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A History of Ancient Mathematical Astronomy

Otto Neugebauer (auth.)

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"This monumental work will henceforth be the standard interpretation of ancient mathematical astronomy. It is easy to point out its many virtues: comprehensiveness and common sense are two of the most important. Neugebauer has studied profoundly every relevant text in Akkadian, Egyptian, Greek, and Latin, no matter how fragmentary; [...] With the combination of mathematical rigor and a sober sense of the true nature of the evidence, he has penetrated the astronomical and the historical significance of his material. [...] His work has been and will remain the most admired model for those working with mathematical and astronomical texts.

D. Pingree in Bibliotheca Orientalis, 1977

"... a work that is a landmark, not only for the history of science, but for the history of scholarship. HAMA [History of Ancient Mathematical Astronomy] places the history of ancient Astronomy on a entirely new foundation. We shall not soon see its equal.

N.M. Swerdlow in Historia Mathematica, 1979

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Springer Berlin Heidelberg

Series:

Studies in the History of Mathematics and Physical Sciences 1

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Studies in the History of Mathematics and Physical Sciences  1  Editors M.J. Klein G.J. Toomer  O. Neugebauer  A History of Ancient Mathematical Astronomy  In Three Parts with 9 Plates and 619 Figures  Springer-Verlag New York Heidelberg Berlin 1975  Otto Neugebauer Brown University, Providence, Rhode Island 02912, USA  ISBN-13: 978-3-642-61912-0 e- ISBN-13: 978-3-642-61910-6 DOl: 10.1007/978-3-642-61910-6 Library of Congress Cataloging in Publication Data. Neugebauer. Otto. 1899-. A history of ancient mathematical astronomy. (Studies in the history of mathematics and physical sciences; v. I). Includes bibliographies and indexes. Contents: pI. 1. The Almagest and its direct predecessors. Babylonian astronomy. - pt. 2. Egypt. Early Greek astronomy. Astronomy during the Roman Imperial period and late antiquity. - pI. 3. Appendices and indices. 1. Astronomy, Ancient - History. 2. Astronomy - Mathematics - History. I. Title. II. Series. QBI6.N46. 520'.93. 75-8778. Springer is a part of Springer Science+ Business Media springeronline.com This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, spocifically those of translation. reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with .the publisher. © by Springer-Verlag Berlin Heidelberg 1975. Softcover reprint of the hardcover lst edition 1975  SPIN: 11016977 - 41/3111  To the Owl and the rabbit  The opposite of an Introduction is a Contradiction Owl (The House at Pooh Corner)  Pendant que j'etudie I'astronomie,je ne pense ni Ii Balkis. ni Ii quoi que ce soit au monde. Les sciences sont bienfaisantes: elles empechent les hommes de penser. A. France, Balthasar (CEuvres IV, p.141)  Preface This work could properly go under the title which Petrarch,;  in 1367, gave to one of his latest writings: "De sui ipsius et multorum ignorantia." By ignorantia I do not mean the obvious fact that only a small fraction of ancient astronomical theory can be restored from the scattered fragments that have survived. What I mean is the ignorantia auctoris in comparison with the scholarship of the 18th and 19th centuries and the ignorantia multorum to whom such a work might be addressed. In many years of study I have tried to become familiar with the ancient methods of mathematical astronomy, to realize their problems, and to understand their interconnections and development. Perhaps I may say that my approach is nearest to Delambre's in his Histoire de l'astronomie (ancienne: 1817, Moyen age: 1819, moderne: 1821) though I fully realize that I do not have by far the professional competence of Delambre. Yet I have tried to come as close as possible to the astronomical problems themselves without hiding my ignorantia behind the smoke-screen of sociological, biographical and bibliographical irrelevancies. The general plan of the following is simple enough. I begin with the discussion of the Almagest since it is fully preserved and constitutes the keystone to the understanding of all ancient and mediaeval astronomy. Then we go back somewhat in time to the investigation of earlier periods, in particular to Babylonian astronomy, for which we have a fair amount of contemporary original sources. Next comes the most fragmentary and most complex section: the investigation of early Greek astronomy and its relation to Babylonian methods. Finally Book V brings us back to safer ground, i.e. to material for which original sources are again extant: Hellenistic astronomy as known from papyri, Ptolemy's minor works and the "Handy Tables". The appendices (Book VI) contain details concerning technical terminology and descriptions of chronological, astronomical, and mathematical tools. The present work covers only about the first half of a much more ambitious plan (laid out in the early 1950s). I had hoped to be able to carry the discussion down to the latest aspects of .. ancient" astronomy. i.e. the astronomy of Copernicus, Brahe, and Kepler. I did not feel it was necessary to eliminate all traces of this overly optimistic plan. As for all books on a complex scientific subject there exists only one ideal reader, namely the author. Topics are selected, viewpoints taken, and answers formulated as they appeal to his taste and prejudices. In the course of more than twenty years, students, friends and collaborators have been exposed to my way of looking at the history of astronomy, and in turn they have influenced my views while adopting and developing some of mine in their own work. This gives me some hope that also in the future sympathetic readers might exist who are willing to penetrate the jungle of technical details and become fascinated by the kaleidoscopic picture which I have tried to unfold here of the history of the first and oldest natural science.  VIII  Preface  It is with feelings of sincere gratitude that I acknowledge my indebtedness to the generous support of my work that I have enjoyed for many years from Brown University and from the Institute for Advanced Study in Princeton. At the Institute I also had the help of Mrs. E. S. Gorman and of Miss Betty Horton whose patience and accuracy greatly facilitated the preparation of the manuscript. My thanks are due in no small measure to the Springer Verlag whose initiative made the publication of this work possible, just as it did that of my first book, fifty years ago, and repeatedly thereafter. Finally I want to acknowledge with gratitude the work of my good friends and associates Janet Sachs and Gerald Toomer for their persistent efforts to improve my English usage and to modify untenable positions in some topics. What remains uncorrected is entirely my responsibility. By deciding to put this manuscript into print, the moment has come when these pages themselves turn into a part of the past. I can only ask for the indulgence of my younger colleagues and friends, and of their pupils, when they see that I have overlooked or misinterpreted what a new generation now can see more clearly in this never-ending process. KlXi o[rc~ &nepXOJLIXI ... ~ JL11bB &p~fJ.JLeVor;.l  Providence, June 1975 1  'Appa.r; llrxJlPci) (Migne, PO 65, 369 II').  O.N.  Table of Contents Part One Introduction § I. Limitations. . . . . . . . . . § 2. The Major Historical Periods, An Outline A. The Hellenistic Period B. The Roman Period. C. Indian Astronomy . D. The Islamic Period . E. Epilogue . . . . § 3. General Bibliography A. Source Material .  1  2 3 5  6 7 14 15 15 16 17  B. Modern Literature C. Sectional Bibliographies.  Book I The Almagest and its Direct Predecessors A. Spherical Astronomy . § I. Plane Trigonometry. . I. Chords . . . . . . 2. The Table of Chords 3. Examples. . . . . 4. Summary . . . . . § 2. Spherical Trigonometry I. The Menelaos Theorem. 2. Supplementary Remarks § 3. Equatorial and Ecliptic Coordinates I. Solar Declinations . . . . . . 2. Right Ascensions. . . . . . . 3. Transformation from Ecliptic to Equatorial Coordinates. § 4. Geographical Latitude; Length of Daylight . I. Oblique Ascensions. . . 2. Symmetries . . . . . . 3. Ascensional Differences . 4. Ortive Amplitude . . . 5. Paranatellonta. . . . . 6. Length of Daylight; Seasonal Hours 7. Geographical Latitude; Shadow Table § 5. Ecliptic and Horizon Coordinates. . . 1. Introductory Remarks . . . . . . 2. Angles between Ecliptic and Horizon 3. Ecliptic and Meridian.  21 21 21 22  24  25 26 26 29  30 30  31 32 34 34 35 36 37 39 40 43 45 45 46 47  x  Table of Contents 4. Ecliptic and Circles of Altitude. 5. The Tables (Aim. II, 13) .  B. Lunar Theory . . . . . . . . § 1. Solar Theory. . . . . . . 1. The Length of the Year . 2. Mean Motion . . . . . 3. Anomaly . . . . . . . A. Eccenter and Epicycles. B. Determination of Eccentricity and Apogee C. The Table for the Solar Anomaly and its Use § 2. Equation ofTime . . . . . . . . . . . . 1. The Formulation in the Almagest (III, 9). 2. Examples. . . . . . . . . . . . . . 3. Proof of Ptolemy's Rule. . . . . . . . 4. The Equation of Time as Function of the Solar Longitude § 3. Theory of the Moon. First Inequality; Latitude 1. Introduction. . . . . . . . 2. Mean Motions. . . . . . . . . . 3. Period of the Lunar Anomaly . . . 4. Radius and Apogee of the Epicycle . A. Summary of the Method . . . . B. Numerical Data and Results . . C. Check of the Mean Anomaly; Epoch Values 5. The Tables for the First Inequality . . . . . 6. Latitude . . . . . . . . . . . . . . . . A. Mean Motion of the Argument of Latitude B. Epoch Value for the Argument of Latitude C. The Lunar Latitude; Example. . . § 4. Theory of the Moon. Second Inequality . 1. Empirical Data and Ptolemy's Model . 2. Determination of the Parameters. . A. Maximum Equation; Eccentricity B. .. Inclination". . . . . . . . . C. Critical Remarks . . . . . . . 3. Computation of the Second Inequality; Tables 4. Syzygies . .  § 5. Parallax. . . . . . . . . . 1. Introduction. . . . . . . 2. The Distance of the Moon. 3. Apparent Diameter of the Moon and of the Sun A. Ptolemy's Procedure. . . B. Criticism. . . . . . . . 4. Size and Distance of the Sun. A. Hipparchus' Procedure. . B. Historical Consequences . 5. The Table for Solar and Lunar Parallax (Aim. V, 18) 6. The Components of the Parallax . . § 6. Theory of Eclipses. . . . . . . . . . 1. Determination of the Mean Syzygies 2. Determination of the True Syzygies. 3. Eclipse Limits. . . . . . 4. Intervals between Eclipses. 5. Tables (VI, 8) . . . . . 6. Area- Eclipse- Magnitudes 7. Angles of Inclination . .  48 50  53 53 54  55 55  56 57 58  61 61 62  65 66 68 68  69 71 73 73 76 78  80 80 80 81 83  84 84  86 86  88  91 93  98 100  100 101 103 104 106 109 109 111 112 115 118 118 122 125 129 134 140  141  Table of Contents C. Planetary Theory .  § 1. Introduction . I. General . . 2. Distances and Eccentricities 3. Ptolemy's Introduction to Almagest IX 4. Parameters of Mean Motion. §2. Venus . . . . . . . . . . . . . . 1. Eccentricity and Equant. . . . . 2. Mean Motion in Anomaly. Epoch 3. The Observational Data. § 3. Mercury. . . . . . . . . I. Apogee . . . . . . . .  2. 3. 4. 5.  Eccentricity and Equant. Perigees . . . . . . . Mean Motion in Anomaly.Epoch. Minimum Distance and Motion of the Center of the Epicycle.  § 4. The Ptolemaic Theory of the Motion of an Outer Planet I. The Basic Ideas . ... . . . . . . . . . . 2. Refinement of the Model . . . . . . . . . 3. Determination of the Eccentricity and Apogee A. Eccentricity from Oppositions. . . B. Approximative Solution . . . . . C. Separation of Equant and Deferent D. Results . . . . . . . 4. The Size of the Epicycle. . 5. Mean Motion in Anomaly. 6. Epoch Values . . . § 5. Planetary Tables . . . I. The General Method  2. Numerical Data . . 3. Examples. . . . . A. Ephemeris for Mars B. Ephemeris for Venus. § 6. Theory of Retrogradation 1. Stationary Points . . A. Mean Distance . .  B. Maximum Distance C. Minimum Distance D. Numerical Data . . 2. Tables for Retrogradations A. Epicycle at Extremal Distances B. Epicycle at Arbitrary Distances; Tables. C. Examples . . § 7. Planetary Latitudes . 1. The Basic Theory 2. Numerical Data. A. The Outer Planets. B. The Inner Planets . 3. The Tables Aim. XIII, 5 . A. Outer Planets. . . B. Inner Planets. . . C. Extremal Latitudes D. Transits . . . . . § 8. Heliacal Phenomena ("" Phases ") . 1. Maximum Elongations . . . . . . . . . . . . . . . . . . . . . .  XI  145 145 145 146 148 150 152 152 156 158 158 159 161 163 165 168 170 170 171 172  173 174 175 177  179 180 182 183 183 184 186 186 187 190 191 192 193 196 197 202  202 204  205  206  207 207 208 212  216 218 221  226 227  230 230  Table of Contents  XII A. Venus . . . . . . . . .  2. 3. 4.  5.  B. Mercury . . . . . . . . C. The Tables (Aim. XII, to). The "Normal Arcus Visionis" A. Ptolemy's Procedure. . . B. Numerical Details. . . . Extremal Cases for Venus and Mercury A. Venus . . . . . . . . B. Mercury . . . . . . . The Tables (Aim. XIII, 10). A. Example . . . . . . . B. Method of Computing the Tables The Planetary Phases in the Handy Tables and Other Sources  D. Apollonius. . . . . . . . . . . . . . .  231 232 233 234 234 236 239 239 241 242 243  244  256 262  § 1. Biographical Data. . . . . . . . . .  262  § 2. Equivalence of Eccenters and Epicycles. 1. Transformation by Inversion. . . 2. Lunar Theory . . . . . . . . . . § 3. Planetary Motion; Stationary Points. . 1. Apollonius' Theorem for the Stations 2. Empirical Data  263  E. Hipparchus . . . . . . . . . . . .  264 265 267 267 270 274  § 1. Introduction . . . . . . . . . .  274  § 2. Fixed Stars. The Length of the Year 1. Stellar Coordinates. Catalogue of Stars A. Stellar Coordinates . . . . . . . B. Hipparchus' and Ptolemy's Catalogue of Stars. C. Catalogue of Stars. Continued. . D. Stellar Magnitudes . . . . . . 2. The Length of the Year. Precession . A. Tropical and Sidereal Year. . . B. Intercalation Cycles. . . . . . C. Constant of Precession; Trepidation .  277 277 277 280 284 291 292 293 296 297  § 3. Trigonometry and Spherical Astronomy . 1. Plane Trigonometry; Table of Chords. 2. Spherical Astronomy .  299 299 301  § 4. Solar Theory. . . . . . . . .  306  § 5. The Theory of the Moon. . . . I. The Fundamental Parameters A. Period Relations . . . B. The Draconitic Month. C. The Epicycle Radius. 2. Eclipses. . . . . . . . . A. Tables . . . . . . . . B. Eclipse Cycles and Intervals 3. Parallax . . . . . . . . . . 4. Size and Distance of Sun and Moon A. Distance of the Sun . . B. Hipparchus' Procedure.  308 309 309 312 315 319 319 321 322 325 325 327  § 6. Additional Topics. 1. The Planets . 2. Astrology. . . 3. Geography . . A. Geographical Latitude.  329 329 331 332 333  Table of Contents  XIII  337 338 339  B. Longitudes. . . . . . . . 4. Fragments . . . . . . . . . § 7. Hipparchus' Astronomy. Summary  BookH Babylonian Astronomy Introduction . . . . . . . . . . . . . . . . . § 1. The Decipherment of the Astronomical Texts § 2. The Sources . . . . .  § 3. Calendaric Concepts. . . . 1. The 19- Year Cycle . . . 2. Solstices and Equinoxes. 3. Sirius Dates. . 4. Summary . . . . . § 4. Length of Daylight . . I. Oblique Ascensions. 2. Length of Daylight . § 5. Solar Motion. . . . . § 6. Mathematical Methodology 1. System B . 2. System A . . A. Planetary Theory. . § I. Basic Concepts . § 2. Periods and Mean Motions. § 3. System A . . . . . . . . §4. Dates . . . . . . . . . . § 5. Subdivision of the Synodic Arc; Daily Motion. I. Subdivision of the Synodic Arc. A. Jupiter. . B. Mars . . . . . . . . . . C. Mercury . . . . . . . . . 2. Subdivision of the Synodic Time; Velocities A. Summary; Jupiter . B. Mars . . 3. Daily Motion A. Jupiter. . B. Mercury. § 6. The Fundamental Patterns of Planetary Theory I. System A . . . . . . . . . . . A. Numerical Data . . . . . . . B. Subdivision of the Synodic Arc C. Approximate Periods 2. System B . . . . . . . 3. Historical Reminiscences § 7. The Single Planets. I. Introduction. 2. Saturn . . . A. System A B. System B. C. Subdivision of the Synodic Arc; Daily Motion.  347 348 351 353 354 357 363 365 366 368 369 371 373 374 375 380 380 388 392 394 397 398 398 399  401  404 404 406  412 413  418 420 421 422 422 426 427 431 434 434  436 437  439 439  XIV  Table of Contents 3. Jupiter . . . A. System A B. System B. C. Subdivision of the Synodic Arc D. Daily Motion. . . 4. Mars . . . . . . . . A. Periods; System A. B. System B . . . . . C. Subdivision of the Synodic Arc; Retrogradation. 5. Venus . . . . . A. Periods . . . B. Ephemerides 6. Mercury . . . . A. Periods . . . B. System AI to A3  B. Lunar Theory . . § 1. Introduction . § 2. Lunar Velocity 1. System B . 2. System A . 3. Daily Motion 4. Summary . . § 3. The Length of the Synodic Months 1. System B, Column G . . . . 2. System A, Columns 41 and G. . A. The Function 41. . . . . . B. Column G near the Extrema C. The Function G. . . . . . 3. System A, Column J . . . . . 4. System A, Columns C', K, and M . 5. System B, Columns H to M A. Summary . . . B. Columns Hand J . . C. ColumnM . . . . . § 4. The" Saros" and Column 41 1. The Functions 41* and F* 2. The Saros. . . . . . . 3. 41, Friends and Relations A. Summary . . . . . B. Mathematical Methodology C. Numerical Details . . . . . § 5. Lunar Latitude. . . . . . . . . 1. Retrogradation of the Lunar Nodes. 2. System A, Column E . . 3. The Saros. . . . . . . 4. Other Latitude Functions § 6. Eclipse Magnitudes 1. System A . 2. System B . . . § 7. Eclipse Tables . . § 8. Solar Mean Motion and Length of Year § 9. Variable Solar Velocity 1. Type A and B . . 2. System A and A' . 3. System B . . . .  441 444 446  447 452 454  454 457 458 460 460  461 466 466 468 474 474 476 477  478 480 481 482 483 484 484 485 487 488 490  492 492 492 496 497 499 502 505 505 506 507 514 514 514 517 520 521 522  523 525  528 530 530  531 533  Table of Contents  § 10. Visibility. . . . . . . . . 1. The Date of the Syzygies 2. First Visibility. . . . . 3. Last Visibility and Full Moons. 4. Visibility Conditions . . . . . C. Early Babylonian Astronomy. . . . . § I. Calendaric Data, Celestial Coordinates . § 2. The Moon . § 3. The Planets . . . . . . . . . . . .  xv 533 534 535 538 539 541 541  547 553  Part Two Book III Egypt § 1. Introduction and Summary.  559  § 2. The 25-year Lunar Cycle . § 3. Concluding Remarks  565  563  566 566  § 4. Bibliography . . . . . . A. General . . . . . . . B. Demotic and Coptic Texts.  567  800kiV Early Greek Astronomy Introduction  . . . . . . . . . .  A. The Beginning of Greek Astronomy . § 1. Chronological Summary . I. The Early Period. . . . . . 2. More Recent Period . . . . § 2. Sphericity of the Earth; Celestial Sphere and Constellations .  § 3. Geminus. . . . 1. Date. . . . . 2. The Isagoge. . 3. The Parapegma § 4. Babylonian Influences . 1. The Sexagesimal System. 2. The Ecliptic and its Coordinates A. Aries 80 as Vernal Point . . B. Other Norms for the Vernal Point. 3. Mathematical Astronomy . A. Lunar and Solar Theory . B. Planetary Theory . . . . 4. Ancient Tradition; Summary A. "Schools" and Astronomers B. Parapegmata. . . C. Summary . . . . . . . B. Early Lunar and Solar Theory . . . § 1. Luni-Solar Cycles; Lunar Theory 1. Early Greek Cycles. . . . .  571  573 573  573 574 575  578 579  581  587  589 590  593  594  598  601 601 604  607 610 612 613  615 615  619  XVI  Table of Contents 2. The Metonic and Callippic Cycle . 3. Lunar Theory .  § 2. Solar Theory. . . 1. Solar Anomaly 2. Solar Latitude . 3. The Trepidation of the Equinoxes § 3. Sizes and Distances of the Luminaries 1. Aristarchus . . . . . . . . . A. Aristarchus' Assumptions . B. Mathematical Consequences C. Numerical Consequences. D. Aristarchus' Procedure. E. Summary . . . . . . 2. Archimedes . . . . . . . A. The "Sand-Reckoner" . B. Cosmic Dimensions. . 3. Posidonius . . . . . . . A. Measurement of the Earth B. Size and Distance of the Moon C. Size and Distance of the Sun . 4. Additional Material . . . . . . A. Apparent Diameter of Sun and Moon B. Distances of Sun and Moon . C. Actual Sizes of Sun and Moon § 4. Eclipses . . . . . . § 5. The" Steps" (fJa.8poi) C. Early Planetary Theory § 1. Eudoxus. . . . .  1. General Data . 2. The Homocentric Spheres . A. The Eudoxan Model. B. Numerical Data . . . C. Later Modifications . 3. The" Eudoxus Papyrus" A. The Text . . . . . . B. Summary of Contents § 2. Other Planetary Hypotheses 1. Arrangement of the Planets 2. Cinematic Hypotheses . . § 3. The Inscription of Keskinto. . D. The Development of Spherical Astronomy § 1. Arithmetical Methods; Length of Daylight; Climata 1. Length of Daylight. 2. Oblique Ascensions. A. SystemA B. SystemB . . . . 3. Climata . . . . . . A. Climata and Rising Times B. Early Mathematical Geography . § 2. Shadow Tables . . . . . . 1. Arithmetical Patterns. . . . . . . A. Greek Shadow Tables . . . . . B. Late Ancient and Medieval Shadow Tables 2. Shadow Lengths in Greek Geography. . . . .  622 624 626 627 629 631 634  634 635 636 637 639 642 643 643 647 651 652 654 655 657 657 659 662 664 669 675 675 675 677 677  680 683 686 686 687 690 690 693 698 706 706 708  712  715 721 725 727 733 736 737 737  740 746  Table of Contents  § 3. Spherical Astronomy before Menelaus 1. Authors and Treatises 2. Figures in the Texts 3. Spherics . . . . . . A. Polar Days. . . . B. Directional Terms. C. Non-Intersecting Semicircles D. "Interchange" of Hemispheres 4. Fixed Star Phases . . . . . . . 5. Rising Times, Length of Daylight, Geographical Data . 6. Later Developments § 4. Plane Trigonometry. . . . . . . . . . . . . . . . .  XVII 748 748 751 755 757 758 758 759  7(fJ  763 767 771  Book V Astronomy during the Roman Imperial Period and Late Antiquity Introduction .  779  A. Planetary and Lunar Theory before Ptolemy § 1. Planetary Theory . . 1. Introduction . . . . . . . . 2. Planetary Tables. . . . . . . A. Arrangement and Contents. B. Notation. . . . . . . . . C. Historical Questions. . . . 3. Planetary Theory in Vettius Valens . A. Solar Longitudes . B. The Outer Planets. . . C. Venus and Mercury . . 4. Incorrect Epicyclic Theory. A. Pliny . . . . B. Pap. Mich. 149 . . . .  781 781 781 785 785 788 789 793 794 794 796 801 802 805  § 2. Lunar Theory . . . . . . . 1. P. Ryl. 27 and Related Texts. A. P. Ryl. 27 . . . . . . . B. P. Lund Inv. 35a . . . . C. The 25-year Cycle and the Epoch Dates D. India . . . E. PSI 1493 . . . . . . . . . . 2. Vettius Valens. . . . . . . . . A. Lunar Longitudes and Phases. B. Lunar Latitude . § 3. Visibility Problems 1. Moon . . . . . . 2. Planets. . . . . .  808 808 809 813  815 817 822 823 824 826 829 829 830  B. Ptolemy's Minor Works and Related Topics § 1. Biographical and Bibliographical Data . 1. The" Almagest". . . . . . . . 2. Later Tradition . . . . . . . .  834  § 2. The" Analemma" and its Prehistory . 1. Introduction. . . . . . 2. Diodorus of Alexandria. A. Biographical Data. .  839 839  834 836 838  840 840  XVIII  Table of Contents B. The Determination of the Meridian Line . C. Pappus' Commentary to the" Analemma"  3. Vitruvius . . . . . . . . . . . . . . . 4. Great Circle Distance between Two Cities . A. Heron . . . . . . B. "Dioptra 35". . . . . . . . 5. Spherical Coordinates . . . . . A. Ptolemy's Coordinate System. B. The" Old" System of Coordinates . 6. Construction of the Ptolemaic Coordinates A. The Hektemoros B. The Six Angles . C. Graphic Solution D. Tables . . . . . E. Application to Sun Dials. 7. The Origin of the Conic Sections .  841 843 843  845 845 847  848 849 849 850 850 851  852 854 855 857  § 3. The" Planisphaerium" . 1. Introduction. . . . 2. Auxiliary Theorems 3. Right Ascensions. . 4. Oblique Ascensions. 5. The Greatest Always Invisible Circle 6. Ecliptic Coordinates . . . . 7. Historical Remarks; Synesius . A. Introduction . . . . . . . B. Earliest History; Hipparchus C. Vitruvius and the Anaphoric Clock D. Ptolemy . . . . . . . . . . . . E. Synesius. . . . . . . . . . . . F. Theon, Severus Sebokht, Philoponus .  857 857  § 4. Map Projection. . . . . . . . . .  879 879  1. 2. 3. 4. 5.  The Marinus' Projection . . . . . Ptolemy's First Conic Projection. . Ptolemy's Second Conic Projection . Visual Appearance of a Terrestrial Globe Appendix. Precession-Globe (Aim. VIII, 3) .  860 861 864  865 866 868 868 868 869 870 872  877  880 883 889  § 5. Optics. . . . .  890 892  § 6. The Tetrabiblos .  896  § 7. "Planetary Hypotheses" and "Canobic Inscription" 1. Introduction. . . . . . . . . 2. Sun and Moon . . . . . . . . 3. Planets; Periods and Longitudes 4. Planetary Latitudes. . . A. Angles of Inclination B. Precession . . C. Epoch Values. . . . D. Tables . . . . . . . 5. The Canobic Inscription 6. The" Ptolemaic System" 7. Book II of the Planetary Hypotheses  900 900  § 8. Additional Writings of Ptolemy .  1. The" Phaseis". . . . . . A. Aim. VIII, 6 . . . . . . B. The" Phaseis" Book II. . 2. Astronomy in the "Harmonics"  901 905 908 908 909 910 913 913 917 922  926 926 927 928 931  Table of Contents 3. The" Geography" . . . A. Spherical Astronomy B. Historical Remarks . 4. Philosophy; Fragments . C. The Time from Ptolemy to the Seventh Century. § 1. Introduction . . . . . . . . . . § 2. The Time from Ptolemy to Theon . 1. Chronological Summary . 2. Papyri and Ostraca. . . . 3. Second and Third Century. A. Artemidoros . . . . . B. Theon of Smyrna and Adrastus C. Achilles . . 4. Fourth Centur)( . . . . . . . . A. Astrology. . . . . . . . . . B. Astronomical Considerations . 5. Cleomedes . . . . . . . . . . A. The Date of Cleomedes . . . B. Geography and Spherical Astronomy C. Moon and Sun D. Planets . . § 3. Pappus and Theon §4. The Handy Tables. 1. Introduction. . A. Arrangement . B. Variants in the Handy Tables. C. Bibliography. . . . . . . . D. Appendix. Notes on Manuscripts 2. Spherical Astronomy. . . . . . . A. Rising Times. . . . . . . . . B. Seasonal Hours; Ascensional Differences. C. Ortive Amplitudes. 3. Theory of the Sun . . A. Solar Longitude. . B. Equation of Time . C. Precession . . . . 4. Theory of the Moon . A. The Tables for Mean Motions. B. Epoch Values . . . . . C. The True Moon. . . . D. Parallax and Prosneusis E. Eclipses . . 5. The Planets. . A. Longitudes. B. Latitudes . C. Visibility (" Phases") . 6. Appendix. Supplementary Material . A. Royal Canon. . . . . . . B. Reference Stars. . . . . . § 5. The Time from Theon to Heraclius I. Chronological Summary 2. Fifth to Seventh Century A. Popularization . . B. The Latest Schools C. Fragments . 3. Ephemerides . . . .  XIX  934 935 937 940  942 942 943 943 944  948 948 949 950 952 952 955 959 960  961 962 964  965 969 969 971 973 976 976 979 979 980 982 983 983 984 986 986 986 987 988 990  999 1002 1002 1006 1017 1025 1025 1026 1028 1028 1029 1029 1031 1051  1055  xx  Table of Contents  Part Three Book VI  Appendices and Indices. Figures and Plates A. Chronological Concepts. . . . § 1. Years and Julian Days . . . § 2. Special Calendars and Eras . 1. The Egyptian Calendar 2. The Seleucid Calendar 3. Synopsis of Eras . . . 4. The "Era Dionysius" . § 3. The Reckoning of Days .  1. Epoch . . . . . . . 2. Hours and Other Divisions 3. Astronomical Time Units . § 4. The Foundations of Historical Chronology § 5. Literature . . . . . . . 1. General. . . . . . . 2. Chronological Tables.  B. Astronomical Concepts . . § 1. Spherical Coordinates . 1. The Horizon System 2. The Equator System 3. The Ecliptic System 4. Relations Between the Systems. 5. Equation of Time . 6. "Polar" Coordinates § 2. Years, Months 1. The Year 2. Months . . § 3. Fixed Stars. . 1. Proper Motion 2. Yearl y Parallax 3. Names and Constellations. § 4. Geocentric Planetary Motion . § 5. Planetary and Fixed Star Phases 1. Planetary Phases. 2. Fixed Star Phases . . 3. Tables . . . . . . . § 6. Lunar and Solar Eclipses. § 7. Kepler Motion 1. Definitions . 2. Parameters . 3. Kepler's Laws 4. Approximations 5. Eccenter Motion . 6. "Elliptic" Orbits . § 8. The Inequalities of the Lunar Motion 1. Longitude. . . . . . . . . . . 2. Latitude . . . . . . . . . . . 3. Bibliographical and Historical Remarks . A. Evection. B. Variation . . . . . . . . . . . .  1061 1061 1064 1064 1064 1065 1066 1067 1067 1069 1070 1071 1074 1074 1075 1077 1077 1077 1078 1078 1079 1081 1081 1082 1082 1083 1084 1084 1085 1087 1088 1090  1090 1090 1091 1092 1095 1095 1096 1097 1098 1100  1102 1103  1106  1107 1108 1108  1109  Table of Contents  C. Annual Equation . . D. Latitude and Nodes . E. Bibliographical Notes  XXI 1110 1111 1112  C. Mathematical Concepts . . . . § 1. Sexagesimal Computations . § 2. Square Root Approximations. § 3. Trigonometry. . . . . . . . § 4. Diophantine Equations; Continued Fractions. 1. Euclidean Algorithm . . . . . 2. Linear Diophantine Equations. 3. Continued Fractions . . . §5. Tables . . . . . . . . . . . 1. SexagesimaJ Computations 2. Trigonometric Functions  1113 1113 1114 1115 1116 1116 1117 1120 1126 1126 1129  D. Indices . . . . . . . . . . . . § 1. Subject Index. . . . . . . . § 2. Bibliographical Abbreviations. § 3. Notations and Symbols . 1. Calendar, Chronology . . 2. Spherical Astronomy. . . 3. Lunar and Planetary Motion 4. Planetary and Fixed Star Phases § 4. Greek Glossary .  1133 1133 1165 1204 1204 1204 1205 1206 1206  E. Figures and Plates .  1209  Part One  Introduction Book I The Almagest and its Direct Predecessors Book II Babylonian Astronomy  Introduction § 1. Limitations  ..... excel/ens philosophus, cosmographus, mathematicus, historicus, stultus, linguarum non ignarus, sed nullius ad unguem peritus."  J. J. Scaliger  Many things are omitted here. The reader who wants to hear about Archimedes taking a bath or about the silver nose of Tycho Brahe can find innumerable books which dwell on these important biographical matters. Nor do I enumerate the pros and cons concerning the place or movement of the earth and the substance of the spheres. The history of these discussions has filled volumes which succeeded, by their sheer mass, almost completely to remove from sight the crucial mathematical arguments which are needed for the prediction of planetary positions, of eclipses and of secular changes, or for the determination of sizes and distances of the celestial bodies. In short: I shall deal here neither with early cosmogony nor with philosophy, but exclusively with mathematical astronomy, i.e. with the numerical, geometrical, and graphical methods devised to control the mechanism of the planetary system. It was my goal to convey to the reader some insight into the different aspects these problems have taken during their wanderings from culture to culture, beginning in the latest period of Mesopotamian civilization and ending with the discovery of the elliptic orbit of Mars, thus preparing the way for Newton's dynamics which brought to its conclusion all ancient mathematical astronomy. But even in this restricted area of mathematical astronomy much is omitted which would have deserved fuller treatment. Besides the personal impossibility to cover evenly such an enormous variety of topics, the lack of easily accessible sources shows its influence. What is badly needed is a systematic corpus of mediaeval sources, Indian, Islamic, and Western alike, a task which would require a great cooperative effort not likely to be made in the foreseeable future. Thus it remains accidental, to a large measure, what can be said in any discussion of mediaeval astronomy. An omission for which I alone am responsible is the theory of instruments. The reason is simply lack of competence. The theory of sun dials, of astrolabes, clocks, planetary computers, etc. constitutes a large field of its own with which I am not sufficiently familiar to bring it into significant relation to the topics discussed here. This is regrettable but I cannot do better than to admit the fact.  2  Introduction 2. Historical Periods  Again by reason of incompetence I have omitted all discussion of the history of astronomy in China. Its influence upon the Islamic and Western development is probably not visible earlier than the creation of Mongol states in Western Asia. Thus the damage done by omitting China is perhaps not too great and at any rate alleviated by ignorantia. No relation whatsoever exists between our study and Maya astronomy. Consequently no reference to this field of study will be found in the following pages. I have also refrained from more than casual comparisons of ancient with modem numerical data. Not only can such data be easily obtained from any good modem textbook and from modem tables (in particular Tuckerman's planetary tables for the period from -600 to + 1650~ but such comparisons are usually rather misleading. For example, the often found references to gross "errors" in ancient solstitial or equinoctial dates are of no value without a careful discussion of the influence on the solar theory as a whole. The reader of the subsequent pages will not need constant reminders of these rather obvious facts. I shall not explain how" good" or how" bad" ancient astronomy was but I will try to describe what seemed to be the essential problems and the methods developed toward their solution.  § 2. The Major Historical Periods, An Outline The history of astronomy falls into three sharply distinct periods: (a) prehistory until about 700 B.C. when (probably) Mesopotamian astronomy begins, (b) the ancient and mediaeval period (in the customary sense of political history) to the middle of the 17th century, and (c) modem astronomy beginning with the time of Newton. These three periods are as sharply distinct from one another as a stone-age settlement, mediaeval Paris, and modem New York. Figuratively speaking we are dealing here exclusively with the history of mediaeval Paris. The main chronological fixed-points for our discussion can be given in a few sentences. Not much before 300 or 400 B.C. there originated in Mesopotamia arithmetical methods for very accurate predictions of lunar and planetary phenomena. Perhaps inspired by these successes, but only iIi a very small measure depending upon them, cinematic models were developed by Greeks, notably by Apollonius, around 200 B.C. Careful systematic observations by Hipparchus (about 150 B.C.) made it clear, however, that the actual motions were more complicated and that further progress would not be easy. Indeed, only after much groping in the dark, about two centuries later (about A.D. 1(0) the important tool of spherical astronomy was put on a sound mathematical basis by Menelaos, while a satisfactory planetary theory and an imprOVed theory for the lunar motion had to wait until Ptolemy (about 150 A.D.). His monumental work remained the foundation for all mathematical astronomy until Kepler (around 1600). Late antiquity, Byzantium, and Islam upheld theoretical astronomy on an intelligent level, even improving on important mathematical and some astronomical details. But it is only in the late Renaissance that a real renewal of astronomy  Introduction 2A. Hellenistic Period  3  took place, most clearly expressed in Tycho Brahe's observational program (about 1560 to 1600) which provided the foundation for Kepler's" Astronomia  Nova".  Since the subsequent chapters are not following a strictly chronological order, it will be useful to provide the reader for his orientation with a schematic summary of the main lines of development and their interrelations. Such a summary is of necessity not only very incomplete but also much too dogmatic in formulation. It is only meant as a kind of preliminary scaffolding that has to be removed as the building itself progresses.  A. The Hellenistic Period The" Babylonian" astronomy of the Seleucid and Arsacid period is of primary importance for our investigations. Here we have authentic source material, representing a substantial amount of astronomical records, exactly as they were written. Only for papyri could the same be said, but their number and deplorable state of preservation places them far below the cuneiform tablets. It is particularly fortunate that these Babylonian documents were preserved because they allow us to study a type of mathematical astronomy the existence of which we would never have deduced from our Greek sources. From these we knew of systematic observational records in Mesopotamia and of the use of some of the resulting basic parameters by Greek astronomers. This was not very surprising in itself and found confirmation in the official reports on astronomical phenomena (i.e. omina) in the Assyrian royal archives. But it was only from the tablets discovered in Babylon and Uruk that it became clear that a theoretical astronomy existed from the Persian time to the first century A.D., operating with methods entirely different from the Greek ones which were based on the combination of circular motions. From the cuneiform texts we learned that ephemerides had been computed exclusively by means of intricate difference sequences which, often by the superposition of several numerical columns, gave step by step the desired coordinates of the celestial bodies - all this with no attempt of a geometrical representation, which seems to us so necessary for the development of any theory of natural phenomena. It is a historical insight of great significance that the earliest existing mathematical astronomy was governed by numerical techniques, not by geometrical considerations, and, on the other hand, that the development of geometrical explanations is by no means such a "natural" step as it might seem to us who grew up in the tradition founded by the Greek astronomers of the Hellensitic and Roman period. Beyond this, our knowledge of Babylonian methods has become a valuable tool for the discovery of historical connections between the Mediterranean world and India and the Islamic centers because remnants of undoubtedly Babylonian methods were discovered in Sanscrit and in Arabic sources. We know only very little about the prehistory of this Babylonian astronomy. In the extant texts from the Hellenistic period almost all methods appear fully developed. On the other hand it is virtually certain that they did not exist at the  4  Introduction 2A. Hellenistic Period  end of the Assyrian period. Thus one must assume a rather rapid development during the fourth or fifth century B.c. The same two centuries witness also the first steps in Greek astronomy. The beginning is made by the "school" of Meton and Euctemon (around 430 B.c.) with observations concerning the length of the solar year and with the formulation of the" Metonic" 19-year cycle which mayor may not be independent of the contemporary Mesopotamian discovery of the same cycle. In general the whole astronomy of the century from Meton to Eudoxus is dominated by calendaric problems. The variation of the length of daylight or the variation of the length of the shadow during the day are represented by the simplest arithmetical schemes. The rising and setting of stars during the year is related to the seasons and used for weather prognostication. Obviously the mathematical tools did not yet exist which are indispensible for a numerical description of the motion of the celestial bodies. On the other hand we know that Eudoxus had gained profound insight in the theory of irrational quantities (in geometrical formulation); his demonstration that the planetary retrogradations can in principle be explained as the result of superimposed rotations of inclined concentric spheres opened the road to the search for geometric representations of the planetary system. How little equipped Greek mathematics was at that time to handle actual astronomical problems is, however, drastically shown by Aristarchus' clumsy procedure (about 300 B.c.) to solve one right triangle in order to determine (by a most inaccurate method) the relative distances of sun and moon. Similarly in Euclid's" Phaenomena" a first attempt is made to give a mathematical formulation for the most elementary consequences of the introduction of a "celestial sphere" as reference system - obviously a novel and by no means trivial hypothesis. While spherical astronomy got slowly on its way a mathematical theory of great elegance and sophistication for the motion of the planets was developed by Apollonius (around 200 B.c.) and determined the direction of all subsequent efforts of cinematic planetary theories. All basic ideas for the description of planetary motion by means of eccentric or epicyclic circular motions were fully developed by Apollonius. Here we see for the first time the full force of highly developed mathematical methods applied to the solution of specific astronomical problems; the theory of stationary points is perhaps the most outstanding example. Unfortunately our information is so fragmentary that we have no idea about Apollonius' attitude toward the numerical part of astronomical problems. For us the influence of Babylonian data, accompanied of course by the sexagesimal number system, is first clearly visible with Hipparchus (around 150 B.c.). Now astronomy becomes a real science in which observable numerical data are made the decisive criterium for the correctness of whatever theory is suggested for the description of astronomical phenomena. It is not surprising that as a result of this attitude no new model of planetary or lunar motion can be ascribed to Hipparchus. His importance lies clearly in the new methodology of exactness, both in definitions and in observational techniques which he introduced. Hipparchus' role is best compared with the role of Tycho Brahe, just as Ptolemy parallels Kepler in the profound theoretical evaluation of his predecessors' material.  Introduction 2B. The Roman Period  5  Babylonian influence in the second century B.c. is not restricted to the methodology of Hipparchus. In the same period astrology begins to flourish in Hellenistic Egypt. The roots of astrology are undoubtedly to be found in Mesopotamia, emerging from the general omen literature. Yet, we know much less about the history of Babylonian astrology than is generally assumed. Only that much seems clear that it was a far less developed doctrine than we find in Greek astrological literature. The real center of ancient astrology, from which it eventually spread over the whole world, is undoubtedly Hellenistic Alexandria. The claim of astrology that it is based on age-old wisdom or on revelation by a deity could only reduce the incentive for a search for better astronomical methods. Consequently we see that astrological practice is often based on antiquated methods, on a conglomerate of diverse, even contradictory, parameters - facts most welcome for the analysis by the historian but very detrimental for the progress of astronomy.  B. The Roman Period More than the hellenistic monarchies the Roman empire contributed to the spread of hellenistic science and hellenistic astrology far beyond its political frontiers, though Rome itself seems not to have made any significant contribution. Seneca tells us 1 that the explanation of the planetary phases had reached Rome "only recently", i.e. around the beginning of our era, a century after Apollonius or Hipparchus. Recent discoveries by D. Pingree have shown that hellenistic astrology, and through it the Babylonian arithmetical methods, reached India as early as the second century A.D.2 This early date is of fundamental importance for the understanding of the fact that pre-Ptolemaic strata become visible centuries later in Indian astronomy, and consequently also in Islamic and finally in Western astronomy. Alexandria in the second century A.D. saw the publication of Ptolemy'S remarkable works, the Almagest and the Handy Tables, the Geography, the Tetrabiblos, the Optics, the Harmonics, treatises on logic, on sundials, on stereographic projection, all masterfully written, products of one of the greatest scientific minds of all times. The eminence of these works, in particular the Almagest, had been evident already to Ptolemy's contemporaries. This caused an almost total obliteration of the prehistory of the Ptolemaic astronomy. Ptolemy had no successor. What is extant from the later time of Roman Egypt is rather sad: huge commentaries on the Almagest or the Handy Tables by Pappus (about 320), by Theon (about 370) and by his daughter Hypatia. 3 The style of these commentaries is the style later customary in mediaeval schools; as A. Rome has formulated it, they were written "non pour apprendre aux eleves Ii raisonner, mais pour les empecher de reflechir".4 But it is probably through Seneca, Nat. Quaest. VII 25.5 (Loeb II, p.279/281). Cf., e.g., Pingree [1963. 1]. 3 Nothing of her writings is preserved but it is unlikely to have been essentially different from her father's. 4 Rome [1948]. p.518. 1  2  6  Introduction 2C. India  the editions of these schoolmasters that much of the ancient scientific classics remained in circulation, thus providing the basis for the Arabic translations as well as for the studies of the Byzantine scholars.  C. Indian Astronomy I think it is fair to say that practically all fundamental concepts and methods of ancient astronomy, for the better or the worse, can be traced back either to Babylonian or to Greek astronomy. In other words, none of the other civilizations of antiquity, which have otherwise contributed so much to the material and artistic culture of the world, have ever reached an independent level of scientific thought. Only into astrology were incorporated two remnants of prescientific astronomical lore from other than Mesopotamian or Greek background: the 36 Egyptian "Decans" and the 28 Indian "Lunar Mansions" (nakshatra). Both are the result of crude qualitative descriptions of the most immediate astronomical observations: the decans reflect the steady shift of the risings and settings of the stars during the course of the year, the nakshatras represent the lunar motion in the sidereal month. Both concepts reached accurate numerical definition only after being assimilated with (and in part transformed by) the ecliptic coordinate system that was developed in Babylonian astronomy. But while the nakshatras were known in India since the first millennium B.C., the contact with western astronomy dates only from the Roman imperial period. That the preceding Greek occupation of the Punjab should have brought astronomical knowledge to India is not very likely, simply because the Greeks themselves at that period seem not to have had any knowledge of a mathematical astronomy. However, we know so little about the early phase of Greek and of Mesopotamian astronomy at the beginning of the hellenistic era that the argument from negative evidence could be deceptive. So much, however, can be considered as fairly certain, namely that the first and lasting impact of western astronomy on India came via Greek astrological texts, operating with Babylonian arithmetical methods. 5 These methods were well known in Alexandria at least at the beginning of our era, as is attested through Greek and demotic papyri and from the astrological literature. This agrees well with the appearance of similar works in India by the middle of the second century A.D. when a branch of the Saka dynasty ruled over a large area of western India with Ujjain as its capital,6 a locality which since then defined the zero meridian for Hindu astronomy. The Saka era, however, (epoch: A.D. 78 March 15) is probably the era starting with the regnal years of the Kushana ruler Kanishka. 7 The first Sanskrit astronomical treatises still preserved were written in the late fifth and sixth centuries A.D., notably the Aryabhatiya of Aryabhata and works by Variihamihira. It is of great interest to see that at that time there existed Cf. for the details of transmission the masterful paper by D. Pingree [1963,1]. Mentioned in Ptolemy'S Geography (II, 1 §63) under the name '0(""" with qJ=20. Actually Ujjain is located at qJ = 23;11 Nand 75;50 east of Greenwich. 7 Cf. Nilakanta-Sastr~ CHI II, p.2331T. S  6  Introduction 2D. The Islamic Period  7  already the need for a historical survey of Hindu astronomy, thus causing Varahamihira to explain in .his Paiicasiddhantika the systems followed by five different siddhantas (thus the name), among others the early version of the Siiryasiddhiinta, a famous work which has come down to us only in a much later version. Of particular influence on Islamic astronomy became the KhaQQakhadyaka of Brahmagupta, written in A.D. 665, known as the al-Arkand to the Muslims. Parameters from this work and characteristic Hindu modifications of the Greek planetary theories found their way, through Islamic intermediaries, in particular through al-Khwarizmi, into Western European astronomy. The use of the zero meridian of" Arin" ( = Ujjain) is the most obvious example of this influence. In spite of the pioneering work done by H. T. Colebrooke (1765-1837), G. Thibaut (1848-1914) and others the study of Hindu astronomy is still at its beginning. The mass of uninvestigated manuscript material in India as well as in Western collections is enormous. May it suffice to remark that many hundreds of planetary tables are easily accessible in American libraries. So far only a preliminary study of this material has been made revealing a great number of parameters for lunar and planetary tables. s The planetary tables themselves are of great extent and based on methods so far not encountered in western material, the basic idea being that the planetary positions are computed for a whole year as function of the initial conditions at the entry of the sun into Aries. When these methods were developed we do not know - the extant texts suggest dates of the 14th century A.D.  D. The Islamic Period The reception of western methods by the Indians proceeded in two steps, an earlier one mainly based on ultimately Babylonian arithmetical methods, soon to be followed by the Greek cinematic procedures. The first impact on Islamic science was made by Indian astronomy (by that time, the ninth century, already a mixture of peculiarly modified Babylonian and Greek methods), shortly followed by the direct reception through Greek sources of the Ptolemaic system. This complex origin of Islamic astronomy left its traces wherever its influence was felt, from Persia to Spain and from Byzantium to Italy, France, and England. The details of the sudden beginning of Islamic astronomy at the Abbasid court in Baghdad are no longer discernable. It is customary to assume Syrian works as intermediary, besides the well attested Hindu-Iranian contacts. Although some pre-islamic Syriac astronomical works are known to have existed proof seems lacking that they exercised any influence on the Islamic treatises and tables. Of course, Syrians, knowing Greek as well as Arabic, were instrumental as translators of Greek originals but this is very different from postulating the existence of independent Syrian (or, in particular, Nestorian) works which would have been translated into Arabic. As long as Syria and Egypt were still under Byzantine rule there was very little need for the translation of Greek theoretical 8  Cf. Poleman, A Census of Indic Manuscripts in the United States and Canada. Am. Oriental Series 12,  1938 (e.g. p. 236); Pingree, SATUS; also Neugebauer-Pingree [1967].  8  Introduction 2 D. Islam  works into Syriac. Only with the gradual diminution of the Greek ruling class after the Islamic conquest does a translation into the language of the new rulers make some sense. I think one may safely assume that the Greek as well as the Indian astronomical treatises were transmitted directly to the scholars of Baghdad and Damascus. 9 Quite another matter is the problem of Persian influences. There can be no doubt about the existence of a substantial body of astronomical and astrological literature in pre-islamic Persia. We know of Pahlavi translations of such first and second century astrological writings as Teucer and Vettius Valens 10 and of the presence of" Indian books" as well as of the" Roman J1.6yiuni" around A.D. 250 under Shapur 1. 11 Under Khosro I Anosharwan (i.e. "of immortal soul") was revised, around A.D. 550, the famous Zij ash-Shah, which has been shown 12 to be greatly dependent on Hindu sources. Its zero meridian was Babylon, probably because of the proximity of Ktesiphon, the Sasanian capital. When Baghdad was founded by al-Man~ur in 762 the propitious moment 13 was determined by the Persian astrologer Naubakht and the converted Jew Masha'allah, the latter of international mediaeval fame as attested, e.g., by Chaucer's version of his treatise on the astrolabe (1391). Miishii'alliih died around 815/820. Of even greater fame and influence was Abu Ma'shar (787 14 to 886) whose writings were spread in many Greek and Latin versions widely over Europe. That Biriini had, rightly, a low opinion about Abu Ma'shar's astronomical competence 15 did not do any damage to his fame. To the same period, however, also belongs a group of competent astronomers, e.g., al-Khwiirizmi (who died probably before 850), the authors of the Mumtal}an zij (the "Tabulae probatae" of the West) prepared under al-Ma'mun (813-833),16 I:Jabash "the computor" (died 862), Thiibit b. Qurra (died 901), and others. During the ninth century both the Almagest and the Handy Tables became available in Arabic translations. The greatest astronomers of Islam lived in the next two centuries: al-Battani (858 to 929), a~-~ufi (903 to 986), Abu'l WaIa (940 to 997/8), Ibn Yunis (died 1009), and the universal scholar al-Biriini (973 to 1048). The disintegration ofthe Muslim world into a great number of independent states is reflected in the geographical dispersion of these men. Battiini worked in ar-Raqqa on the upper Euphrates,17 Sufi and AbU-I WaIa in Baghdad, then under Buyid domination, Ibn Yunis in Cairo under the fiitimid ruler al-I:Jiikim, while Biruni, born in Khwiirazm, had to follow the conqueror of his country, Mabmud of Ghazna, to Afghanistan 18 The situation seems to be dilTerent in the case of Greek and Syriac medicine but there is no necessity to assume that dilTerent fields follow identical patterns. 10 Cf. e.g., Nallino, Scritti VI, p.295. 11 Cf. e.g., Bailey, Zor. Probl., p.86. 1Z Kennedy [1958] and Biriini, Transits; also Pingree [1963, 1], p.242. 13 The date is fixed by the horoscope, as given in Biriini's Chronology (trsl. Sachau p.263 with the omission ofl) in 'Y' 26;40) as 762 July 31, not July 24 as usually stated. 9  14  His horoscope was dated by D. Pingree (1962], p.487 n.6 to 787 Aug.10.  IS  Cf. Kennedy, Survey, p. 133, No. 63. Cf. Vernet [1956]. qJ = 35;57 (36;0 according to Battiini~ 39;3 East of Greenwich. Ghazna is located southwest of Kabu~ at qJ = 33;33 (Biriini found 33;35) and 68;28 east of Greenwich.  16  17  18  Introduction 2D. Islam  9  and then to India. Biruni's greatest astronomical work is dedicated to the son and successor of Mabmiid, the Sultan Mas'iid, therefore called Qiiniin al-Masciidi. Shortly before the Qiiniin Biriini had completed his work on India (10301 a real mine of information on Indian astronomy and culture. Biriini's earliest great work, the "Chronology of Ancient Nations" (completed about A.D. 1000) belongs still to his life in Khwiirazm. The total of Biruni's writings on a great variety of subjects in all branches of exact and natural sciences, as well as literature and philosophy, comes to some 180 titles. 19 Biriini's work is in all its aspects of outstanding individuality and belongs to a class all its own, both from a purely literary point of view and with respect to the wealth of information it contains for the modem historian. This concerns not only the contemporary oriental civilizations and their history but also the history of Greek mathematics and astronomy, much of which was still accessible to Biriini but is now lost. Compared with Biriini, Battiini's work appears very pedestrian. But by consistently following the procedures of the Almagest and giving a clear account of the underlying empirical data, Battiini's tables became one of the most important works for the astronomy of the Middle Ages in the Orient and later in the Renaissance. A~-~ufi became a great historical influence in a different way. In his "Book on the Fixed Stars" he collected whatever he could find of Arabic names of constellations and tried to establish accurate boundaries for each of them, in many cases arbitrarily fixing a rather fluid tradition. 20 In this way a definitive terminology was established which remained the norm for the iconography of the constellations in Islamic astronomy. On the basis of Sufi's catalogue of stars in combination with Ptolemy's catalogue (Almagest VII, 5 and VIII, 1) Bayer in his" Uranometria" (1603) established the modem terminology, again with some arbitrary modification of the historical data. 21 Abu'l Waia's contributions belong mainly to the field of mathematics in so far as he modernized spherical trigonometry, e.g. by replacing the Menelaos theorem by the sine theorem for spherical triangles. The replacement of the Greek chord-function by the half-chords, i.e. by the sine-function, took place centuries earlier in India. 22 Parallel to the progress made by the Islamic scholars in spherical geometry numerical techniques were advanced. Improvements in the arrangement and refinement of tables constitute a definite step beyond the level of classical antiquity. In the concluding phase of Islamic astronomy the construction of colossal observational instruments was undertaken. Thus we see a development in the same direction which became of decisive importance in Europe in the 16th century. We are only very insufficiently informed about the influence of Islamic on Byzantine astronomy. This is because of the almost complete neglect by modern 19 Boilot [1955] and Ene. of Islam(2) I, p.1236-1238. 20 Cf. Kunitzsch, Sternnomenklatur. 21 Cr., e.g., Boll, Sphaera, p. 450. 22 On the other hand Copernicus still speaks only about "half-chord of the double arc" (cr., e.g.,  De Revol. I, 12).  10  Introduction 2D. Byzantium  scholarship of Byzantine astronomical treatises though they are still extant in a quite respectable number. But even on the basis of the available fragmentary information it seems evident that from the 10th century on Islamic material had been studied and, in part, translated into Greek. The Byzantines were in a position to observe the scientific activities of their neighbors closely. Under Mas'ild began the steady progress of Turkish tribes under the Seljuks. He was defeated by the Turks in 1040 (and murdered the year following); the Seljuks succeeded in occupying Persia, eliminated the Bilyids from Baghdad, and reached the Mediterranean on the south coast of Asia Minor. Under the able rule of Jalal-ad-Din Malikshah (1072-1092) a well staffed observatory came into being probably in Isfahan 23 (1074). It seems to have reached almost the canonical age of 30 years of existence, deemed desirable for a planetary observational program-obviously because of Saturn's 30-year period. With Malikshah's name is also connected a famous calendar reform which introduced (instead of the inconvenient Arabic lunar years and instead of the very convenient Persian calendar years) accurate tropical years as the basis of the socalled Jalali calendar. References to this calendar and its patron, MeA-lea, appear repeatedly in Byzantine manuscripts. To a son of Malikshah, the Sultan Sanjar (1118 to 1157) was dedicated the famous Sanjari zij, translated into Greek and often mentioned in Byzantine astronomical writings. Its author, al-Khiizini, was himself a Greek freedman of a judge in Marv (Turkestan). In 1255 the Mongol prince Hulagil Khan, grandson of Genghis Khan, succeeded in taking the mountain stronghold of the Assassins, al-Alamilt, which had been a major obstacle for a Mongol domination of Persia and Iraq. Among the prisoners taken by Hulagil in al-AlamOt was the astronomer Na~ir ad-Din at-Tilsi, who had been held as honored prisoner and astrological adviser by the "Old Man of the Mountain," the Grand Master of the Assassins. At-Tilsi once more became the trusted adviser of his master, now Hulagil, and see~s to have played in this capacity a somewhat sinister role in the end of the last Abbasid Caliph when Baghdad was taken by the Mongols in 1258. 24 In the following year the construction of an observatory was begun 25 at Maragha,26 sout1! of Tabriz, under the directorship of a!-Tilsi. Its main product was the Persian Ilkhani tables (known as 6A-KClVij in Byzantium), completed in 1271, six years after Hulagil's death. This observatory functioned for an unusual length of time, until the end of the Ilkhani dynasty itself, i.e. until about  1336. 27  A!-Tilsi's work ranges wide over all branches of astronomy and astrology, optics, mathematics, and the construction of instruments. One small incident may be mentioned here, namely Tilsi's discovery that a simple harmonic motion can be produced by the rolling of a circle of diameter 1 inside of a circle of radius 1. 23  24  2S 26 27  Sayili, 01, p. 164.  Cf. Boyle [1961]. cr. Sayili 01, p.I90. cp =37;25 (37;20 in the Khiqini zij) and 46;130 east of Greenwich. Sayili 01, p.211.  Introduction 2D. Byzantium  11  This device was used in the theory of the motion of Mercury in order to replace a crank mechanism used by Ptolemy, intended to move Mercury's epicycle periodically nearer to, or away from, the observer. At-Tiisi's methods were further developed by his pupils, e.g. by Ibn ash-Shlitir (1306-1375), and appear finally in Copernicus' De Revolutionibus. 28 A!-Tusi died in 1274, his name being known from Byzantium to China. The last important Islamic observatory was founded in 1420 by Ulugh Beg in Samarkand, the city which his grandfather Timur (who died in 1405) had made the capital of his empire. A group of prominent scholars, among them al-Kashi (died 14291 were entrusted with the planning and the development of the observatory; its colossal ruins are still extant. The tables which were produced there became very famous, in particular the catalogue of stars which was, e.g., incorporated into Flamsteed's "Historia coelestis Brittanica" (1725). Gauss, in 1799, applied his new method of least squares to the equation of time as given in Ulugh Beg's tables. 29 Tycho Brahe, however, seems not to have known these tables. 30 Modern scholars became aware of one particular case of transmission of oriental astronomy, from Tabriz to Constantinople, through a Greek version of "Persian Tables", probably the Sanjari zij and the Zij al-'Alali, and canons written by the physician Gregory Chioniades in Constantinople about 1300. 31 Manuscripts of the version by Georgios Chrysokokkes, which reached European collections in comparatively large numbers, and excerpts concerning planetary periods, were made known by Boulliau (1605 to 1694) in his "Astronomia  Philolaica" (1645).  I do not think, however, that we are dealing here with a very significant incident. The direct tradition from Ptolemy and Theon had never been broken in the Eastern Roman Empire. The interest in astronomy of the emperor Heraclius (610 to 641) is well attested. One ofthe most famous manuscripts of the "Handy Tables" (Vat. gr. 1291) was written under Leon V (813 to 820). References to Islamic works which were composed from the 9th to the 14th centuries are frequently found in Byzantine treatises. There is no reason to assume that there is any period in which Islamic astronomy was not known in Constantinople. Nevertheless there are certain periods where we know that astronomy became fashionable in the highest circles of Byzantine society, e.g. under the Emperor Manuel Comnenus (1143 to 1180). At the end of the 13th century Theodoros Metochites (died 1332) wrote a commentary to the Almagest; from the 14th century we have treatises on the theory of the astrolabe by Nicephoros Gregoras and by Isaac Argyros. The "Tribiblos" of Theodoros Meliteniotes belongs to the end of the 14th century. Our chronological summary has thus reached the time of the final collapse of the Byzantine empire. The story has often been told how this affected the West and contributed to the revival of Greek literature and science. As we slowly begin Roberts [1957]; Kennedy [1966]; Hartner [1969, 2], [1971]. cr. also below p. 1035. Gauss, Werke 12, p.64-68. 30 The first (incomplete) western publication seems to be Joh. Gravius, Epochae celebriores ... , London 1650. For discussion, cr. Ideler [1832]. 31 Pingree [1964]. 28  29  12  Introduction 2D. Spain  to obtain some knowledge of the contents of the astronomical material that reached in this way the European collections of Greek manuscripts, we can say that Byzantine and Islamic astronomy from the Near East was of a higher level of competence and inner consistency than the material that originated from Spain which is reflected in such famous works as the Toledan or Alfonsine Tables. Nevertheless there can be little doubt that Spain exercised a far greater influence on the revival of astronomy than Byzantium. The earliest work that became widely known in the Latin West is a version of al-Khwiirizmi's tables, prepared by MasIama ibn A~mad al-Majri!i (who died in 1(08), translated in the 12th century into Latin, presumably by Adelard of Bath. Thus England and France became influenced by a work reflecting by its mixture of Hindu and Greek elements the earliest phase of Islamic astronomy at a time when this period had long been outdated by eastern Islamic works. Majri!i also revised an Arabic translation of Ptolemy's" Planisphaerium" and thus preserved for us, in a Latin translation,32 this treatise which represents the basis of the theory of the astrolabe. This is not the only case in which the Spanish tradition connects us directly with Greek astronomy. We know from papyri that there existed a category of astronomical tables in Graeco-Roman Egypt that can be described as "Almanacs," i.e. tables computed for specific calendar years, giving the positions and phenomena of the celestial bodies. It is this approach which appears again in mediaeval Spain, e.g. in the "Perpetual Almanac" of Zarqiili (epoch year 1089~ a competent astronomer from whom we have observations for the years around 1060. He also wrote on instruments and is closely related to the preparation of the "Toledan Tables" of which translations and adaptations, e.g. by Gerard of Cremona (1114 to 1187), were widely distributed in Europe. We even have a translation of these tables into Greek, preserved in a Vatican codex of the 14th century.33 These continuous adaptations certainly did not contribute to the reliability of the tables, which from the very beginning were of a composite character. 34 Several of the important Spanish scholars were Jews. Ibn Ezra (born about 1090, died 1167) wrote on the astrolabe, on astrology, translated into Hebrew a commentary by Ibn al-Muthanna to al-Khwarizmi, etc. He had travelled extensively, lived some years in Italy and France, visited Cyprus, was in London 1158/9 and thus contributed much to the spread of Islamic astronomy in Europe. Highly competent in astronomy (and strictly opposed to astrology) was Maimonides (1135 to 1204) who was expelled in his youth from Spain, during one of the comparatively rare waves of Muslim intolerance, and became famous as physician and scholar at the Egyptian court under Saladin and his son. In Book III,8 of his "Code" Maimonides instructed his co-religionaries in the Ptolemaic theory of the lunar motion in order to compute in advance the date of the visibility of the first crescent. Philosophically, however, he objected to the Ptolemaic methods as not based exclusively on circular motions. In the "Guide of the Perplexed" he therefore advocated in principle the system of al-Bitruji (died 1204) Cf. Heiberg in Ptolemaeus, Opera minora, Prolegomena, p. CLXXXVII; also Suter, MAA, p.77. Below VB3. 33 Vat. gr. 212 (Mercati-Cavalieri CVG I, p.270). 34 cr. Toomer [1968, 2J.  32  Introduction 2D. Spain  13  which was based on concentric spheres rotating about inclined axes. 35 Michael Scot's translation of Bitruji's "De motibus celorum," completed 1217 in Toledo, made this astronomically hopeless theory known in Europe. Even Regiomontanus studied it seriously and wrote a refutation. 36 Nevertheless it found in Fracastoro 37 (1478-1553) again a convinced advocate. The Toledan Tables were superseded by the Alfonsine Tables whose epoch, 1252 June 1, is the day of coronation of Alfonso X; actually the tables were completed about 20 years later. These tables reached a very wide distribution, eventually through many printed editions, but astronomically they are not essentially superior to the Toledan Tables. Alfonso, who ruled from 1252 to 1284, instigated, or wrote himself, a great number of works. The famous Libro del Saber (1276/9) is a compilation on instruments; the Libros de las Cruces with its endless enumeration of trivial combinations of astrological influences reveals an unusual dullness of mind. At the end of Spanish astronomy we find again a Jewish scholar, Abraham Zacut, born about 1452 in Salamanca, who wrote, among others, a "perpetual almanac" and published tables used by Columbus and Vasco da Gama. Zacut was expelled from Spain in 1492 and died about 1522 in Damascus. The rapid spread of astronomical interest in the late Middle Ages is illustrated by the fact that a Jewish scholar from Tarascon, Immanuel Bonfils composed, about 1370, an astronomical treatise in Hebrew, commonly called "Six Wings" from its six chapters. 38 This work was translated to Greek and a commentary written in 1435 by Michael Chrysokokkes 39 under the title Hexapterygon. 4o The work has a rather restricted scope since it concerns only the computation of lunar and solar eclipses. The mean conjunctions are found by means of the Jewish lunar calendar. The true conjunctions and the corrections for parallax can be found from very extensive tables with a minimum of computation. It is probably for this reason that this work became so popUlar. Already at the Council of Basle (1431 to 1448) the question of the "reform" of the julian calendar was put on the agenda. The bishop (later cardinal) Nicholas of Cusa (1401 to 1464) was one of the main supporters of the plan, having made detailed suggestions of his own for the procedures for the shift to a new calendar. He himself was an ardent, if not very successful, student of astronomy. At the Council he met in 1434/5 the metropolitan, also later cardinal, Isidore of Kiev, who led the embassy sent by Andronicus III Palaiologus which should negotiate the" Union" of the Eastern and the Western churches. In Basle Isidore showed Cusanus "Persian Tables" in Greek, a fact proudly recorded by the latter, who translated certain sections into Latin.41 The result proved as ineffective as the calendar reform,42 the Union, and the Council. 35 It has been shown by B. Goldstein that Bi~riiji's model is not a simple modification ofthe EudoxanCallippic model but is based on an independent idea (cf. Bi~ruji, Astron., ed. Goldstein Vol. 1). 36 Cf.Rosen [1961]' 37 The physician, often mentioned for his poem on syphilis. 38 The original title was Eagle Wings. 39 He is not to be confused with Georgios Chrysokokkes, mentioned on p. 40 Cf. the summary of the contents in Solon [1970]. 41 Cf. Stegemann, Cusanus, p. 54/55 and p. 117, note 11. 42 It had still to wait until 1582.  14  Introduction 2E. Epilogue  E. Epilogue Under the influence of the tradition of the 18th and 19th centuries, astronomy seems to us almost synonymous with continuous systematic observation of the celestial objects. The historical development, however, shows a quite different pattern. Both Babylonian and Greek astronomy are based on a set of relatively few data, like period relations, orbital inclinations, nodes and apogees, etc. The selection of these data undoubtedly required a great number of observations and much experience to know what to look for. Nevertheless, a mathematical system constructed at the earliest possible stage of the game was generally no longer systematically tested under modified conditions. This attitude can be well defended. Ancient observers were aware of the many sources of inaccuracies which made individual data very insecure, e.g. the finite diameters of slits and pins at instruments, and of the human eye, the inaccuracy of shadow- and water-clocks, etc. On the other hand period relations, e.g. time intervals between planetary oppositions and sidereal periods, can be established within a few decades with comparatively high accuracy because the error of individual observations is distributed over the whole interval of time. If a theory was capable of guaranteeing correct periodic recurrence of the characteristic phenomena then intermediate deviations would matter little, in particular when one had no means to distinguish clearly between theoretical and observational causes of the errors. Thus we notice an outspoken tendency in the development of astronomy to take refuge behind mathematical schemes, rather than to embark on systematic observational programs which are so characteristic for astronomy since the invention of the telescope. Yet we know of countless observations recorded between the 9th and the 16th centuries and practically every astronomer claimed, probably correctly, to have made (some) observations of his own. Nevertheless the results of these observations turned out to be insignificant. They were not designed to test over and over again the validity of the accepted mathematical idealizations, or to follow up all the ramifications of a change in anyone of the basic parameters. For example, Islamic astronomers persisted in the measurement of the inclination of the ecliptic and of other elements of the solar orbit, like eccentricity and apogee. Yet the effects of an improved determination of the solar orbit on the elements of the planetary orbits were not investigated simultaneously. Thus it became possible that more and more inconsistent elements were contributing to mediaeval tables until it no longer could be doubted that a revision of all its foundations had become necessary. Rightly Tycho Brahe called his observational program the foundation of an Astronomia Nova. Ancient Astronomy is overwhelmingly mathematical astronomy. A reader who ventures beyond this point must be warned that he should not except to find in the following a "History of Ideas". All I intended to do is to illustrate some of the enormous technical difficulties which had to be overcome before astronomy could become fully associated with the progress of mathematics after both fields had laid stagnant for some 1300 years. How the self-appointed custodians of "ideas" reacted to astronomical and mathematical developments  Introduction 3A. Sources  15  seems to me of little interest. Nevertheless it might be usefu~ if not amusing, to see how the crowning achievement in classical astronomy is reflected in a famous philosopher's mind: "Es wird von den Mathematikem seiber zugestanden, daB die Newtenischen FormeIn sich aus den Kepplerischen Gesetzen ableiten lassen. Die ganz unmittelbare Ableitung aber ist einfach diese: 1m dritten Kepplerischen Gesetz ist Constante. Dies als  A;2A  2  gesetzt und mit Newton  :2  A: das  T  die allgemeine Schwere  genannt, so ist dessen Ausdruck von der Wirkung dieser sogenannten Schwere im umgekehrten Verhaltnisse des Quadrats der Entfemungen vorhanden ".(Hegel, System der Philosophie. II. Die Naturphilosophie. Samtliche Werke IX (Stuttgart 1929), p. 124.)43  § 3. General Bibliography A list of all bibliographical abbreviations used throughout in the present work will be found at the end of Book VI (cf. p.1165fT.). Several sections in the following text are supplemented by bibliographical notes; cf. the list given below p. 18. Also the subject index (p. 1133 fT.) may be consulted for the literature on specific topics.  A. Source Material The following references are meant as a first guide to the primary sources, i.e. to the editions of texts or to works where such references can be found; completeness is not attempted. For the pre-Greek period the situation is simple: the sources for Babylonian mathematical astronomy are collected in Neugebauer, ACT (1955); for Egypt cf. p.566f. For classical antiquity and the Middle Ages no systematic collection of mathematical or astronomical treatises exists. No attempt has ever been made to compile basic collections comparable to the Loeb Classical Library or the Bude collection, or Migne's Patrologia, the Monumenta Germaniae Historica, the Bonn Corpus of Byzantine historians, etc. Except for the Teubner editions of Greek and Latin authors no set of texts and translations exists 1 and one must find one's way from author to author through handbooks and bibliographies. This fact alone suffices to show that the so-called" History of Science" is still operating on an exceedingly primitive level. Similarly Schopenhauer, in his "Farbenlehre", about Newton's optics (Samtliche Werke 6, Wiesbaden 1947, p. 210. 1 In the year A.D. 1964 I was informed that the editors of the Loeb Classical Library "felt that they had discharged their duties toward the Loeb Library as well as toward Greek Mathematics" (and by inference astronomy) by publishing a two-volume anthology (for which cf. my review in AJP 64, 1943, p. 452-457). 43  16  Introduction 3 B. Modern Literature  In this situation the five huge tomes of Sarton's "Introduction" (1927 to 1948), ending at 1400, are a useful tool, although being hardly more than a compilation from all available encyclopaedias. Unfortunately this method resulted in an overburdening with irrelevant material, not to speak about the absurdly rigid chronological arrangement and the historical introductions which are reminiscent of the mentality of Isidore of Seville. For Greek and Latin authors one will naturally consult first the "RealEncyclopadie" ofPauly-Wissowa which, however, does not include the Byzantine period. For this period one has now a bibliography in Vol. IV, 2 of the second edition of the Cambro Med. Hist. (1967).2 Unpublished manuscripts, extant in vast quantities, can only be identified through the descriptions in library catalogues, few of which are sufficiently detailed, in particular when scientific subjects are involved. 3 For the location of Greek manuscripts and their catalogues one has an excellent guide in Richard, Repertoire (1958, 1964). The 12 volumes of the CCAG contain in many cases also information about purely astronomical sections in "astrological" manuscripts. For Greek papyri cf. the bibliographies Neugebauer [1962,1] and Neugebauer-Van Hoesen [1964]. For the Islamic period one has, of course, the standard work of Brockelmann, GAL, and the articles in the Encyclopaedia of Islam (second edition, beginning 1960). For the exact sciences Suter, MAA (1900, Nachtr. 1902; also Renaud [1932]) is fundamental, for astronomy Kennedy, Survey (1956). A mine of information is Nallino, Battani, and his Scritti, Vol. V and VI. For Indian astronomical texts cf. 'Pingree SATUS and Pingree [1963]. The worst conditions prevail for the mediaeval european scientific literature in Latin. The best guide to the sources and their interrelation is Haskins SMS. Thorndike's Magic (in 8 volumes, 1923 to 1958, reaching to the 17th century) is not fruitful for the astronomical literature which was beyond the author's competence. Steinschneider's unsurpassed mastery of mediaeval Arabic and Hebrew sources has produced his great bibliographical works which are an indispensible tool for all studies in mediaeval sciences; for the Spanish area cf. the publications of Millas Vallicrosa (cf. VI D 2).  B. Modern Literature At the end of the 18th century France was undoubtedly the center of mathematical and astronomical progress. At that period ancient or oriental astronomical data were still of current practical interest and it is therefore not surprising to see the modem historical study of astronomy originate in France at the tum of the century. Here we have the monumental works, in part strictly historical, in part introductory to astronomy itself, by Montucla (published between 1758 and 1802), by Bailly (between 1775 and 1782), by Lalande (between 1764 and 1803), 2 By K. Vogel; the text, covering all sciences on 40 pages is oflittle use but the bibliography (p.452-456, 463-469) is the best one has at present; omission: Pingree [1964]. 3 Descriptions like" astronomical tables" is usually all the information one can hope for.  Introduction 3 B. Modern Literature  17  by Delambre (between 1817 and 1827), and many historical excurses in the contemporary works, e.g. in Laplace's" Exposition du systerne du monde" (1796). Much more elementary but very useful for bibliographical matters are the publications of Joh. Fridr. Weidler" Historia astronomiae" (1741) and" Bibliographia astronomica" (1755), both published in Wittenberg. Work on the history of astronomy became stagnant during the main part of the 19th century. Its modern revival may perhaps be reckoned from P. Tannery who published as fruits of his researches a "Histoire de l'astronomie ancienne" in 1893. More or less in the same style of a historical outline is Dreyer's "History of the Planetary Systems from Thales to Kepler" (1906) whereas Heath, "Aristarchus" (1913) deals mainly with earliest Greek astronomy or cosmogony. These works suffer from the overemphasis on philosophical speCUlations during the prescientific period and (unavoidable at that time) from the ignorance of oriental sources. The same tendency still prevails in Duhem's "Systeme du monde" (1913 to 1917) which provides very little insight into the history of technical astronomy. An outstanding summary of the history of Islamic astronomy was given by Nallino in his lectures in Cairo 1909/10, published in an Italian version in Vol. V (p. 88-329) of his Scritti. The professional attitude of Delambre was resumed by N. Hertz in his "Geschichte der Bahnbestimmung" (1887, 1894) which concerns mathematical astronomy from Eudoxus to the time of Halley and Cassini. The impact of Kugler's discoveries in Babylonian astronomy (1900 to 1914) is first felt in Pannekoek's "History of Astronomy" (1951), to become the dominant theme in my "Exact Sciences" (1951, 1957) and in van der Waerden's "Anfange der Astronomie" (1966). The works which we have mentioned in connection with the primary sources (above p. 15f.) deal, of course, frequently with the general historical background. Beyond this exist special reference works of which shall be mentioned Lalande, Bibliographie (1803) and his Astronomie (1764 to 1792), the bibliographical references in Wolfs" Handbuch" Vol. II (1872) and in his "Geschichte der Astronomie" (1877); finally Houzeau "Vademecum" (1882) which is particularly useful for technical details, parameters and empirical constants, and Houzeau-Lancaster, "Bibliographie genera Ie " (1887 to 1892). For tables, designed for historical purposes, should be mentioned: for chronology: Schram, Tafeln; for the computation of all types of astronomical phenomena: P. V. Neugebauer, Tafeln, and Astron. Chron.; also Baehr, Tafeln, and Ahnert, Tafeln; for the positions of sun, moon, and planets: Tuckerman, Tables (from -600 to + 1649); Goldstine, New and Full Moons (from -1000 to + 1651). Chronological Summaries are found below ::::-; -800 to -300 ::::-; -300 to 0 ::::-;0 to 150 ::::-; 150 to 400 ::::-; 400 to 650  in the following sections: p.573 p.574f. p.779f. p.943 p. 1028.  18  Introduction 3 C  c. Sectional Bibliographies The following sections contain bibliographical information pertaining to their specific topic: chronological tables: p.1075f. chronology in general: p.l074f. demotic and coptic texts: p. 567 f. eclipses: p.l093 Egyptian astronomy in general: p. 566f. fixed star phases: p. 1092, p. 589 Handy Tables: p.976f. lunar inequalities: p.1l08ff. papyri: p. 787 f. parapegmata: p. 589 planetary phases: p.l091 f. Ptolemy's" Geography" and geography in general: p. 940 stars, names and catal. of stars: p. 1087. The Subject Index (p.1l33-1l65) is of central importance for the use of this work. The arrangement of the references is intended to furnish access to interconnections and topics which are not expressly treated as main subjects in the text. The Bibliographical Abbreviations (p.1l65-1203), extensive as they are, are not meant to be taken as a complete bibliography of ancient astronomy.  Book I The Almagest and its Direct Predecessors Addidit Ptolomeus in astronomia plus quam esset id totum quod ante se scriptum in venit. Anon., Ashrnole 191. II fol. 138' margo  A. Spherical Astronomy La sphere inspire les meditations des geometres par Ie nombre de ses proprietes; quand elle procede de La nature physique, elle en acquiert des qualites nouvelles. Anatole France, L'tle des Pingouins (C£uvres, Vol. 18, p.70)  E'l tie; A£l'tl (he; TO TOU Kvpiov i~ ftVIX(1TcXUtWe; UOJ/l1X 1Xi8i;pIOV Tt KIXI UeplXlpOtlOee; np UX~/lIXTI ... &vcX9t/l1X  eUTW.  Concilium Constantinopolitanum (A.D. 543); Canones concilii V adversus Origenem. X. (Joannes Dominicus Mans~ Sacrorum Conciliorum ... collectio IX, ed. nov., Florentiae 1763, col. 400)  § 1. Plane Trigonometry 1. Chords The Chaps. 10 and 11 of Book I of the Almagest contain the ancient theory of plane trigonometry and the resulting tables. The basic function, however, is not the sine function but its equivalent, the length of the chord subtended by the given angle in the unit circle. We shall use the notation crd a if the radius of the unit circle is 1, but Crd a if the radius R of the circle is the sexagesimal unit R= 1,0=60. Ptolemy uses Crd ex exclusively but we shall frequently replace it by crd ex which only implies a shift in the sexagesimal place value. Since one trigonometric function (e.g. sin a) suffices to express all the other ones and since crd ex and sin ex are related by the simple rules (cf. Fig. 1) crd ex = 2 sin ;  sin a =  ~ crd 2 ex  22  lAt, 1. Chords  it is obvious that there can be no essential difference between the methods for the solution of elementary trigonometric problems when using chords or modern trigonometric functions. The really important difference between ancient and modern procedures lies in a practical point, namely in the absence of tables for the equivalent of the function tan a, i.e. of the quotients crd a/crd (180-a). Consequently every problem which in modern terminology involves tan a required in antiquity two entries in the tables to find separately crd a and the value of the co-function crd (180-a) and finally the computation of the quotient. Obviously it is the smallness of the number of people who were interested and able to undertake productive work in theoretical astronomy that is responsible for the slow progress in the mechanisation of procedures. Whenever a large number of practitioners is involved, as, e.g., in calendaric or in astrological computations, we notice the tabulation of a variety of sometimes very complicated functions taking place. But trigonometric computations only occur in the derivation of new procedures and this does not happen between Ptolemy and the Abbasid period. In our description of the ancient procedures we shall often use modern trigonometric functions but only so far as it does not influence the method itself. In other words we will never use trigonometric identities for which there does not exist a counterpart within ancient procedures.  2. The Table of Chords Before giving some examples for the operation with chords we shall describe shortly the procedure which led to the computation of a table of chords in steps of 1/2°. The basic idea is simple: several chords are easily determined since they are the sides of constructible regular polygons. Using an important theorem on quadrilaterals inscribed in a circle one then can find crd (a + p) and crd (aI2) and in this way reach the chords of all arguments which are multiples of 314°. Finally an interpolation procedure is employed to obtain steps of only 112°. The method described in the Almagest is based on parameters which we modify only in so far as we use a circle of radius r = 1 instead of Ptolemy's R = 1,0. Let Sn be the side of the regular n-gon, then (cf. Fig. 2) =crd (360ln) = crd an s~ = crd (180-an) = v"4-;r2'---s-"';. Sn  The following values are known  a3 = 120° a4=90 as=72 a6=60 a 1 0=36  S3 =V3  = 1;43,55,23  s4=V2= 1;24,51,10 (Euclid XIII, 10; cf. Fig. 3)  S6= 1 (slO+s6)/s6=s6Is10 ("golden section"; cf. Fig. 3).  (1)  23  I A 1,2. Chords  y2  V3  Ptolemy does not say how the approximations for and were found but we know that at least the above given value for was used as early as in Old Babylonian times. Obviously this whole set of relations was considered as well known. The next step consists in the proof of the following theorem (cf. Fig. 4):  V2  (2) One constructs on the diagonal dl a point Q such that the angle P3 Pz P4 = angle  I\ P2 Q. Then one has to notice that angles which are marked in our figure  by equal symbols are equal, either because they subtend the same chord, or by construction. This shows that we have two pairs of similar triangles Pl Q P2 ~ P4 P3 Pz and P2 QP3 ~ Pl P2 P4 . Thus two relations which combine to (2). It is now easy to show that this theorem leads to the desired relations by specializing it to the case where one side of the quadrilateral coincides with the diameter d of the circle. (A.) From given Sl = crd (Xl and S2 = crd (X2 (cf. Fig. 5) one can find x = crd «Xl - (X2) linearly from  Thus crd 12° = crd (72 - 60) can be computed, using (1). (B.) From given S = crd (X one can find linearly x 2 = crd 2 «X/2) because (Fig. 6)  Y  xd + x d2 -  S2  =  S  Yd  2-  x Z•  Hence, starting with crd 12° one can obtain the chords of 6°, 3°, 3/2°, and 3/4°. (C.) From given Sl and S2 (Fig. 7) one can find x=crd «Xl +(X2) from the quadratic equation For (B) and (C) Ptolemy does not use the above given procedures. In case (C) he avoids the determination of x from a quadratic equation by finding first the chord x' (cf. Fig. 8) from  Y  and then x from d2 - x' 2. In case (B), however, Ptolemy does not use the quadrilateral theorem at all but follows a different procedure which goes back to Archimedes, as we know from a section in Thiibit ibn Qurra's translation of Archimedes' treatise on the heptagon 1. Since BC=CD (cf. Fig. 9) we have also PI =P2' If we now make AF=AB we have also FC=CB=CD=x. If one draws CE as altitude in the I  Schoy, AI-Bir. p.81 (No. 14); cf. also below p.776.  IA 1,3. Chords  24  triangle FDC one can find y = DE from y = 1/2 DF = 1/2 (d - Vd2 - S2). But x: y = d: x and therefore x = crd 1X/2 can be computed. The fact that here an Archimedean construction replaces the quadrilateral theorem might be used as an argument for a more recent date of the latter theorem which, for us, is for the first time attested in the Almagest. (D.) Ptolemy now states, without proof, that crd(1/2°) cannot be found "by geometric construction," 2 as is indeed the case. 3 He therefore proves the inequality 4/3 crd 3°/4> crd 1° > 2/3 crd 3°/2. Within an accuracy of two sexagesimal places the values obtainable from the parameters (1) for 4/3 Crd 3°/4 and 2/3 Crd 3°/2 are both found to be 1;2,50. Thus one has also Crd 1°= 1;2,50. This completes the list of fundamental parameters needed to compute a table of chords in steps of half degrees. Almagest I, 11 gives this table, for arguments from 1°/2 to 180°. Tabulated are the chords to minutes and seconds of units in which the radius R of the circle is 60. There follows a column of coefficients of interpolation for minutes of arc. Repeatedly the last digit of these coefficients does not agree exactly with the 30th part of the difference in the preceding column. It seems as if the underlying table of chords was computed to one more sexagesimal place.  3. Examples No.1. Find the length s of the noon shadow of a gnomon of length 60 for the summer solstice at Rhodes (AIm. II, 5). We assume the geographical latitude to be qJ= 36°, the obliquity of the ecliptic 8=23;51,20°. Thus (Fig. 10) lX=qJ-8= 12;8,40  21X = 24;17,20  2P= 155;42,40.  From the table of chords Crd21X s=g Crd2p  25;14,43 2 1,0 157.18 , , ,51 ~ 1 ;54,42.  In modem terms one would have said s=gtanlX and which agrees with Ptolemy's result.  tanlX=0.21519~0;12,54,41  No. 2. We assume that the sun S moves on a circle of radius R=60 with center M (cf.Fig.ll). We take as eccentricity OM=e=2;30 and as position of the apogee A = ][ 5;30. The angle K, i.e. the distance of S from A, is called the mean eccentric anomaly. We wish to find the value of K if S is seen from 0 in ::a. 0° (AIm. III, 7). blli ,rov YPr1.J.lJlWv, meaning "rigorous" methods (cf. below p.771 n. I). If one could construct crd(l/2°~ one could find, by virtue of the preceding steps, the chord for 1°, thus for 2°, 4°, 4 + 6 = 10°, hence also for 20" and finally for 40" which is the side of the regular 9-gon. But Gauss has shown (Disquis. arithm § 365, Werke I, p. 461) that the construction of a regular n-gon by ruler and compass is only possible when n is a prime number of the form 21 + 1 (k an integer). 1  3  IA 1,3. Chords  25  We know the angle y=A.A =65;30, hence 2y= 131;0. Furthermore  R  e  MK= 2,0 Crd 2y as well as  MK= 2,0 Crd 2c;  hence from the table of chords  e  2;30  Crd 2c="R Crd 2y =l,O' Crd 131 =2;30· 1;49,11,44~4;33. Again from the tables 2c=4;20 thus c=2;10 and  180-K=y-c=63;20 hence K= 116;40. No.3. Find the angle c, the so-called "equation," to given mean eccentric anomaly K= 30 (AIm. III, 5). Apogee and eccentricity are the same as in the preceding example. Make (cf. Fig. 12) OK perpendicular to SMK. Then  OK=~Crd2K= 2;30 Crd60= 2;30. 10 =1'15 2,0  2,0  2,0"  .  From the table of chords e 2'30 2'30 MK= 2,0 Crd (180-2K)= ~,O Crd 120= ~,O ·1,43;55~2;10.  Consequently we have in the right triangle SOK with SM=R=60 SK=SM+MK=I,2;1O thus  SO=v'SK2+0K2~1,2;114  and therefore, if ON =NS= 1/2 SO ON OK= 2,0 Crd2c  or Crd2c=  1,0·OK SO  2,30 1,2;11 ~2;25.  From the table one finds for the corresponding angle 2c=2;18, thusc= 1;9. This value is also found tabulated for K = 30 in the table for the solar anomaly (AIm. III, 6). The examples Nos. 2 and 3 deal with complementary problems. In the first case the mean longitude K had to be found from the (observable) true longitude K; in the second case the mean longitude is assumed to be known and the true longitude is asked for, to be found from K = K- c.  4. Summary For the sake of convenience I write down the three basic formulae for the solution of right triangles in the form in which they appear in ancient trigonometry, assuming the norm 2R = 120 for the function Crd II. I also give the equivalent 4 Using the procedure described below p.lll4 we get for y'1,2;IQz + 1;15 z =Yl,4,26;15,25 as first approximation IXI = 1,2;10. Thus PI = 1,4,26;15,25/1,2;1O~ 1,2;12. Hence IXz = 1,:2;11.  26  I A 2, 1. Spherical Trigonometry  formulae for R = 1 and the modern trigonometric functions (cf. Fig. 13): ; =  2~0 Crd21l=1/2crd21l=sinll  b 1 -=-Crd (180-211)= 1/2 crd(180-21l)=cosll C 2,0  a b  Crd 211 Crd (180 - 211)  crd 211 crd (180- 211)  tan 11.  For the general triangle the equivalent of the sine theorem is a direct consequence of the fact that the angle in the center is twice the angle at the periphery over the same chord (cf. Fig. 14):  a: b:c=Crd 21l:Crd 2{3:Crd 2y = sin 11: sin {3: sin y. This theorem is repeatedly used, e.g. in the determination of the eccentricity of the planets (cf. p. 174, Steps 1 and 2). Fig. 13 also illustrates the reason for a terminology which is frequently encountered in the solution of right triangles. Where we would say a = sin 11 ancient trigonometry has to express the fact that a is the chord to the angle 211 at the center of the circumscribed circle. This then is expressed in the form 5 angle CAB = 11  as 4 right angles are 360  or=21l as 2 right angles are 360. Now one enters the table of chords with the argument 211 and finds a=crd 211.  § 2. Spherical Trigonometry 1. The Menelaos Theorem The trigonometry of the right spherical triangle (cf. Fig. 15) rests on the following four formulae 1 . sin a SlDll=-.(1) SlDC  tanb  cos 11=-tane  (2)  tan a tan 11=-sin b  (3)  cos e=cos a cos b.  (4)  Cf., e.g., ed. Heiberg I, p. 317, 22f., et passim. Two additional formulae (or their counterparts for chords), namely cos a=cos a/sin fJ and cos c= cot a cot fJ which express the sides by means of the angles never occur in ancient spherical trigonometry, although it was known that a spherical triangle is determined by its angles (Menelaos I, 18; Krause, p. 138). The equivalent of (1), (4) and cos a =coSlX/sin fJ is proved by Copernicus (De revol. I, 14 Theorems 3 and 4) but he has still no formula in which a tangent occurs. 5  1  I A 2, 1. Menelaos Theorem  27  With decreasing sides the formulae (1) to (3) tend toward the defining relations for the trigonometric functions in the plane right triangle whereas (4) is the equivalent of the Pythagorean theorem c2 = a2 + b2• These relations were not discovered before the Islamic period. In Greek astronomy relations were utilized which involve six parts in a certain configuration of which the triangle under consideration is a part and therefore called by the Arabs "the rule of the six quantities." The discovery of this theorem is due to Menelaos whose date is known through a reference in the Almagest 2 to an observation of his made in the time of Trajan (in A.D. 98). The proof of this " Menelaos Theorem" is preserved not only in the Almagest (I, 13) but also in Menelaos' original work on "Spherics," extant in an Arabic version. It proceeds in two steps. First it is shown to hold for a plane configuration and then it is associated with a similar spherical arrangement. In order to present the idea of the proof as simply as possible we introduce the following terminology. We call a "Menelaos Configuration" a configuration as represented in Fig. 16. Two "outer parts" AV and BV meet in the "vertex" V, two "inner parts" BC and AD intersect in K. In this way six line segments are defined which we denote as follows  ml +m2=m nl> n2 and n1 +n 2=n  outer parts: ml> m2 and inner parts: r1 , r 2  and r1 +r 2=r  SI,S2  and S.+S2=S,  For these six parts two theorems are established. If c is parallel to r. (cf. Fig. 16) then m r r r1 - = - = - . - and !:!=~. hence  Theorem I.  ml  c  c  r. c  S  m r S2 _=_e_  in which two outer elements are expressed by means of inner ones. Similarly, if we make d parallel to S. (Fig. 16) we have r.  n2  --=rl +e n  thus  m m.  r2  n n  2 - 2= _ . -  or  r.  Theorem II. Here two inner parts are expressed by means of outer ones. We now prove that similar relations hold for a Menelaos configuration made up by great circle arcs on a sphere (cf. Fig. 1 Let A be the vertex, AB and Ar  n  2  VII,3 (Heib. II, p.30, 18).  28  IA2, I. Menelaos Theorem  the outer parts, BE and r LI the inner parts, intersecting in Z. We consider the plane triangle ArLi and the intersections A, K, e of the radii of the sphere HE, HZ, HB, respectively with the plane of this triangle. Thus A lies on the side Ar, K on rLl, e on ALI (extended). But A, K, e do not only belong to the plane ofthe triangle ArLI but also to the plane (shaded in Fig. 17) of the great circle EZB with center H. Thus A, K, e lie on the straight line in which the plane ofthe triangle ArLI intersects the plane of the great circle EZB. This shows that associated with the spherical Menelaos configuration of vertex A is a plane Menelaos configuration with the outer elements Ae, AT and with the inner elements e A, r LI which intersect in K. The final transformation of the two above theorems from the plane configuration to a spherical configuration rests on the relation EA Er  crd tX crdp  --=--  (1)  which can be easily verified for the two cases represented in Figs. 18 and 19. This relation (1) permits us to replace the ratios which enter the Menelaos theorems for the plane configuration by ratios between corresponding arcs of the spherical configuration. If, e.g., in Fig. 17 AT represents the outer part m in the plane configuration, J.I. in the spherical configuration, then it follows from (1) that (cf. Fig. 20) ml crd(2J.1.1) -= (2) m2 crd (21l2)"  If we now introduce, for the sake of simplicity, the notation crd (2tX)=2 sin tX=(tX) then we can write, because of (2) and of similar identities for the other parts, the two Menelaos theorems in the form  Theorem I. Theorem II.  (r2)  (rtl =  (m2) (m l )  •  (n) (n 2 )  where m, n, r, s, etc., now represent great circle arcs. It is, of course, always possible in a Menelaos configuration to replace the outer part m by n by interchanging the inner parts accordingly. This leads to equivalent forms for both theorems  Theorem I.  (n) (nl)  (r 2 )  Theorem II.  (S2) (m) (n 2) (SI) = (m2) . (ntl·  =(,1.  (s) (SI)  I A 2, 2. Menelaos Theorem  29  In all the above formulae we may interpret the symbols (m~ (n~ etc., as sines instead of as chords, because  (0) = crd 20=2 sin 0  and therefore  crd 2a  (a)  sin a  (1))= crd 2b = sin b . In this interpretation the Menelaos theorems appear as the "theorems of the six quantities" in Islamic spherical astronomy. We shall use the same device if we wish to compare Ptolemy's results with the corresponding solutions given by modem trigonometry.  2. Supplementary Remarks Before turning to the astronomical applications of the Menelaos theorems we shall make two additional remarks. One is due to Theon of Alexandria who realized that Theorem I can be derived from Theorem 11. 3 The other remark concerns the fact that the four basic formulae (1) to (4) (above p. 26) for the right spherical triangle are the exact equivalents of the Menelaos Theorems I and II. To show that Theorem I can be derived from Theorem II we consider the Menelaos configuration AV1C (Fig. 21). Then it follows from Theorem II that (m2) (ml)  ('2)  (n2)  =-VtY' Tn)'  (1)  We now extend the great circles AV1 and AB until they intersect in the point A which is diametrically opposite to A. Then we have in CV2 A a new Menelaos configuration in which the previously considered inner parts 51> 52 now play the role of outer parts VI' V2 and conversely nl =0"1' n2 =0"2' We have furthermore ml =Jll> '1 =PI, m2 = 180- Jl (thus 2m2 = 360- 2Jl or I(m2) I= 1{Jl)1) and '2 = 180- P (thus 1('2)1=1(P)I). Substituting the new elements in (1) one thus obtains (Jl)  (p)  (0"2)  {Jld = (PI) . (c;f which is Theorem I for the new configuration. Since every Menelaos configuration can be completed such that it plays the role of the second configuration in the above proof, Theorem I will hold for it by virtue of the fact that Theorem II holds for all Menelaos configurations; q.e.d. We now shall prove that the fundamental formulae (1) to (4)

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