kindi represented what was good in the surrounding culture. He made that student proud to study in it, for there he was, in the midst of what was perceived to be the bastion of Western Science, studying the "native" culture and trying in every way he could to return it to his students. One must add that, for the average AUB undergraduate, the texts that Kindi assigned in Islamic Astronomy and Mathematics were all in comprehensible and shrouded in mystery. The situation, 1 am afraid, has remained very much the same up to this very day. Kindi, the student of Islamic science par excellence, the collector of Lebanese proverbs, was the American on the AUB campus who knew more about Arabie scientific works than his Arabic-speaking colleagues and students. In short, he was the American who got involved, and a professor as much a surprise to AUB as to his students. 1 suspect that, when they hired Kindi, AUB wanted to pave a good mathematics teacher for their Freshman Class, and that indeed they had, but they also had a star performer who almost embarrassed them in his sincerity and devotion. By his Middle Eastern students, however, and his Lebanese students in particular, he was welcomed as one of their own. He was, and still is, completely at home with them. Being a student of Kindi, on the other hand, was sheer joy. 1 am not talking merely about his remarkable sense of humor, his generosity, his unselfish sharing of personal files and microfilms (for he had all of these qualities), but rather the manner in which he allowed us to see the solutions of the most difficult pro~s as if they were our own, his gui ding hand extended all the while in a most unassuming way. It is his humility and the confidence he imparts to his students that make him much more than a teacher. He allows the student to feel the thrills and the excitement of being at the verge of discovery. Every manuscript might bring something new, and each folio might advance our understanding of Islamic astronomy and mathematics in a major way. But 1 do not think there is any student of Kindi who has not experienced his overpowering goodness. To describe this goodness in words, one cannot fail to call it a continuous act of giving, be it from his time on sabbatical at Brown University, or from his summer vacation amongst his pine trees on top of a Lebanese mountain: he was always available to any stumbling student who needed his help.
EDlTORS' PREFACE
xi
OAK
It was as a graduate student at Yale that 1 first met Ted Kennedy in the Fall of 1969. Ted was enjoying one of those precious years of research at Brown. It was both strange and exciting to meet the man who had written the numerous articles on Islamic science that had been my required reading. Our first encounter was a day that 1 shaH never forget, for it nas pr6foundly influenced my life since. Ted was always ready to welcome anyone to the fold who had discovered the excitement of working on the primary sources and realized the richness of the Islamic scientific heritage: he immediately agreed that 1 could come to AUB for a year to work on my thesis there. The Kennedys were not in Beirut when my wife and 1 arrived, but, typically, they let us move into their apartment during their abseJlCe. During the year 1970-71, Ted shared with me, as with George, books, photos, microfilms, and • ilm. He allowed me to read through his notes on zrjes, to use his parameter file, to copy his microfilms, and to photocopy from his collection of offprints. His intellectual generosity knew no bounds and was exceeded only by the hospitality of both of the Kennedys. We travelled with the Kennedys in Lebanon and Syria, and have been privileged to witness with our own eyes the Kennedys on the move. On one occasion we went on a bus trip to various historical sites in Syria with students and faculty from AUB. The Kennedys followed our rented Leb~se mountain bus in their Volkswagen. We were driving north from Qasr al-Hayr al-Sharqi to Palmyra when night fell and we became hopelessly lost. We strayed onto a Syrian military airfield and within minutes were surrounded by jeeps, trucks, and soldiers. 1 shall never forget that scene in the middle of the Syrian desert with thirty foreigners trying to talk their way off the airfield with a group of sceptical Syrian officers. What sticks in my mind most of aIl is the truck bearing an anti-aircraft gun trained not at the bus but at the Kennedys' Volkswagen. The Kennedys' hospitality was again available to us when we moved to Cairo in 1972. Beirut for Cairenes in those days was a haven and not yet in chaos. AIso, the Kennedys came to Cairo, and Ted experienced what he hoped was his first and last visit to the Egyptian National Library. But in 1976 they returned to Cairo for two years, and Ted, when eventually released from quarantine in Alexandria, was to put in many hours reading Persian astronomicaltexts in the Lîbrary. Ted and 1 colJabQrated on a preliInlnary revision of his zr; survey and w,orked through a few zfjes together, an activity which Dave Gordon at AUB used to call "zijjing about." Having the Kennedys as next-door neighbors is one of our happiest mem.ories of those Cairo days.
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ANNALS NEW YORK ACADEMY OF SCIENCES
The prospect of having them as neighbors in Frankfurt was no minor factor in our move there in 1985. GS and OAK
We count ourselves fortunate indeed to have known the Kennedys. Ted has stood for us as the model scholar, strictly interested in the Quellen und Studien approach to the history of science. But in addition we have savored his humanity and his hum or, as weIl as his concern for justice and peace and his love for the Near East. It will have become obvious to the reader who has not had this privilege that Ted Kennedy is a man truly respected by his friends and colleagues. But in dedicating this Festschrift to him, we also pay tribu te to his wife, MaryHelen, who accompanied him at every step and drove a major part of the way. We also pay tribute to the idea that a Western scholar can devote the major part of his life to working in the Near East in the service of its people and their cultural legacy.
.r ACKNOWLEDGMENTS
The suggestion that this volume of studies be submitted to the New York Academy of Sciences for publication was made by Prof. Walter Scott of New York University. We wish to express the gratitude of ourselves and all of the contributors to the Academy for agreeing to publish the work in the Annals of the Academy. We are also grateful to Bill Boland, Joyce Hitchcock and Cook Kimball of the Academy's Editorial Department for their collaboration. To Professor J.L,Berggren and Dr. J.-P. Hogendijk we express our thanks for their assistance in editing sorne of the papers on mathematics. The contributions of Dr. Debarnot, Dr. Sesiano, and Prof. Vernet were translated by Patricia C. King. GEORGE SALI BA
DAVID
A.
KING
E.S. Kennedy: A Brief Biography DWA.:&D STEWART KEN'N,EDY, known to his friends as Ted and to his closest coneagues also as al-Kindi, was born in San An,g el, D. E, Mexico, on January 3, 1912. His father was a building contractor and his mother a missionary. The family was thrown out of Mexico during the Pancho Villa revolution when the United States intervened. They took up residence in Easton, Pennsylvania, where the father resumed bricklaying for a living. Five brothers were born in the States. Ted attended the public schools of Easton up to high school, and then Lafayette College in Easton, working as a church janitor to pay his way through college. He graduated in 1932 at the height of U;le Depression with a degree in electrical engineering, but was uncertain about becoming an electrical engineer. Lafayette had an agreement with Alborz College in Tehran whereby they would support there one American teacher: Ted applied and was accepted for his first assignment in the Near East. The trip out from Easton was via Berlin and then overland through the USSR to Baku, and thence by boat to Iran. As was often the case in church-supported colleges, new appointees were assigned to any classes that were not otherwise provided for. Ted taught economies, matbematics, inter 'Ilia, and aIso serviced the college generator.. He served as basketball coach and as Boy Scout leader. During his four yeai-s in Iran he started to work on the history of mathematics and also masteréd Persian, but as yet had no plans for the kind of research he was to conduct on medieval Persian scientific manuscripts in later years. In 1936 Ted decided to return to the USA for doctoral studies. Back in Easton, he and two brothers invested $30 in a Ford and commuted to Lehigh University in Bethlehem, PA, where Ted studied mathematics. His thesis dealt with the Lambert Series. His intention was to get the Ph.D. and to return to Iran, but political events prevented this. In the faU of 1939 he received an invitation to teach mathematics at the University of Alabama. In 1941, with war imminent, Ted signed up for dut y in the US Army. On Oecember 7 of that year he was in Hawaii on his way to Iran. He slept through thè attack on Pearl Harber, assuming tha t the alert was outside his immediate C:0ncern. He learned of the cause of the alert later in the day fre~ a newspaper. After the US engagement in the Pacifie Ted returned to WashingtoD, DC, but Was eventually sent back to Iran, this time by way of South America and Africa. The remaining war years were spent in Tehran as Assistant Military Attaché at the US Embassy. He learned Russian because at that time the USA and the USSR were working together, and he had to be able to'communicate with Ms oounœrpart in the Soviet Embassy. Ted arrived as a lieutenant and
E
xiii
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ANNALS NEW YORK ACADEMY OF SCIENCES
departed as a major, and he tells a good story about hunting wild boar. His superior was a hunter and required the "protection of his assistants" on weekend trips into the Alborz Mountains and the plain below. In later years, Iran remained Ted's favorite country: for him, it had the friendliest people, the best cuisine, the most beautiful language, and the most exquisite poetry. Immediately after the war, Ted investigated the possibilities for studying Arabic in the USA. He ended up at Harvard, and, for what it is worth, he had a desk in a room next to George Sarton. That year (1945-46) he met Otto Neugebauer for the first time, an encounter which was to shape the rest of his scholarly life. In the faU of 1946 he joined the faculty of the American University of Beirut (AUB). Along with teaching mathematics, he continued the study of Arabie during a daily shave with the campus barber. The academic year 1949-50 was spent at Brown University and the Institute for Advanced Study at Princeton, the two places where Neugebauer was conducting his research. Ted returned to a tenured position at AUB, this time cycling from Rotterdam to Naples. Tenure was accepted with the proviso that every fourth year could be taken alternatively as leave without pay or sabbatical, in order to work with colleagues in the USA. I;r1950-51 he renewed the acquaintance of Mary-Helen Scanlon and the couple was married in Aleppo in June, 1951. They spent the year 1951-52 in Iran where Ted was a Fulbright Fellow: this was the year of Mosadegh and numerous demonstrations, and it provided a foretaste of the kind of problems they were to witness years later in Lebanon. From 1952 to 1976 the Kennedys were based in Beirut, with Ted teaching at AUB. They had three children: Anna (providence, 1953), Michael (Beirut, 1954) and Nora (Beirut, 1957). The Kennedys planned their retirement in the land they loved: they purchased a house and sorne land in the forests above Ainab overlooking Beirut. Yet theJrstay in Beirut was of necessity punctuated by numerous visits to Brown Uruversity and/or the lnstitute for Advanced Study. It was mainly during these periods in the USA that Ted carried out the independent research which led to the list of publications presented below. His interest in al-Kashï came early, resulting from his own personal associations with Iran. He started the zïj survey that first year with Neugebauer. A large (2m X 1 l/zm) cloth with many pockets served as a repository for index-cards with information on the various zïjes. It hung by the Kennedys' front door and was quite a conversation piece. In Beirut Ted was preoccupied with teaching introductory mathematics and computer science. When a course on the history of mathematics was finally put on the schedule at AUB, it was in the form of a one-semes ter introductory course for mathematics majors offered every second year. A one-semes ter seminar in which students did "independent" work in the intervening years
xv
E. S. KENNEDY BIOGRAPHY
resulted in an impressive number of joint publications with students and several important theses whieh Ted supervised. He put his own knowledge of computer science to good use, initiating the use of the computer in the analysis of medieval Islamie astronomie al tables. Over the years the family made numerous camping trips around the Near East and through Europe en route to or from the USA. In September, 1971, they put tents and sleeping bags in their Volkswagen and drove to Afghanistan to visit each of the sites where al-Bïrunï had lived and worked. By 1976 the Kennedy children had left home in various directions and the Kennedys were free to accept an invitation to participate in the Smithsonian Institution Project in Medieval Islamie Astronomy based at the Ameriean Research Center in Egypt. During 1976 and 1977, they lived in Zamalek in Cairo next door to the Kings, and Ted joined the ranks of the fe~ardy scholars who have survived the rigors of working winter days in the Egyptian National Library. Beginning in 1976, Ted began an association with the newly-founded Institute for the History of Arabie Science at the University of Aleppo, and in 1977 he assumed the editorship of the new journal published there. Together with Mary-Helen he was responsible for the excellence of the first few issues of that journal. They commuted between Cairo and Aleppo and later between Ainab and Aleppo for sorne five years, spending periods at the Institute of between three weeks and three months. Their devotion to the new ente~rise in Aleppo continued under conditions that became steadily worse and more precarious as the situation in Lebanon and in Syria deteriorated. It was during this period that their summer home in Ainab was converted to a year-round full-sized house. In 1979 the Kennedys settled down in Ainab to begin their retirement, with occasional trips to Aleppo. Ted was able to devote full time to his research, with time off for log-splitting and pieking grapes and not least for practicing the French horn. The last activity was a new love acquired at the age of 56, and prompted a colleague to compose: Der Kindi spielt den Doppelhorn
sowie von hinten aIs ven vern.
Ted had always been interestedin music, but he much prefers performing by liimse1f Or in a small group. He had plasred the trumpet, guitar and Bute, but ,now at last could affard to buy a French ,homo ln 1981 the Kennedys drove fr-oQ'l Amab to the International Congress for the History of Science in Bucharest, inevitably via, Rotterdam, and this in spite of Ted's famillarity with mathematical geography. ln 1982 Alnab and the area cu:ound their house was shelled by the Israelis, and there followed a year of Occ~pation by the Israelis and the puppet militia which they had bmught in.
xvi
ANNALS NEW YORK ACADEMY OF SCIENCES
Soldiers who wandered onto the Kennedys' property were surprised to find there American citizens who thought their presence was as outrageous as most l..ebanese did. One of Ted's hobbies was reading military history, but as a pacifist he was heart-broken to witness such history being made in his adopted land. Hiding in the cellar of their Ainab home du ring one of the shelling sprees, he and a friend found solace in recounting the military campaigns of the Andent Persians against the Greeks. When one day the Kennedys found that what was left of the village of Ainab had been deserted, they moved down to Beirut. In May, 1984, with the outbreak of random kidnappings of Americans, the Kennedys finally decided to leave their adopted land. The fact that the victims included friends of theirs who were men dedieated to the cause of peace in the Near East in general and to the l..ebanon in particular, caused them particular grief. The Kennedys have found a temporary haven at the newly-founded Institute for the History of Arabie and Islamic Sciences in Frankfurt-am-Main, where they are at this time. Ted is stiU churning out papers, encouraging colleagues and students, and playing the French horn with a smaU group of friends. ThyKennedy chiIdren live in three different continents and now the1 E:: ri
t""
~ .... Ul
tT1
> ~
..Cl
z
...
194
ANNALS NEW YORK ACADEMY OF SCIENCES
Suter, and translated into English and analyzed by O. Neugebauer. ~ It contains a table for crescent visibility which has been investigated by E.S. Kennedy and M. Janjanian .:Zo The table displays values of a funetion f("-) computed for each "face" of eaeh zodiacal sign, that is, each 10° of À.. The values are symmetrical about A = 180 ° and no latitude is mentioned. In the chapter on crescent visibility in al-Majri~ï's work the instructions state that the crescent will be seen if ~A > f(Am). Clearly neither instructions nor table are due ta al-Khwarizmï. The en tries in the table, as published by H. Suter, are displayed in TABLE 2. The same table, with minor variants, occurs in the Zïj of Ibn Is}:taq alTunisï, who worked in Tunis in the early 13th century.21 His Zïj is extant in the unique copy MS Hyderabad Andra Pradesh State Library 298 (ca. 200 fols., ca. 800 Hijra ::;:; A.D. 1400),22 where the table occurs as no. 160 of the 360 tables (see PLATE 4). The table is here and also in the 31st chapter of Ibn Is}:taq's introduction attributed to an individual named al-Qallas, on whom 1 have no infor;rnation whatsoever b eyond that his name indicates that he was a maker of skull-caps. Ibn IsJ:taq d o es not mention that the table is compiled for a specifie latitude. ('Iwo other tables for crescent visibility in his Zïj are respectively specifieally for Tunis and universal, and are attributed ta the Ilthcentury Andalusian scholar Ibn Mu ' adh, and the 12th-century Andalusian scholar and student of Maimonides, al-Sabtï. 23 These tables do not concern the present study but do merit detailed investigation.) For other tables associated with Ibn Is}:laq, see 1 and J on page 203. In their analysis of al-Majri~rs table, B.S. Kennedy and M. Janjanian were unable ta account for the symmetry of I(À.) in terms of the mathematical methods used by early Muslim astronomers. However, they cUd establish that the entrics correspond fairly closely to recomputation using the Indian parameter of 12° for the time between sunset and moonset (with an artificial condition to pro duce symmetry) and a latitude of about 42 °. But they were unable ta explain why the latitude underlying the table corresponds to the Pyrenees and Cantabrica rather than sorne center of Islamic culture such as Toledo. My investigations of the table reveal the following : (1) The table was clearly computed for the latitude of the fifth climate. The precise value of the latitude q> is related ta that of the obliquity E according ta the following scheme E
= 23;33 ° 23;35
q>
= 41;17° 41;14
E
= 23;51 24; 0
q>
=40;52 40;41
Recomputed values of the visibüity ronction vary no more th an a minute or two with any of these pairs of parameters. (2) The faet that entries are given fol' each "face" of each sign, corresponding to each 10° of ecliptic longitude, leads me to suspect that tables of p(À.) for
TABLE 2 Lunar Crescent Visibility Table in Adelard's Translation of al-Majriti's Recession of al-Khwarizmï's 2fj (A) and Attributed ta al-QalIas in the 2fj of Ibn Is};laq al-lunisî (B)
A: Su ter, 1914, 168, edited from three Latin manuscripts B: MS Hyderabad State Central Library 298, table no. 160. Faces/Signs
Visibility Function
Signs/ Faces 1 2 3
9;26° 9;25 9;21
9;26° 9;25 9;21
3 2 1
XII
II
1 2 3
9;19 a 9;18 b 9;21
9;19 9;18 9;21
3 2 1
XI
III
1 2 3
9;33 9 ;57 10;37c
9;33 9;57 10;37Î
3 2 1
X
IV
1 2 3
11;29 12;48 14;15
3 2 1
IX
V
1 2 3
15;58d 17;31e 19;11
11;29 12;48; 14;15 15;58 h
3 2 1
VIII
17;31 e 19;11
1 2 3
20;20 21; 4 21;17'
20;20 21; 4& 21;17'
3 2 1
VII
VI
,.,
".
a A has one reading 29. b B: 19. cA has one reading 9;37; B: 10;57. dB: 12;38. e B: 14;31. , Suter has 21;57, which Neugebauer (1962, 103) emends ta the ather attested value 21;17; B: 21;57. g A: 20;4. h B: 15;28. j B: 14;43. j B: 10;57.
Recomputation
CIl
CIl
m
Z n
m
n ...
o'T'l
-
n
~
o-< ::ç
Z m ~
CIl
> t'"
> Z Z
\() (Xl
' II
XI
X
IX
VIII
VII
VI
2
.
"N
l it la
10,l~_
"'5 1
111~1
9,51 11,11 11140
N
10'2b
ABCDEFGtf't
11,24 11'29
III'" 11147
10143
111 24
11 1 J
101 2 101 l
A
H BCGI"" E OF
10111
BCDEFGI1
•
N
101 9 111 q
_L-.!'~
...."91 '1
3:5128
lU11
9,28 10' 9 1012'1
9J1B
91 9
N
H
L J
BCDEFBIK
"CHI A
H B
11127
1" • 91S2
LN K
CDEFBI" J
A
11110 11120 11122
111 ..
91 •
11125
11134
N
lICDEF8 •
Il.26
11127
L
1112.
- ··
I J K
' " SI. HI 4 1111' lU17
AH
III 3
··
E
•
311.2
16112
L
'lue
BCG DE
11, 2
N
CO
·
H
",21
101 B
H
1 0 1~
DEF I
AH
ABCDEF~
"E 14.52
1
If
Il
" • 91~1
~121
\ BCIlEFB
"
AH
BDEF
caIN
"
.
.
1 ABCDEF_
ADfHIN BCG E
"
CSH AB CCSH IN llEFIO I
ABCDEFBHI ....
AIICllEFIIU let
9128 '12'1 9131
9111 9, 19
913:5
.,3:5 III 0
9115 9119
1113 1
9.1S 111 0 1114'
15153
A
"142
Il. 0 13, 3 1 14,:51 20.31
J N BCOCE1F9I"
14112 14121
DCEl
13139
BCIlfI
lBI 7 18.50
13, 2 lB' 2
ABCFIlN
14.50 1611b 16117 17116 17117
20':1:5
H
F N
"8
L
21128
17153
20113
... ~
H
J H I ADEFl
KING: EARLY ISLAMIC TABLES
203
on fol. 192v at the end of the main tables of the zïj and copied in the same hand. The table is not contained in MS Cairo Dar al-Kutub mïqat 1081 of a Maghribi version of Zacuto's tables. The eighth copy (H) of this table is found on the title folio of MS Cairo Tarat mïqat 119, a copy of the astrological treatise al-Masel'il fi 'l-a1:zkam of 'Umar ibn al-Farrukhan al-Tabarï (fi. Baghdad, ca. A.D. 800), prepared in 1185 Hijra (= A.D. 1771/72), apparently in Cairo. 35 The table bears no title and there is no indication of its purpose (see PLATE 6). Yet another source (1) is the Zïj of Ibn IsJ:taq al-Tunisï (see page 194). in which tables for three of the climates, namely, the third, fourth, and fifth, are displayed separately (see PLATE 4). In the Hyderabad manuscript they are tabulated alongside the table attributed to al-Qallas (see ab ove) . In his introduction Ibn IsJ:taq do es not mention the na me of any c~piler for the tables, and states simply that one should enter in the table for the appropriate climate with the sign of the moon and compare the entry with the longitude difference between the sun and moon, which is to be computed for six hours after midday on the 29th day of the lunar month. The entries in Ibn IsJ:taq's table correspond more or less to those in the Andalusian table for the second (not thirdl), fourth, and fifth climates. A single table for the fourth climate (J), which differs from those described above, is attributed to Ibn IsJ:taq in MS Paris B.N. ar. 2520,2, fol. 128r in the margin of a passage on crescent visibility theory in a recension of the zi} of Ibn al-Sha~ir by the Damascus astronomer Ibn Zurayq.36 The table is entitled Jadwal li-ru'yat al-hilal fi 'l-iqlïm al-rabi' li-Abï Is1:zaq al-Tunisï, "table for crescent visibility in the fourth climate by Abu [sic] IsJ:taq al-Tunisï." The entries are displayed separately in TABLE 4. The first four entries correspond to variant readings in the Andalusian table for the fourth climate, and the last four entries are close to but not the same as the last four values in the Andalusian table for the third climatel Another copy of the table for die seven climates is contained on fol. 165v of this same manuscript. The hand is different from that of the rest of the manuscript. This is the only copy in which the zodiacal sign is the horizontal argument and the climate is the vertical argument. The values from this copy of the table are not included in TABLE 3. Our next source (K and L) is a treatise on folk astronomy entitled alBayan or al-Tibyan bi-watJ.i1:z (?) al-burhan written by a Moroccan astronomer named Abu 'Abd Allah MuJ:tammad ibn •Abd Allah ibn •Amr al-Mu'addib ("the teacher").37 This work is extant in the unique MS Oxford Bodleian Pococke 249 (63 fols., copied ca. 800 Hijra ::::: A.D. 14007). On fol. 43v of the treatise the author presents a simple table for visibility- see (h), page.217 - together with another pair which concerns us here-see PLATE 7.
.UI "
t
, ~~,
~~\ -AJ\WI ~\W\
J/~\
,~ ,
j».1v .), ~ a.- J
Jo,. L ; \, .f r-" e..\ll ~ ~ -' '- lst '- JI. ~ \, \, ~ )jJl JJ' JA., .lAJ ,." ~ l - tr a- b \,..... \, -\ i 1
t-=--f---t---1I-=-----1~~-~ . _a~
._ct
~
_•
J --"" '.-1.)J \. "" t ù \, ~.jÙtr' ~ JI. /.1 ù b .).J ~ 1- t. ~ l.- ~ !i ~~~l ~.J'\t 1./ i:i- f ~~ ~ r .{.,., I ~~1
.lit
.-.1
, ..;s
).J...s
~
~~ ~:I- f jJ
. 4W \.
~~
\t jJ ~
'1
t ~r
-t_ .)J
ü
~~ ~
.-1
_ U~.
~_~' t
~ ~ ~ t~ ~ -=' 1a.--\, _c.r'~ ~
\; ...It Jt ".... \, f \, \st ~
~- .. -G -j" J ; /
0
\, i.J ~ .~
-;-~ u-- ~ lAt~
.
\,
~
~
\,
l,f.. ,~I
\, .1 .~
~,
\;-~. >r \,t~~~~ï:
6 . The table for the seven climates as it occurs on fol. Ir of MS Cairo Tal'at mïqat 119 of an astrological treatise by Ibn Farrukhan al-Tabarï:. (Courtesy of the Egyptian National Library.)
PLATE
204
KING: EARLY ISLAMIC TABLES
205
TABLE
4
Crescent Visibility Table for the Fourth Climate by Ibn Isl;laq al-Tünisl MS Paris Bibliothèque Nationale ar. 2520, fol. 128r. O' 1 2 3 4 5 6 7 8 9 10 11
10; 6° 10;21 10;12 14;32 14; 9 15; 1 18;20 14;50 13;39 11;15 11;10 11;11
The tables are labeled ru'yat al-ahilla bi-'l-'ashiyat / bi-'l-ghadat li-'l-iqlfm al-rabi', "lunar crescent visibility in the evenings / mornings for the fourth climate ." The city of Baghdad is specifically mentioned, although it lies roughly 3° south of the middle of the fourth climate. This is the only one of our sources containing a table for last visibility in the mornings - the entries for sign n in this table correspond roughly to those for sign n + 6 in the table for evening visibility. The en tries in the two tables, often mutually inconsistent, are displayed (labeled K for evening and L for morning) in TABLE 3. The last source (M and N), which came to my attention only in April, 1984, is a collection of astronomical and astrological treatises of ca. 150 folios in a manuscript in the private collection of Rabbi Kafah of Jerusalem. 38 The manuscript is clearly of Yemeni provenance and was copied ca. 1800 (7). There are two tables for lunar crescent visibility, one for the evenings (M) and the other for the mornings (N), each serving the seven climates. The entries are displayed in TABLE 3, and those in M and N are often mutually inconsistent. Clearly the entries for the fourth climate in sources Land N are related. In an introductory note it is implied that the tabulated function is a function of the lunar longitude (wa-khudh ma tajid qibalat burj al-qamar . . . ). My reconstruction of the table is only tentative. 1 have not been able to determine precisely how it was compile d, nor am 1 able to account for the inordinate number of copyists' errors. TABLE 3 shows the accurately computed values for E = 23;51,20° when the argument is the lunar longitude. In very few cases, values given in or reconstructed from the text agree closely (to less th an. ±6') with recomputation: these are indicated with asterisks in TABLE 3.
J
/'
'it'
~
"rp
~
« ~
t ~.
206
KING:
207
EARLY ISLAMIC TABLES
This table takes the prize as the most corrupt table in the known medieval Islamic sources. MISCELLANEOUS CRESCENT VISIBILITY TABLES 1 THE TABLE OF
ABU
JA'FAR AL-KHAZIN
Abü Ja in for the lunar longitudes in the nth sign then the crescent will be seen. Clearly the values in should correspond to a specifie latitude, but since the values are invariably given in integral degrees it is impossible to derive the underlying latitudes with any certitude. For each of the sources labeled (a) to (p), the values are displayed in TABLE 9. The most remarkable feature of the seventeen different sets of values discussed in this section is that they are aIl differentr (a) In MS Escorial ar. 927, fol. 6r, of the introduction to the anonymous Mosul recension of the Mumta1:zan Zfj (see ab ove, page 190), immediately preceding the visibility table attributed in other sources to al-Khwarizmisee I;LATE 2 - there is a section that translates as follows:
214
ANN ALS NEW YORK ACADEMY OF SCIENCES
9. al-Ladhiqls table for Lattakia as it appears on fol. 22v of MS Cairo Tal'at niiqat 92 of his Zïj. Note that there are two sets of values tabulated-see pages 210-211,213 and (n) on page 218. (Courtesy of the Egyptian National Library.) PLATE
In (signs) 0 (and) I l (the crescent) will be se en from I l degrees. (In signs) 1 (and) 10 it be seen from 12 (degrees). (In sigôs) 2 (and) 9 it will be seen from 12 de~s, 30 minutes. (In signs) 3 (and) 8 it will be seen from 14 degrces, 30 minutes. (In sign) 4 it will be seen from 15 degrees. (In signs) 5, 6, (and) 7, it will be seen from 18 degrees. Know this, if God Almighty wills.
'wm
The values given in the text are clearly independent of the values in the main table (cf. TABLE 1). (b) The treatise on theoretical astronomy and astrology entitled AI-Mudkhal compiled in 365 Hijra = A.D. 975/76 by Abu Na~r al-QummJ:51 contains in Maqala 2, Fa!? 1 12 a set of values of ln for an unspecified latitude. 1 have examined MSS Cairo Taleat mlqat 222,2 (fols. 60r-177r, 617 Hijra = A.D. 1220/21) of this work, in which the values occur in the text of fols. 99v-100r, and also the later copy MS Cairo Dar al-Ku tub Mu~~afâ Fâc;lil mïqiit 208 (91
loIs., ca. 1150 Hijra::::: A.n. 1750), where the sam.e values arearranged in tabular form on fol. 28v. al..Qu:mmï's treati.se contains a table of oblique ascensions for
KING:
215
EARLY ISLAMIC TABLES
TABLE 9 Crescent Visibility Tables for Various Latitudes a (a)
(b)
(c)
(d) (e)
(f)
(h)
(g)
(i)
0 1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
11 a
11 °
10
12
11
12
14 16 18 18 16 14 12
11
12 '/2 14 '/2 15 18 18 18 14'/2 12 '/2 12
12
11
10
8 10 12 14 16 17 19
11
8 8 9 11 13
16 16 15 12 11
8 8
10 9110 10 10 13 13 14 15 15 15 15 18 17 19 18 17 19 18 15 19 15 15 15 15 14 15 12 11 10 9 11 10 9
10 10
11 11 13
11 13
16 19 19 16 13 11
10 10
12 13 14 15 16 17 17 16 15 14 13
12
(k)
(j)
Signs
10 10 13
14 15 19 19 15 14 12 10 10
(1)
(m)
(n)
(a)
(p)
10 10
9 10
12 13
9 8 8 9 10 12
13 12 14 14 16 16 15 16 14 14 12
(1) 12
(2) 12
13
13
14 15 17 18 18 16 15 14
14 13 11 15 14 14 16 16 15 18 19 19 18 19 18 16 15 16 15 14 15 14 13 / 14 -"12 10 11 12 10 12 10
13 12
14 15 16 18 18 16 15 14 13
12
13 12 11
9 8 9
11
See Miscellaneaus Crescent Visibility Tables II.
36;0 ° (fol. 78v of the Tal
