Studies in the History of Mathematics and Physical Sciences
1
Editors M.J. Klein G.J. Toomer 0. Neugebauer
A History of Ancient Mathematical Astronomy
In Three Parts with 9 Plates and 619 Figures
Springer-Verlag Berlin Heidelberg GmbH 1975 Otto Neugebauer Brown University, Providence, Rhode lsland 02912, USA
ISBN 978-3-642-61912-0 ISBN 978-3-642-61910-6 (eBook) DOI 10.1007/978-3-642-61910-6
Additional material to this book can be downloaded from http://extras.springer.com
Library of Congress Cataloging in Publication Data. Neugebauer. Otto. 1g99-. A history of ancient mathematical astronomy. (Studies in the history of mathematics and physical sciences; v. 1). lncludes bibliographies and indexes. Contents: pt. 1. The Almagest and its direct predecessors. Babylonian astronomy. - pt. 2. Egypt. Early Greek astronomy. Astronomy during the Roman Imperial period aod late antiquity. - pl. 3. Appendices and indices. 1. Astronomy, Ancient - History. 2. Astronomy - Mathematics - History. 1. Title. Il. Series. QB16.N46. 520'.93. 75-8778.
springeronline.com
This work is subject to copyright. Ali rights are reserved, whether the who1e 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 payab1e to the publisher, the amount of the fee to be determined by agreement with .the publisher. © by Springer-Verlag Berlin Heidelberg 1975. Originally published by Springer-Verlag New York Heidelberg Berlin in 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 quej'etudie l'astronomie,je ne pense ni d Balkis, ni d quoi que ce soit au monde. Les sciences sont bienfaisantes: elles empechent les hommes de penser. A. France, Balthasar ( 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, modeme: 1821) though I fully realize that I do not have by far the professional competence ofDelambre. 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 Coper• nicus, 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 tum 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 kaleido• scopic 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 Insti• tute 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 tum 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. Kai oirwx; a:TtBPX.OJI.IXI ... Wt; wtlJ8 &p~rXJI.BVor;. 1 Providence, June 1975 O.N. 1 'Appa.~; flaJJPfJJ (Migne, PG 65,369 r(). Table of Contents Part One Introduction § I. Limitations...... 1 § 2. The Major Historical Periods, An Outline 2 A. The Hellenistic Period 3 B. The Roman Period . 5 C. Indian Astronomy . 6 D. The Islamic Period . 7 E. Epilogue .... 14 § 3. General Bibliography 15 A. Source Material . 15 B. Modern Literature 16 C. Sectional Bibliographies . 17 Book I The Almagest and its Direct Predecessors A. Spherical Astronomy . 21 § I. Plane Trigonometry . . 21 I. Chords ...... 21 2. The Table of Chords 22 3. Examples . . . . . 24 4. Summary ..... 25 § 2. Spherical Trigonometry 26 I. The Menelaos Theorem . 26 2. Supplementary Remarks 29 § 3. Equatorial and Ecliptic Coordinates 30 I. Solar Declinations ...... 30 2. Right Ascensions...... 31 3. Transformation from Ecliptic to Equatorial Coordinates . 32 § 4. Geographical Latitude; Length of Daylight . 34 I. Oblique Ascensions. . . 34 2. Symmetries ...... 35 3. Ascensional Differences . 36 4. Ortive Amplitude . . . 37 5. Paranatellonta . . . . . 39 6. Length of Daylight; Seasonal Hours 40 7. Geographical Latitude; Shadow Table 43 § 5. Ecliptic and Horizon Coordinates . . . 45 l. Introductory Remarks ...... 45 2. Angles between Ecliptic and Horizon 46 3. Ecliptic and Meridian . 47 X Table of Contents 4. Ecliptic and Circles of Altitude . 48 S. The Tables (Aim. II, 13) . so B. Lunar Theory ...... 53 § 1. Solar Theory ...... 53 1. The Length of the Year . 54 2. Mean Motion . . . . . 55 3. Anomaly ...... 55 A. Eccenter and Epicycles . 56 B. Determination of Eccentricity and Apogee 57 C. The Table for the Solar Anomaly and its Use 58 § 2. Equation ofTime ...... 61 1. The Formulation in the Almagest (III, 9). 61 2. Examples ...... 62 3. Proof of Ptolemy's Rule ...... 65 4. The Equation of Time as Function of the Solar Longitude 66 § 3. Theory of the Moon. First Inequality; Latitude 68 1. Introduction...... 68 2. Mean Motions ...... 69 3. Period of the Lunar Anomaly . . . 71 4. Radius and Apogee of the Epicycle . 73 A. Summary of the Method . . . . 73 B. Numerical Data and Results .. 76 C. Check ofthe Mean Anomaly; Epoch Values 78 S. The Tables for the First Inequality . . . . . 80 6. Latitude ...... 80 A. Mean Motion of the Argument of Latitude 80 B. Epoch Value for the Argument of Latitude 81 C. The Lunar Latitude; Example. . . 83 § 4. Theory of the Moon. Second Inequality . 84 1. Empirical Data and Ptolemy's Model . 84 2. Determination of the Parameters . . 86 A. Maximum Equation; Eccentricity 86 B. "Inclination" ...... 88 C. Critical Remarks ...... 91 3. Computation of the Second Inequality; Tables 93 4. Syzygies . . 98 § S. Parallax ...... 100 1. Introduction...... 100 2. The Distance of the Moon . 101 3. Apparent Diameter of the Moon and of the Sun 103 A. Ptolemy's Procedure . . . 104 B. Criticism ...... 106 4. Size and Distance of the Sun . 109 A. Hipparchus' Procedure .. 109 B. Historical Consequences . lll S. The Table for Solar and Lunar Parallax (Aim. V, 18) 112 6. The Components of the Parallax . . 115 § 6. Theory of Eclipses ...... 118 1. Determination of the Mean Syzygies 118 2. Determination of the True Syzygies . 122 3. Eclipse Limits ...... 125 4. Intervals between Eclipses . 129 S. Tables (VI, 8) . . . . . 134 6. Area-Eclipse-Magnitudes 140 7. Angles oflnclination . . 141 Table of Contents XI C. Planetary Theory . 145 § l. Introduction . 145 l. General. . 145 2. Distances and Eccentricities 146 3. Ptolemy's Introduction to Almagest IX 148 4. Parameters of Mean Motion . 150 §2. Venus ...... 152 1. Eccentricity and Equant . . . . . 152 2. Mean Motion in Anomaly. Epoch 156 3. The Observational Data. 158 § 3. Mercury ...... 158 l. Apogee ...... 159 2. Eccentricity and Equant. 161 3. Perigees ...... 163 4. Mean Motion in Anomaly. Epoch. 165 5. Minimum Distance and Motion of the Center of the Epicycle . 168 § 4. The Ptolemaic Theory of the Motion of an Outer Planet 170 l. The Basic Ideas . . · ...... 170 2. Refinement of the Model ...... 171 3. Determination of the Eccentricity and Apogee 172 A. Eccentricity from Oppositions . . . 173 B. Approximative Solution . . . . . 174 C. Separation of Equant and Deferent 175 D. Results ...... 177 4. The Size of the Epicycle . . 179 5. Mean Motion in Anomaly. 180 6. Epoch Values .... 182 § 5. Planetary Tables . . . . 183 1. The General Method . 183 2. Numerical Data . . . 184 3. Examples ...... 186 A. Ephemeris for Mars . 186 B. Ephemeris for Venus. 187 § 6. Theory of Retrogradation 190 1. Stationary Points . . 191 A. Mean Distance . . 192 B. Maximum Distance 193 C. Minimum Distance . 196 D. Numerical Data. . . 197 2. Tables for Retrogradations 202 A. Epicycle at Extremal Distances 202 B. Epicycle at Arbitrary Distances; Tables. 204 C. Examples . . 205 § 7. Planetary Latitudes . 206 1. The Basic Theory 207 2. Numerical Data . 207 A. The Outer Planets . 208 B. The Inner Planets . 212 3. The Tables Aim. XIII, 5 . 216 A. Outer Planets. . . 218 B. Inner Planets . . . 221 C. Extremal Latitudes 226 D. Transits . . . . . 227 § 8. Heliacal Phenomena ("'Phases'") . 230 1. Maximum Elongations ...... 230 XII Table of Contents A. Venus ...... 231 B. Mercury ...... 232 C. The Tables (Aim. XII, 10). 233 2. The "Normal Arcus Visionis" 234 A. Ptolemy's Procedure . . . 234 B. Numerical Details . . . . 236 3. Extremal Cases for Venus and Mercury 239 A. Venus ...... 239 B. Mercury ...... 241 4. The Tables (Aim. XIII, 10). 242 A. Example ...... 243 B. Method of Computing the Tables 244 5. The Planetary Phases in the Handy Tables and Other Sources 256 D. Apollonius...... 262 § 1. Biographical Data ...... 262 § 2. Equivalence of Eccenters and Epicycles . 263 1. Transformation by Inversion. . . 264 2. Lunar Theory ...... 265 § 3. Planetary Motion; Stationary Points. . 267 1. Apollonius' Theorem for the Stations 267 2. Empirical Data 270 E. Hipparchus ...... 274 § 1. Introduction ...... 274 § 2. Fixed Stars. The Length of the Year 277 1. Stellar Coordinates. Catalogue of Stars 277 A. Stellar Coordinates ...... 277 B. Hipparchus' and Ptolemy's Catalogue of Stars . 280 C. Catalogue of Stars. Continued. . 284 D. Stellar Magnitudes ...... 291 2. The Length of the Year. Precession . 292 A. Tropical and Sidereal Year . . . 293 B. Intercalation Cycles ...... 296 C. Constant of Precession; Trepidation . 297 § 3. Trigonometry and Spherical Astronomy . 299 1. Plane Trigonometry; Table of Chords . 299 2. Spherical Astronomy . 301 § 4. Solar Theory ...... 306 § 5. The Theory of the Moon . . . . 308 1. The Fundamental Parameters 309 A. Period Relations . . . 309 B. The Draconitic Month . 312 C. The Epicycle Radius . 315 2. Eclipses ...... 319 A. Tables ...... 319 B. Eclipse Cycles and Intervals 321 3. Parallax ...... 322 4. Size and Distance of Sun and Moon 325 A. Distance of the Sun . . 325 B. Hipparchus' Procedure. 327 § 6. Additional Topics . 329 1. The Planets . 329 2. Astrology . . . 331 3. Geography . . 332 A. Geographical Latitude . 333 Table of Contents XIII B. Longitudes ...... 337 4. Fragments ...... 338 § 7. Hipparchus' Astronomy. Summary 339 Book II Babylonian Astronomy Introduction ...... 347 § 1. The Decipherment of the Astronomical Texts . 348 § 2. The Sources . . . . . 351 § 3. Calendaric Concepts . . . . 353 1. The 19-Year Cycle . . . 354 2. Solstices and Equinoxes . 357 3. Sirius Dates . . 363 4. Summary ..... 365 § 4. Length of Daylight . . 366 1. Oblique Ascensions. 368 2. Length of Daylight . 369 § 5. Solar Motion . . . . . 371 § 6. Mathematical Methodology 373 1. System B . 374 2. System A .. 375 A. Planetary Theory . . 380 § 1. Basic Concepts . 380 § 2. Periods and Mean Motions . 388 § 3. System A ...... 392 §4. Dates ...... 394 § 5. Subdivision of the Synodic Arc; Daily Motion. 397 1. Subdivision of the Synodic Arc . 398 A. Jupiter .. 398 B. Mars ...... 399 C. Mercury ...... 401 2. Subdivision of the Synodic Time; Velocities 404 A. Summary; Jupiter. 404 B. Mars .. 406 3. Daily Motion 412 A. Jupiter .. 413 B. Mercury. 418 § 6. The Fundamental Patterns of Planetary Theory . 420 1. System A ...... 421 A. Numerical Data ...... 422 B. Subdivision of the Synodic Arc 422 C. Approximate Periods 426 2. System B ...... 427 3. Historical Reminiscences 431 § 7. The Single Planets. 434 1. Introduction. 434 2. Saturn ... 436 A. System A 437 B. System B. 439 C. Subdivision of the Synodic Arc; Daily Motion. 439 XIV Table of Contents 3. Jupiter . . . 441 A. System A 444 B. System B. 446 C. Subdivision of the Synodic Arc 447 D. Daily Motion. . . 452 4. Mars ...... 454 A. Periods; System A. 454 B. System B ..... 457 C. Subdivision of the Synodic Arc; Retrogradation . 458 5. Venus .... . 460 A. Periods .. . 460 B. Ephemerides 461 6. Mercury ... . 466 A. Periods .. . 466 B. System A1 to A3 468 B. Lunar Theory . . 474 § l. Introduction . 474 § 2. Lunar Velocity 476 l. System B . 477 2. SystemA . 478 3. Daily Motion 480 4. Summary .. 481 § 3. The Length of the Synodic Months 482 1. System B, Column G . . . . 483 2. System A, Columns .P and G . . 484 A. The Function .P ...... 484 B. Column G near the Extrema 485 C. The Function G...... 487 3. System A, Column J . . . . . 488 4. System A, Columns C', K, and M . 490 5. System B, Columns H to M 492 A. Summary ... 492 B. Columns H and J . . 492 C. Column M ..... 496 §4. The "Saros" and Column .P 497 l. The Functions .P* and F* 499 2. The Saros ...... 502 3. .P, Friends and Relations 505 A. Summary ..... 505 B. Mathematical Methodology 506 C. Numerical Details . . . . . 507 § 5. Lunar Latitude ...... 514 l. Retrogradation of the Lunar Nodes . 514 2. System A, Column E . . 514 3. The Saros ...... 517 4. Other Latitude Functions 520 § 6. Eclipse Magnitudes 521 l. System A . 522 2. System B ... 523 § 7. Eclipse Tables . . 525 § 8. Solar Mean Motion and Length of Year 528 §9. Variable Solar Velocity 530 l. Type A and B . . 530 2. System A and A' . 531 3. System B .... 533 Table of Contents XV § 10. Visibility ...... 533 1. The Date of the Syzygies 534 2. First Visibility . . . . . 535 3. Last Visibility and Full Moons . 538 4. Visibility Conditions . . . . . 539 C. Early Babylonian Astronomy . . . . . 541 § I. Calendaric Data, Celestial Coordinates . 541 § 2. The Moon . 547 § 3. The Planets ...... 553 Part Two Book III Egypt § I. Introduction and Summary . 559 § 2. The 25-year Lunar Cycle . 563 § 3. Concluding Remarks 565 § 4. Bibliography ...... 566 A. General ...... 566 B. Demotic and Coptic Texts. 567 Book IV Early Greek Astronomy Introduction ...... 571 A. The Beginning of Greek Astronomy . 573 § I. Chronological Summary . 573 I. The Early Period ...... 573 2. More Recent Period . . . . 574 § 2. Sphericity of the Earth; Celestial Sphere and Constellations . 575 § 3. Geminus . . . . 578 I. Date . . . . . 579 2. The Isagoge . . 581 3. The Parapegma 587 § 4. Babylonian Influences . 589 1. The Sexagesimal System. 590 2. The Ecliptic and its Coordinates 593 A. Aries go as Vernal Point . . 594 B. Other Norms for the Vernal Point . 598 3. Mathematical Astronomy . 601 A. Lunar and Solar Theory . 601 B. Planetary Theory . . . . 604 4. Ancient Tradition; Summary 607 A. "Schools" and Astronomers 610 B. Parapegmata . . . 612 C. Summary ...... 613 B. Early Lunar and Solar Theory . . . 615 §I. Luni-Solar Cycles; Lunar Theory 615 1. Early Greek Cycles . . . . . 619 XVI Table of Contents 2. The Metonic and Callippic Cycle . 622 3. Lunar Theory . 624 § 2. Solar Theory . . . 626 l. Solar Anomaly 627 2. Solar Latitude . 629 3. The Trepidation of the Equinoxes 631 § 3. Sizes and Distances of the Luminaries 634 l. Aristarchus ...... 634 A. Aristarchus' Assumptions . 635 B. Mathematical Consequences 636 C. Numerical Consequences. 637 D. Aristarchus' Procedure. 639 E. Summary ...... 642 2. Archimedes ...... 643 A. The "Sand-Reckoner" . 643 B. Cosmic Dimensions . . 647 3. Posidonius ...... 651 A. Measurement of the Earth 652 B. Size and Distance of the Moon 654 C. Size and Distance of the Sun . 655 4. Additional Material ...... 657 A. Apparent Diameter of Sun and Moon 657 B. Distances of Sun and Moon . 659 C. Actual Sizes of Sun and Moon 662 § 4. Eclipses ...... 664 § 5. The "Steps" (Ptx9Jl0i) 669 C. Early Planetary Theory 675 § l. Eudoxus . . . . . 675 l. General Data . 675 2. The Homocentric Spheres . 677 A. The Eudoxan Model. 677 B. Numerical Data ... 680 C. Later Modifications . 683 3. The "Eudoxus Papyrus" 686 A. TheText...... 686 B. Summary of Contents 687 § 2. Other Planetary Hypotheses 690 l. Arrangement of the Planets 690 2. Cinematic Hypotheses .. 693 § 3. The Inscription of Keskinto. . 698 D. The Development of Spherical Astronomy 706 § l. Arithmetical Methods; Length of Daylight; Climata 706 l. Length of Daylight . 708 2. Oblique Ascensions. 712 A. SystemA 715 B. SystemS ... . 721 3. Climata ...... 725 A. Climata and Rising Times 727 B. Early Mathematical Geography . 733 § 2. Shadow Tables ...... 736 l. Arithmetical Patterns ...... 737 A. Greek Shadow Tables . . . . . 737 B. Late Ancient and Medieval Shadow Tables 740 2. Shadow Lengths in Greek Geography . . . . . 746 Table of Contents XVII § 3. Spherical Astronomy before Menelaus 748 l. Authors and Treatises 748 2. Figures in the Texts 751 3. Spherics ...... 755 A. Polar Days . . . . 757 B. Directional Terms. 758 C. Non-Intersecting Semicircles 758 D. "Interchange" of Hemispheres 759 4. Fixed Star Phases ...... 7(JJ 5. Rising Times, Length of Daylight, Geographical Data . 763 6. Later Developments 767 § 4. Plane Trigonometry ...... 771 BookV Astronomy during the Roman Imperial Period and Late Antiquity Introduction . 779 A. Planetary and Lunar Theory before Ptolemy 781 § l. Planetary Theory . . 781 l. Introduction ...... 781 2. Planetary Tables ...... 785 A. Arrangement and Contents . 785 B. Notation...... 788 C. Historical Questions . . . . 789 3. Planetary Theory in Vettius Valens . 793 A. Solar Longitudes . 794 B. The Outer Planets . . . 794 C. Venus and Mercury .. 796 4. Incorrect Epicyclic Theory. 801 A. Pliny .... 802 B. Pap. Mich. 149 .... 805 § 2. Lunar Theory ...... 808 1. P. Ryl. 27 and Related Texts . 808 A. P. Ryl. 27 ...... 809 B. P. Lund Inv. 35a . . . . 813 C. The 25-year Cycle and the Epoch Dates 815 D. India .. . 817 E. PSI 1493 ...... 822 2. Vettius Valens ...... 823 A. Lunar Longitudes and Phases . 824 B. Lunar Latitude . 826 § 3. Visibility Problems 829 1. Moon ...... 829 2. Planets ...... 830 B. Ptolemy's Minor Works and Related Topics 834 § 1. Biographical and Bibliographical Data . 834 l. The "Almagest" ...... 836 2. Later Tradition ...... 838 § 2. The "Analemma" and its Prehistory . 839 l. Introduction...... 839 2. Diodorus of Alexandria . 840 A. Biographical Data . . 840 XVIII Table of Contents B. The Determination of the Meridian Line . 841 C. Pappus' Commentary to the "Analemma" 843 3. Vitruvius ...... 843 4. Great Circle Distance between Two Cities . 845 A. Heron ...... 845 B. "Dioptra 35" ...... 847 5. Spherical Coordinates . . . . . 848 A. Ptolemy's Coordinate System . 849 B. The "Old" System of Coordinates . 849 6. Construction of the Ptolemaic Coordinates 850 A. The Hektemoros 850 B. The Six Angles . 851 C. Graphic Solution 852 D. Tables ..... 854 E. Application to Sun Dials . 855 7. The Origin of the Conic Sections . 857 § 3. The "Planisphaerium" . 857 1. Introduction. . . . 857 2. Auxiliary Theorems 860 3. Right Ascensions. . 861 4. Oblique Ascensions. 864 5. The Greatest Always Invisible Circle 865 6. Ecliptic Coordinates . . . . 866 7. Historical Remarks; Synesius . 868 A. Introduction ...... 868 B. Earliest History; Hipparchus 868 C. Vitruvius and the Anaphoric Clock 869 D. Ptolemy ...... 870 E. Synesius ...... 872 F. Theon, Severus Sebokht, Philoponus. 877 § 4. Map Projection ...... 879 1. The Marinus' Projection . . . . . 879 2. Ptolemy's First Conic Projection . . 880 3. Ptolemy's Second Conic Projection . 883 4. Visual Appearance of a Terrestrial Globe 889 5. Appendix. Precession-Globe (Aim. VIII, 3). 890 § 5. Optics ...... 892 § 6. The Tetrabiblos ...... 896 § 7. "Planetary Hypotheses" and "Canobic Inscription" 900 1. Introduction...... 900 2. Sun and Moon ...... 901 3. Planets; Periods and Longitudes . 905 4. Planetary Latitudes. . . 908 A. Angles of Inclination 908 B. Precession . . 909 C. Epoch Values. . .. 910 D. Tables ...... 913 5. The Canobic Inscription 913 6. The "Ptolemaic System" 917 7. Book II of the Planetary Hypotheses 922 § 8. Additional Writings of Ptolemy . 926 1. The " Phaseis" ...... 926 A. Aim. VIII, 6 ...... 927 B. The " Phaseis" Book II . . 928 2. Astronomy in the "Harmonics" 931 Table of Contents XIX 3. The "Geography" . . . 934 A. Spherical Astronomy 935 B. Historical Remarks . 937 4. Philosophy; Fragments . 940 C. The Time from Ptolemy to the Seventh Century . 942 § l. Introduction ...... 942 § 2. The Time from Ptolemy to Theon . 943 l. Chronological Summary . 943 2. Papyri and Ostraca . . . . 944 3. Second and Third Century. 948 A. Artemidoros . . . . . 948 B. Theon of Smyrna and Adrastus 949 C. Achilles . . 950 4. Fourth Centur)' ...... 952 A. Astrology...... 952 B. Astronomical Considerations . 955 5. Cleomedes ...... 959 A. The Date of Cleomedes . . . 960 B. Geography and Spherical Astronomy 961 C. Moon and Sun 962 D. Planets .. 964 § 3. Pappus and Theon 965 §4. The Handy Tables. 969 l. Introduction. . 969 A. Arrangement . 971 B. Variants in the Handy Tables . 973 C. Bibliography ...... 976 D. Appendix. Notes on Manuscripts 976 2. Spherical Astronomy ...... 979 A. Rising Times ...... 979 B. Seasonal Hours; Ascensional Differences . 980 C. Ortive Amplitudes. 982 3. Theory of the Sun . . 983 A. Solar Longitude. . 983 B. Equation of Time . 984 C. Precession . . . . 986 4. Theory of the Moon . 986 A. The Tables for Mean Motions. 986 B. Epoch Values ..... 987 C. The True Moon. . . . 988 D. Parallax and Prosneusis 990 E. Eclipses . . 999 5. The Planets . . 1002 A. Longitudes. 1002 B. Latitudes . 1006 C. Visibility(" Phases") . 1017 6. Appendix. Supplementary Material . 1025 A. Royal Canon ...... 1025 B. Reference Stars ...... 1026 § 5. The Time from Theon to Heraclius 1028 l. Chronological Summary 1028 2. Fifth to Seventh Century 1029 A. Popularization . . 1029 B. The Latest Schools 1031 C. Fragments . 1051 3. Ephemerides . . . . 1055 XX Table of Contents Part Three Book VI Appendices and Indices. Figures and Plates A. Chronological Concepts . . . . 1061 § l. Years and Julian Days . . . 1061 § 2. Special Calendars and Eras . 1064 l. The Egyptian Calendar 1064 2. The Seleucid Calendar 1064 3. Synopsis of Eras . . . 1065 4. The "Era Dionysius" . 1066 § 3. The Reckoning of Days . 1067 l. Epoch ...... 1067 2. Hours and Other Divisions 1069 3. Astronomical Time Units . 1070 § 4. The Foundations of Historical Chronology 1071 § 5. Literature ...... 1074 l. General...... 1074 2. Chronological Tables . 1075 B. Astronomical Concepts . . 1077 § l. Spherical Coordinates . 1077 l. The Horizon System 1077 2. The Equator System 1078 3. The Ecliptic System 1078 4. Relations Between the Systems . 1079 5. Equation of Time . 1081 6. " Polar" Coordinates 1081 § 2. Years, Months 1082 l. The Year 1082 2. Months .. 1083 § 3. Fixed Stars . . 1084 l. Proper Motion 1084 2. Yearly Parallax 1085 3. Names and Constellations . 1087 § 4. Geocentric Planetary Motion . 1088 § 5. Planetary and Fixed Star Phases 1090 1. Planetary Phases . 1090 2. Fixed Star Phases . . 1090 3. Tables ...... 1091 § 6. Lunar and Solar Eclipses . 1092 § 7. Kepler Motion 1095 1. Definitions . . 1095 2. Parameters . . 1096 3. Kepler's Laws . 1097 4. Approximations . 1098 5. Eccenter Motion . 1100 6. "Elliptic" Orbits . 1102 § 8. The Inequalities of the Lunar Motion 1103 1. Longitude ...... 1106 2. Latitude ...... 1107 3. Bibliographical and Historical Remarks . 1108 A. Evection . 1108 B. Variation ...... 1109 Table of Contents XXI C. Annual Equation . . 1110 D. Latitude and Nodes . 1111 E. Bibliographical Notes 1112 C. Mathematical Concepts . . . . 1113 § 1. Sexagesimal Computations . 1113 § 2. Square Root Approximations . 1114 § 3. Trigonometry...... 1115 § 4. Diophantine Equations; Continued Fractions . 1116 1. Euclidean Algorithm . . . . . 1116 2. Linear Diophantine Equations . 1117 3. Continued Fractions . . . 1120 § 5. Tables...... 1126 1. Sexagesimal Computations 1126 2. Trigonometric Functions 1129 D. Indices ...... 1133 § 1. Subject Index ...... 1133 § 2. Bibliographical Abbreviations. 1165 § 3. Notations and Symbols . 1204 1. Calendar, Chronology . . 1204 2. Spherical Astronomy . . . 1204 3. Lunar and Planetary Motion 1205 4. Planetary and Fixed Star Phases 1206 § 4. Greek Glossary . 1206 E. Figures and Plates . 1209