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 (<Euvres IV, p.l4l) 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, 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.