The history of Ancient Astronomy. Problems and methods

The History of Ancient Problems $ i and methods Journal IV - Astronomy of Near Eastern Studies, 1945 - p. 1 - 38 } ISI pal NELGESARLL £R, a. VANS D Vergeben O The Mesfora. Ask LEE TMfineBu s aid, T ef Near Lasher, Srs SE Am ceen! 28} KHDa cmd Lo el Le (Jan gys/ Wo neles blog rguh

The History of Ancient Astronomy Problems and Methods Author(s): O. Neugebauer Source: Journal of Near Eastern Studies, Vol. 4, No. 1 (Jan., 1945), pp. 1-38 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/542323 Accessed: 18/01/2010 13:03 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ucpress. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of Near Eastern Studies. http://www.jstor.org

JOURNAL STUDIES EASTERN NEAR Number 1 JANUARY 1945 Volume IV THE HISTORY OF ANCIENT ASTRONOMY PROBLEMS AND METHODS 0. NEUGEBAUER TABLE OF CONTENTS I. INTRODUCTION. .. 2 1. Scope and Character of the Paper .......2 2. Definition of "Astronomy" .2 3. Character of Ancient Astronomy 3 II. EGYPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Egyptian Mathematics .4 5. Egyptian Astronomical Documents 6. Description of Egyptian Astronomy III. MESOPOTAMIA . . . . . . . 4 6 . . . . . . . . . . . . . . . . . . . 12. Greek Spherical Astronomy . . . . 13. Mathematical Geography ....... 14. Astrology . . . . . . . . . 15. Greek Mathematics .........22 16. From Hipparchus to Ptolemy .......23 17. Relations to Mesopotamia . . . . . . . . . . . . . . . 16 . . . . . . . 24 .16 .18 .20 . . . . . . . . . . 18. Social Background 19. Metrology . . . . . .27 20. History of Constellations 21. Chronology . . . . . .28 22. Hindu Astronomy .29 23. Methodology of the History of Astronomy . .26 V. SPECIAL PROBLEMS . 8 8 8 11 7. The Sources of Babylonian Astronomy 8. Mathematical Astronomy in the Seleucid Period. 9. Babylonian Mathematics 10. Earlier Development of Babylonian Astronomy .....13 11. Babylonian Astrology . . . . . . . .14 IV. THE HEIT,TNISTIC PERIOD 4 . . ..26 BIBLIOGRAPHY AND ABBREVIATIONS ......32 ...

JOURNAL OF NEAR EASTERN STUDIES Ce qui est admirable, ce n'est pas que le champ des etoiles soit si vaste, c'est que l'homme l'ait mesure.-ANATOLEFRANCE,Le Jardin d'Epicure. I. INTRODUCTION 1. In the following pages an attempt is made to offer a survey of the present state of the history of ancient astronomy by pointing out relationships with various other problems in the history of ancient civilization and particularly by enumerating problems for further research which merit our interest not only because they constitute gaps in our knowledge of ancient astronomy but because they must be clarified in order to lay a solid foundation for the understanding of later periods. I wish to emphasize from the very beginning that the attitude taken here is of a very personal character. I do not believe that there is any single approach to the history of science which could not be replaced by very different methods of attack; only trivialities permit but one interpretation. I must confess still more: I cannot even pretend to be complete in the selection of topics essential for our understanding of ancient astronomy,1 nor do I wish to conceal the fact that many of the steps which I myself have taken were dictated by mere accident. To mention only one example: without having been brought into contact with a recently purchased collection of Demotic papyri in Copenhagen, I would never have undertaken the investigation of certain periods of Hellenistic and Egyptian astronomy which now seem to me to constitute a very essential link between ancient and medieval astronomy. In other words, though I have always tried to subordinate any particular research problem to a wider program of systematic analysis, the impossibility of elaborate long-range plan1 Also the bibliography, given at the end, is very incomplete and is only intended to inform the reader where he can find further details of the specific viewpoint discussed here and to list the original sources. ning has again and again been impressed upon me. The situation is comparable to entering a vast mountainous region on a single trail; one must simply follow the winding path, trying to give account of its general direction, but one can never predict with certainty what new vistas will be exposed at the next turn. 2. The enormous complexity of the study of ancient astronomy becomes evident if we try to make the first, and apparently simplest, step of classification: to distinguish between, say, Mesopotamian, Egyptian, and Greek astronomy, not to mention their direct successors, such as Hindu, Arabic, and medieval astronomy. Neither geographically nor chronologically nor according to language can clear distinctions be made. Entirely different conditions underlie the astronomy in Egypt of the Middle and New kingdoms than in the periods after the Persian conquest. Greek astronomy of Euclid's time has very little in common with Hipparchus' astronomy only a hundred and fifty years later. It is evident that it is of very little value to speak about a "Babylonian" astronomy regardless of period, riigin, and scope. And, worst of all, the concept "astronomy" itself undergoes changes in meaning when we speak about different periods. The fanciful combination of a group of brilliant stars to form the picture of a "bull's leg" and the computation of the irregularities in the moon's movement in order to predict accurately the magnitude of an eclipse are usually covered by the same name! For methodological reasons it is obvious that a drastic restriction in terminology must be made. We shall here call "astronomy" only those parts of human interest in celestial phenomena which are amenable to mathematical

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS treatment. Cosmogony, mythology, and applications to astrology must be distinguished as clearly separated problems -not in order to be disregarded but to make possible the study of the mutual influence of essentially different streams of development. On the other hand, it is necessary to co-ordinate intimately the study of ancient mathematics and astronomy because the progress of astronomy depends entirely on the mathematical tools available. This is in conformity with the concept of the ancients themselves: one need only refer to the original title of Ptolemy's "Almagest," namely, "Mathematical Composition." 3. The study of ancient astronomy will always have its center of gravity in the investigation of the Hellenistic-Roman period, represented by the names of Hipparchus and Ptolemy. From this center three main lines of research naturally emerge: the investigation of the previous achievements of the Near East; the investigation of pre-Arabic Hindu astronomy; and the study of the astronomy of late antiquity in its relation to Arabic and medieval astronomy. This last-mentioned extension of our program beyond antiquity proper is not only the natural continuation of the original problem but constitutes an integral part of the general approach outlined here. Astronomy is the only branch of the ancient sciences which survived almost intact after the collapse of the Roman Empire. Of course, the level of astronomical studies dropped within the boundaries of the remnants of the Roman Empire, but the tradition of astronomical theory and practice was never completely lost. On the contrary, the rather clumsy methods of Greek trigonometry were improved by Hindu and Arabic astronomers, new observations were constantly compared with Ptolemy's results, etc. This must be paralleled with 3 the total loss of understanding of the higher branches of Greek mathematics before one realizes that astronomy is the most direct link connecting the modern sciences with the ancient. In fact, the work of Copernicus, Brahe, and Kepler can be understood only by constant reference to ancient methods and concepts, whereas, for example, the meaning of the Greek theory of irrational magnitudes or Archimedes' integrations were understood only after being independently rediscovered in modern times. There are, of course, very good reasons for the fact that ancient astronomy extended with an unbroken tradition deep into modern times. The structure of our planetary system is such that it is simple enough to permit the achievement of relatively far-reaching results with relatively simple mathematical methods, but complicated enough to invite constant improvement of the theory. It was thus possible to continue successfully the "ancient" methods in astronomy at a time when Greek mathematics had long reached a dead end in the enormous complication of geometric representation of essentially algebraic problems. The creation of the modern methods of mathematics, on the other hand, is again most closely related to astronomy, which urgently required the development of more powerful new tools in order to exploit the vast possibilities which were opened by Newton's explanation of the movement of the celestial bodies by means of general principles of physics. The confidence of the great scientists of the modern era in the sufficiency of mathematics for the explanation of nature was largely based on the overwhelming successes of celestial mechanics. Essentially the same held for scholars in classical times. In antiquity, mathematical tools were not available to explain any

JOURNAL OF NEAR physical phenomena of higher complexity than the planetary movement. Astronomy thus became the only field of ancient science where indisputable certainty could be reached. This feeling of the superiority of mathematical astronomy is best expressed in the following sentences from the introduction to the Almagest: "While the two types of theory could better be called conjecture than certain knowledge -theology because of the total invisibility and remoteness of its object, physics because of the instability and uncertainty of matter- .... mathematics alone .... will offer reliable and certain knowledge because the proof follows the indisputable ways of arithmetic and geometry."2 II. EGYPT 4. A few words must be said about Egyptian mathematics before discussing the astronomical material. Our main source for Egyptian mathematics consists of two papyri3-certainly not too great an amount in view of the length of the period in question! Still, it seems to be a fair assumption that we are well enough informed about Egyptian mathematics. Not only are both papyri of very much the same type but all additional fragments which we possess match the same picture--a picture which is paralleled by economic documents in which occur precisely those problems and methods which we find in the mathematical papyri. The Egyptian mathematical texts, furthermore, find their direct continuation 2 Almagest I, 1 (ed. Heiberg I, 6, 11 if.). 3 Math. Pap. Rhind [Peet R MP; Chace RMP] and Moscow mathematical papyrus [Struve MPM]. For a discussion of Egyptian arithmetic see Neugebauer [1], for Egyptian geometry Neugebauer [2], and, in general, Neugebauer Vorl. The most recent attempt at a synthesis of Egyptian science, by Flinders Petrie (Wisdom of the Egyptians [London, 1940]), must unfortunately be considered as dilettantish not only because of its disregard of essential source material but also because of its lack of understanding for the mathematical and astronomical problems as such. EASTERN STUDIES in Greek papyri,4 which again show the same pattern. It is therefore safe to say that Egyptian mathematics never rose above a very primitive level. So far as astronomy is concerned, numerical methods are of primary importance, and, fortunately enough, this is the very part of Egyptian mathematics about which we are best informed. Egyptian arithmetic can be characterized as being predominantly of an "additive" character, that is, its main tendency is to reduce all operations to repeated additions. And, because the process of division is very poorly adaptable to such procedures, we can say that Egyptian mathematics does not provide the most essential tools for astronomical computation. It is therefore not surprising that none of our Egyptian astronomical documents requires anything more than simple operations with integers. Where the complexity of the phenomena exceeded the capacity of Egyptian mathematics, the strongest simplifications were adopted, consequently leading to little more than qualitative results. 5. The astronomical documents of purely Egyptian origin are the following: Astronomical representations and inscriptions on ceilings of the New Kingdom,5 supplemented by the so-called "diagonal calendars" on coffin lids of the Middle Kingdom6 and by the DemoticHieratic papyrus "Carlsberg 1."7 Secondly, the Demotic papyrus "Carlsberg 9," which shows the method of determining new moons.8 Though written in Roman 4 The continuation of this tradition is illustrated by the following texts: Demotic: Revillout [1]; Coptic: Crum CO, No. 480, and Sethe ZZ, p. 71; Greek: Robbins [1] or Baillet [11. For Greek computational methods in general, see Vogel [1]. 5 Examples: The Nut-pictures in the cenotaph of Seti I (Frankfort CSA) and Ramses IV (Brugsch Thes. 1) and analogous representations in the tombs of Ramses VI, VII, and IX. 6 Cf. Pogo [1] to [4]. 7 Lange-Neugebauer 8 Neugebauer-Volten

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS times (after A.D. 144), this text undoubtedly refers to much older periods and is uninfluenced by Hellenistic methods. A third group of documents, again written in Demotic, concerns the positions of the planets.9 In this case, however, it seems to be very doubtful whether these tables are of Egyptian origin rather than products of the Hellenistic culture; we therefore postpone a discussion to the section on Hellenistic astronomy.10The last group of texts is again inscribed on ceilings and has been frequently discussed because of their representation of the zodiac." There can be no doubt that these latter texts were deeply influenced by non-Egyptian concepts characteristic for the Hellenistic period. The same holds, of course, for the few Coptic astronomical documents we possess.'2 It is, finally, worth mentioning that not a single report of observations is preserved, in strong contrast to the abundance of observational records from Mesopotamia. It is hard to say whether this reflects a significant historical fact or merely 10Cf. below, p. 24. 9 Neugebauer [3]. 1 I know of the following representations of zodiacs: No. 1 (Ptolemy III and V, i.e., 247/181 B.c.): northwest of Esna, North temple of Khnum (PorterMoss TB VI, p. 118); Nos. 2 and 3 (Ptolemaic or Roman): El-Salamfini, Rock tombs (Porter-Moss TB V, p. 18); mentioned by L'Hote, LE, pp. 86-87. No. 4 (Ptolemaic-Roman; Tiberius): Akhmim, Two destroyed temples (Porter-Moss TB V, p. 20); mentioned by Pococke DE, I, pp. 77-78. No. 5 (Tiberius): Dendera, Temple of Hathor, Outer hypostyle (Porter-Moss TB VI, p. 49). No. 6 (Augustus-Trajan): Dendera, Temple of Hathor, East Osiris-chapel central room, ceiling, west half (Porter-Moss TB VI, p. 99). Nos. 7 and 8 (1st cent. A.D.): Athribis, Tomb (Porter-Moss TB V, p. 32). No. 9 (Titus and Commodus): Esna, Temple of Khnum (Porter-Moss TB VI, p. 116). No. 10 (Roman): Dealer in Cairo, publ. Daressy [1], pp. 126-27, and Boll, Sphaera, P1. VI. Five other representations of the zodiacal signs are known from coffins, all from Ptolemaic or Roman times. On the other hand, the original Egyptian constellations are still found on coffins of the Saitic or early Ptolemaic periods. 12 The only nonastrological Coptic documents known to me are the tables of shadow lengths published by U. Bouriant and Ventre-Bey [1].-P. Bouriant [1] did not recognize that the text published by him was a standard list of the planetary "houses" with no specific reference to Arabic astronomy. 5 that we are at the mercy of the accidents of excavation. Speaking of negative evidence, three instances must be mentioned which play a more or less prominent role in literature on the subject and have contributed much to a rather distorted picture of Egyptian astronomy. The first point consists in the idea that the earliest Egyptian calendar, based on the heliacal rising of Sothis, reveals the existence of astronomical activity in the fourth millennium B.C.It can be shown, however, that this theory is based on tacit assumptions which are very implausible in themselves and that the whole Egyptian calendar does not presuppose any systematic astronomy whatsoever.'3 The second remark concerns the hypothesis of early Babylonian influence on Egyptian astronomical concepts.14 This theory is based on a comparative method which assumes direct influence behind every parallelism or vague mythological analogy. Every concrete detail of Babylonian and Egyptian astronomy which I know contradicts this hypothesis. Nothing in the texts of the Middle and New Kingdom equals in level, general type, or detail the contemporaneous Mesopotamian texts. The main source of trouble is, as usual, the retrojection into earlier periods of a situation which undoubtedly prevailed during the latest phase of Egyptian history. This brings us to the third point to be mentioned here: the assumption of an original Egyptian astrology. First of all, there is no proof in general for the widely accepted assertion that astrology preceded astronomy. But especially in Egypt is there no trace of astrological ideas in the enormous mythological literature which we possess for all periods.15 13 Neugebauer 14 Sponsored [41, Winlock [1], Neugebauer [5]. especially by the "Pan-Babylonian" school. 15It is interesting to observe how deeply imbedded is the assumption that astrology must precede as-

JOURNAL OF NEAR The earliest horoscope from Egyptian soil, written in Demotic, refers to A.D. 13;16the earliest Greek horoscope from Egypt concerns the year 4 B.C.17We shall presently see that the assumption of a very late introduction of astrological ideas into Egypt corresponds to various other facts. 6. It is much easier to show that certain familiar ideas about the origin of astronomy are historically untenable than to give an adequate survey of our real knowledge of Egyptian astronomy. A. Pogo is to be credited with the recognition of the astronomical importance of inscriptions on the lids of a group of coffins from the end of the Middle Kingdom,18 apparently representing the setting and rising of constellations, though in an extremely schematic fashion. The constellations are known as the "decans" because of their correspondence to intervals of ten days. He furthermore saw the relationship between these simple pictures and the elaborate representations on the ceilings of the tombs belonging to kings of the New Kingdom.19 It can be safely assumed that the coffin lids are very abbreviated forms of contemporaneous representations on the ceilings of tombs and mortuary temples of the rulers of the Middle Kingdom. The logical place for these representations of the sky tronomy. Brugsch called his edition of cosmogonic and mythological texts "astronomische und astrologische Inschriften" in spite of the fact that these texts do not betray the slightest hint of astrology. 16 Neugebauer 17 Pap. Oxyrh. [6]. 804. From this time until A.D. 500 more than sixty individual horoscopes, fairly equally distributed in time, are known to me. 1s Cf. n. 6. 19Some of Pogo's assumptions must, however, be abandoned, because they are based on the distinction of different types of such coffin inscriptions. A close examination of these texts (and also unpublished material) shows that all preserved samples belong to the same type. A systematic edition of all these texts is urgently needed if we are to obtain a solid basis for the study of Egyptian constellations. EASTERN STUDIES on ceilings explains their destruction easily enough. The earliest preserved ceiling, discovered in the unfinished tomb of Senmut, the vezir of Queen Hatshepsut,20 is about three centuries later than the coffin lids. Then come the well-preserved ceiling in the subterranean cenotaph of Seti I21 and its close parallels in the tomb of Ramses IV22 and later rulers.23 The difficulties we have to face in an attempt to explain these texts can best be illustrated by a brief discussion of the abovementioned papyrus "Carlsberg 1." This papyrus was written more than a thousand years after the Seti text but was clearly intended to be a commentary to these inscriptions. In the papyrus we find the text from the cenotaph split into short sections, written in Hieratic, which are followed by a word-for-word translation into Demotic supplemented by comments in Demotic. The original text is frequently written in a cryptic form, to which the Demotic version gives the key. We now know, for instance, that various hieroglyphs were replaced by related forms in order to conceal the real contents from the uninitiated reader. How successfully this method worked is shown by the fact that one such sign, which is essential for the understanding of a long list of dates of risings and settings of the decans, was used at its face value for midnight instead of evening.24 It is needless to emphasize what the recognition of such substitutions means for the correct understanding of astronomical texts. A complete'revision of all previously published material is needed in the light of this new 20 Winlock [2], pp. 34 ff., reprinted in Winlock EDEB, pp. 138 ff., and Pogo [5]. The final publication has not yet appeared. 21 Frankfort CSA. 22Brugsch Thes. I opposite pp. 174-75, complete (cf. Lange-Neugebauer [1], p. 90). 23 Cf. n. 5. 24 Sethe, ZAA, p. 293, n. 1, and but in- Lange-Neugebauer

THE HISTORYOF ANCIENTASTRONOMY:PROBLEMSAND METHODS insight into the Egyptian scheme of describing the rising and setting of stars the year round. One point, however, must be kept in mind in every investigation of Egyptian constellations. One must not ascribe to these documents a degree of precision which they were never intended to possess. I doubt, for example, very much whether one has a right to assume that the decans are constellations covering exactly ten degrees of a great circle on the celestial sphere. I think it is much more plausible that they are constellations spread over a more or less vaguely determined belt around the sky, just as we speak about the Milky Way. It is therefore methodically wrong to use these star lists and the accompanying schematic date lists for accurate computations, as has frequently been attempted. The second Demotic astronomical document, papyrus Carlsberg 9, is much easier to understand and gives us full access to the Egyptian method of predicting the lunar phases with sufficient accuracy. The whole text is based on the fact that 25 Egyptian years cover the same time interval as 309 lunations. The 25 years equal 9125 days, which are periodically arranged into groups of lunar months of 29 and 30 days. The periodic repetition of this simple scheme corresponds, on the average, very well with the facts; more was apparently not required, and, we may add, more was not obtainable with the available simple mathematical means which are described at the beginning of this section. The purpose of the text was to locate the wandering lunar festivals within the schematic civil calendar, as is shown by a list of the "great" and "small" years of the cycle, which contain 13 or 12 lunar festivals, respectively.25 Accordingly, calen- 25The "great" and "small" years (already mentioned in an inscription of the Middle Kingdom) have given rise to much discussion (cf., e.g., Ginzel Chron., I, pp. 176-77) which can now be completely ignored. 7 daric problems are seen to be the activating forces here as well as in the decanal lists of the Middle and New Kingdom. The two Carlsberg papyri thus give us a very consistent picture of Egyptian stellar and lunar astronomy and its calendaric relations and are in best agreement with the level known from the mathematical papyri. Before leaving the description of Egyptian science, brief mention should be made of the much-discussed question of the "scientific" character of Egyptian mathematics and astronomy. First of all, the word "scientific" must be clearly defined. The usual identification of this question with that of the practical or theoretical purpose of our documents is obviously unsatisfactory. One cannot call medicine or physics unscientific even if they serve eminently practical purposes. It is neither possible nor relevant to discover the moral motives of a scientist-they might be altruistic or selfish, directed by the desire for systematization or by interest in competitive success. It is therefore clear that the concept "scientific" must be described as a question of methods, not of motives. In the case of mathematics and astronomy, the situation is especially simple. The criterion for scientific mathematics must be the existence of the concept of proof; in astronomy, the elimination of all arguments which are not exclusively based on observations or on mathematical consequences of an initial hypothesis as to the fundamental character of the movements involved. Egyptian mathematics nowhere reaches the level of argument which is worthy of the name of proof, and even the much more highly developed Babylonian mathematics hardly ever displays a general technique for proving its procedures.26 26See the discussion in Neugebauer Vorl., pp.

JOURNAL OF NEAR EASTERN STUDIES Egyptian astronomy was satisfied with a must underline the incompleteness of the very rough qualitative description of the present state of research, which is due to phenomena-here, too, we miss any trace the fact that we do not yet have reliable of scientific method. The first scientific at- and complete editions of the text material. tack of mathematical problems was made The observation reports addressed to the in the fifth century B.C. in Greece. We Assyrian kings were collected by R. C. shall see that scientific astronomy can be Thompson27and in the editions of Assyrfound shortly thereafter in Babylonian ian letters published and translated by texts of the Seleucid period. In other Harper,28 Waterman,29 and Pfeiffer;30 words, the enormous interest of the study much related material is quoted in the of pre-Hellenistic Oriental sciences lies in publications of Kugler,31 Weidner,32 and the fact that we are able to follow the de- others. But Thompson's edition gives the velopment far back into pre-scientific pe- original texts only in printed type, subriods which saw the slow preparation of ject to all the misunderstandings of this material and problems which deeply in- early period of Assyriology, and very litfluenced the shape of the real scientific tle has been done to repair these original methods which emerged to full power for errors. Nothing short of a systematic the first time in the Hellenistic culture. "corpus" of all the relevant texts can proIt is a serious mistake to try to invest vide us with the requisite security for Egyptian mathematical or astronomical systematic interpretation. The great coldocuments with the false glory of scien- lection of astrological texts, undertaken tific achievements or to assume a still un- by Virolleaud33but never finished, conknown science, secret or lost, not found in fronts the reader with still greater diffithe extant texts. culties, because Virolleaud composed complete versions from various fragments and III. MESOPOTAMIA duplicates without indicating the sources 7. Turning to Babylonian astronomy, from which the different parts came. And, one's first impression is that of an enor- finally, the tablets dealing with the movemous contrast to Egyptian astronomy. ment of the moon and the planets were This contrast not only holds in regard to discussed and explained in masterly fashthe large amount of material available ion by Kugler;34 but here, too, a systefrom Mesopotamia but also with respect matic edition of the whole material is to the level finally reached. Texts from necessary.35Years of systematic work will the last two or three centuries B.C. permit be needed before the foundations for a rethe computation of the lunar movement liable history of the development of Babyaccording to methods which certainly lonian astronomy are laid. 8. Kugler uncovered step by step the rank among the finest achievements of ancient science-comparable only to the ingenious methods by which the ephemerworks of Hipparchus and Ptolemy. 27 Thompson 29 Waterman RC. Rep. (1900). It is one of the most fascinating prob30 Pfeiffer SLA. Harper Letters. lems in the history of ancient astronomy 31 Kugler SSB and Kugler MP. to follow the different phases of this de32 Weidner Hdb., Weidner [1], [21, and numerous velopment which profoundly influenced articles in the pre-war volumes of Babyloniaca. 33 Virolleaud A Ch. all further events. Before giving a short 34 Kugler BMR and SSB. sketch of this progress as we now restore 35Such an edition by the present author is in prepit according to our present knowledge, we aration; it is quoted in the following as ACT.

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS ids of the moon and the planets which we find inscribed on tablets ranging from 205 B.C. to 30 B.C. were computed.36It can justly be said that his discoveries rank among the most important contributions toward an understanding of ancient civilization. It is very much to be regretted that historians of science often quote Kugler but rarely read him;37 by doing this, they have disregarded the newly gained insight into the origin of the basic methods in exact science. This is not the place to describe in detail the Babylonian "celestial mechanics," as it might properly be called; that will be one of the tasks of a history of ancient astronomy which remains to be written. A few words, however, must be said in order to render intelligible the relationship between Babylonian and Greek methods. The problem faced by ancient astronomers consisted in predicting the positions of the moon and the planets for an extended period of time and with an accuracy higher than that obtainable by isolated individual observations, which were affected by the gross errors of the instruments used. All these phenomena are of a periodic character, to be sure, but are subject to very complicated fluctuations. All that we know now seems to point to the following reconstruction of the history of late Babylonian as- 9 tronomy. A systematic observational activity during the Late Assyrian and Persian periods (roughly, from 700 B.C. onward) led to two different results. First, the collected observations provided the astronomers with fairly accurate average values for the main periods of the phenomena in question; once such averages were obtained, improvements could be furnished by scattered observational records from preceding centuries. Secondly, from individual observations, for example, of the moment of full moon38 or of heliacal settings, etc., short-range predictions could be made by methods which we would call linear extrapolation. Such methods are frequently sufficient to exclude certain phenomena (such as eclipses) in the near future and, under favorable conditions, even to predict the date of the next phenomenon in question. After such methods had been developed to a certain height, apparently one ingenious man conceived a new idea which rapidly led to a systematic method of long-range prediction. This idea is familiar to every modern scientist; it consists in considering a complicated periodic phenomenon as the result of a number of periodic effects, each of a character which is simpler than the actual phenomenon.39The whole method probably originated in the theory of the moon, where we find it at its highest perfection. The moments of new moons could easily be found if the sun and moon would each move with constant velocity. Let us assume this to be the case and use average values for this ideal movement; this gives us average positions for the new moons. The actual movement deviates from this average but oscillates around it periodically. These deviations were now treated 36The first tentative (but very successful) steps were made by Epping AB (1889). Then follow Kugler's monumental works BMR (1900) and SSB (published between 1907 and 1924), supplemented by Schaumberger's explanation of the determination of first and last visibility of the moon (1935) and continued by the present author with respect to the theory of latitude and eclipses (Neugebauer [8], [9], Pannekoek [2] and van der Waerden [1]). The theory of planets is treated in Kugler SSB, to be supplemented by Pannekoek [1], Schnabel 12],and van der Waerden [2]. All previously published texts and much unpublished material will be contained in Neugebauer ACT. The whole material amounts to about a hundred ephemerids for the moon and the planets, covering the above38 Frequently mentioned in the mentioned two centuries. "reports" to the 37Abel Rey, La Science orientale avant les grecs Assyrian court (e.g., Thompson Rep.). (Paris, 1930), and E. Zinner, Geschichteder Stern39A classic example is the treatment of sounds as kunde (Berlin, 1931), are brilliant examples showing the result of the superposition of pure harmonic vicomplete ignorance of Kugler's results. brations.

JOURNAL OF NEAR EASTERN STUDIES as new periodic phenomena and, for the tion between these fixed points.41It must sake of easier mathematical treatment, be said, however, that the planetary thewere considered as linearly increasing and ory was not developed to the same degree decreasing. Additional deviations are of refinement as the lunar theory; the caused by the inclination of the orbits. reason might very well be that the lunar But here again a separate treatment, theory was of great practical importance based on the same method, is possible. for the question of the Babylonian calenThus, starting with average positions, the dar: whether a month would have 30 or 29 corrections required by the periodic devia- days. For the planets no similar reason for tions are applied and lead to a very close high accuracy seems to have existed, and description of the actual facts. In other it was apparently sufficient merely to words, we have here, in the nucleus, the compute the approximate dates of pheidea of "perturbations," which is so funda- nomena, which, in addition, are frequentmental to all phases of the development of ly very difficult to observe accurately. We cannot emphasize too strongly that celestial mechanics, whence it spread into the essential point in the above-described of exact science. branch every We do not know when and by whom methods lies not in the comparatively this idea was first employed. The consist- high accuracy of the results obtained but ency and uniformity of its application in in their fundamentally new attitude tothe older of the two known "systems" of ward the whole problem. Let us, as a lunar texts point clearly to an invention typical example, consider the movement by a single person. From the dates of the of the sun.42Certain simplb observations, preserved texts, one might assume a date most likely of the unequal length of the in the fourth or third century B.c.40This seasons, had led to the discovery that the basic idea was applied not only to the the- sun does not move with constant velocity ory of the moon (in two slightly modified in its orbit. The naive method of taking this forms) but also to the theory of the plan- fact into account would be to compute the ets. In this latter theory the main point position of the sun by assuming a regularconsists in refraining from an attempt to ly varying velocity. It turned out, howdescribe directly the very irregular move- ever, that considerable mathematical diffiment, substituting instead the separate culties were met in computing the syzytreatment of several individual phenom- gies of the moon according to such an asena, such as opposition, heliacal rising, sumption. Consequently, another velocity etc.; each of these phenomena is treated distribution was substituted, and it was with the methods familiar from the lunar found that the following "model" was theory as if it were the periodic movement satisfactory: the sun moves with two difof an independent celestial body. After ferent velocities over two unequal arcs of dates and positions of each characteristic the ecliptic, where velocities and arcs phenomenon are determined, the inter- were determined in such a fashion that the mediate positions are found by interpola- initial empirical facts were correctly explained and at the same time the compu40 The attempts to determine a more precise date tation of the conjunctions became suffi(Schnabel Ber., pp. 219 ff., and Schnabel [1], pp, 15 ff.) are based on unsatisfactory methods. The generally that Naburimannu was the accepted statement founder of the older system of the lunar theory relies on nothing more than the occurrence of this name in one of the latest tablets in a context which is not perfectly clear. 41 This is shown by a tablet for Mercury, to be published in Neugebauer ACT. The interpolation is not simply linear but of a more complicated type known from analogous cases in the lunar theory. 42 For details see Neugebauer [10] and [9] ? 2.

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS I think it can be justly said that we ciently simple. It is self-evident that the man who devised this method did not have a fairly good knowledge of the charthink that the sun moved for about half a acter of mathematical problems and methyear with constant velocity and then, hav- ods in the Old Babylonian period (ca. 1700 ing reached a certain point in the ecliptic B.C.). Almost a hundred tablets from this suddenly started to move with another, period are published;44they contain colmuch higher velocity for the rest of the lections of problems or problems with to far year. His problem was clearly this: to complete solutions-amounting make a very complicated problem ac- beyond a thousand problems. We know cessible to mathematical treatment with practically nothing about the Sumerian the only condition that the final conse- mathematics of the previous periods and quences of the computations correctly cor- very little of the interval between the Old Babylonian period and Seleucid times. respond to the actual observations-in our example, the inequality of the seasons. We have but few problem texts from the The Greeks43 called this a method "to latter period, but they give us some idea preserve the phenomena"; it is the method of the type of mathematics familiar to the of introducing mathematically useful astronomers of this age. This material is steps which in themselves need not be of sufficient to assure us that all the essential any physical significance. For the first achievements of Old Babylonian times time in history, mathematics became the were still in the possession of the latest leading principle for the structure of phys- representatives of Mesopotamian science. In other words, Babylonian mathematical ical theories. 9. It will be clear from this discussion astronomy was built on foundations indethat the level reached by Babylonian pendently laid more than a millennium mathematics was decisive for the develop- before. ment of such methods. The determination If one wishes to characterize Babyloof characteristic constants (e.g., period, nian mathematics by one term, one could amplitude, and phase in periodic motions) call it "algebra." Even where the foundanot only requires highly developed meth- tion is apparently geometric, the essence ods of computation but inevitably leads is strongly algebraic, as can be seen from to the problem of solving systems of equa- the fact that frequently operations occur tions corresponding to the outside condi- which do not admit of a geometric intertions imposed upon the problem by the pretation, as addition of areas and lengths, observational data. In other words, with- or multiplication of areas. The predomiout a good stock of mathematical tools, nant problem consists in the determinadevices of the type which we find every- tion of unknown quantities subject to where in the Babylonian lunar and plane- given conditions. Thus we find prepared tary theory could not be designed. Egyp- precisely the tools which were later to tian mathematics would have rendered become of the greatest importance for ashopeless any attempt to solve problems of tronomy. Of course, the term "algebra" does not the type needed constantly in Babylonian therefore It for our is essential astronomy. completely cover Babylonian mathematto a brief sketch of give topic Babylonian 44 These texts were published in Neugebauer MK T mathematics. (1935-38) and in Neugebauer-Sachs MCT (1945). A 43 E.g., Manitius, Proclus, 140, 21). Hypotyposis astron. pos. v. 10 (ed. large part of the MKT material was republished in Thureau-Dangin TMB (1939). For a general survey see Neugebauer Vorl.

JOURNAL OF NEAR EASTERN STUDIES ics. Not only were a certain number of geometrical relations well known but, more important for our problem, the basic properties of elementary, sequences (e.g., arithmetic and geometric progressions) were developed.45The numerical calculations are carried out everywhere with the greatest facility and skill. We possess a great number of texts from all periods which contain lists of reciprocals, square and cubic roots, multiplication tables, etc., but these tables rarely go beyond two sexagesimal places (i.e., beyond 3600). A reverse influence of astronomy on mathematics can be seen in the fact that tables needed for especially extensive numerical computations come from the Seleucid period; tables of reciprocals are preserved with seven places (corresponding to eleven decimal places) for the entry and up to seventeen places (corresponding to twenty-nine decimal places) for the result. It is clear that numerical computations of such dimensions are needed only in astronomical problems. The superiority of Babylonian numerical methods has left traces still visible in modern times. The division of the circle into 360 degrees and the division of the hour into 60 minutes and 3600 seconds reflect the unbroken use of the sexagesimal system in their computations by medieval and ancient astronomers. But though the base 60 is the most conspicuous feature of the Babylonian number system, this was by no means essential for its success. The great number of divisors of 60 is certainly very useful in practice, but the real advantage of its use in the mathematical and astronomical texts lies in the place-value 45Incidentally, we also have an example (Neugebauer-Sachs MCT, Problem-Text A) of purely number theoretical type from Old Babylonian times (so-called "Pythagorean numbers"); but it should be added that we do not find the slightest trace of number mysticism anywhere in these texts. notation,46which is consistently employed in all scientific computations. This gave the Babylonian number system the same advantage over all other ancient systems as our modern place-value notation holds over the Roman numerals. The importance of this invention can well be compared with that of the alphabet. Just as the alphabet. eliminates the concept of writing as an art to be acquired only after long years of training, so a place-value notation eliminates mere computation as a complex art in itself. A comparison with Egypt or with the Middle Ages illustrates this very clearly. Operation with fractions, for example, constituted a problem in itself for medieval computers; in place-value notation, no such problem exists,47thus eliminating one of the most serious obstacles for the further development of mathematical technique. The analogy between alphabet and place-value notation can be carried still further. Neither one was the sudden invention made by a single person but the final outcome of various historical processes. We are able to trace Mesopotamian number-writing far back into the earliest stages of civilization, thanks to the enormous amount of economic documents preserved from all periods. It can be shown how a notation analogous to the Egyptian or Roman system was gradually replaced by a notation which developed naturally in the monetary system and which tended toward a place-value notation. The value 60 of the base appears to be the outcome of the arrangement of the monetary 46 Place-value notation consists in the use of a very limited number of symbols whose magnitude is determined by position. Thus 51 does not mean 5 plus 1 (as it would with Roman or Egyptian numerals), but 5 times 10 plus 1. Analogously in the sexagesimal system, five followed by one (we transcribe 5,1) means 5 times 60 plus 1 (i.e., 301). 47 Example: to add or to multiply 1.5 and 1.2 requires exactly the same operations as the addition or multiplication of 15 and 12.

HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS 13 units.48Outside of mathematical texts, the "gnomon"51and the measurement of the place-value notation was always over- length of the day by water clocks.52The lapped by various other notations, and latter problem has caused considerable toward the end of Mesopotamian civiliza- trouble in the literature on the subject betion a modified system became predomi- cause the texts show the ratio 2:1 for the nant. It seems very possible, however, extremal values during the year. A ratio that the idea of place-value writing was 2:1 between the longest and the shortest never completely lost and found its way day, instead of the ratio 3:2, which is through astronomical tradition into early otherwise used,53 would correspond to a Hindu astronomy.49 whence our present geographical latitude absolutely impossinumber system originated during the first ble for Babylon. The discrepancy disaphalf of the first millennium A.D. pears, however, if one recalls the fact that 10. We now turn to the periods pre- the amount of water flowing from a cylinceding the final stage of Babylonian as- drical vessel is not proportional to the tronomy which culminated in the mathe- time elapsed but decreases with the sinkmatical theory of the moon and the plan- ing level.54It is worth mentioning in this ets described above. It is not possible to connection that the outflow of water from give an outline of this earlier development a water clock is already discussed in Old because most of the preliminary work re- Babylonian mathematical texts.55 This mains to be done. A few special problems, whole group of texts, however, leads to however, which must eventually find their nothing more than very approximate replace in a more complete picture, can now sults. This is seen from the fact that the be mentioned. year is assumed, for the sake of simplicity, In our discussion of the methods used to be 360 days long and divided into 12 in the lunar and planetary theories, we months of 30 days each.56This schematic had occasion to mention the extensive use treatment has its parallel in the schemes of periodically increasing and decreasing which we have met in Egyptian astronsequences of numbers. A simple case of this method appears in earlier times in the problem of describing numerically the changing length of day and night during the year. The crudest form is the assumption of linear variation between two extremal values.50 Two much more refined schemes are incorporated in the texts of the latest period, but it seems very likely that they are of earlier origin. Closely related are two other problems: the variability of the length of the shadow of the 48 For details see Neugebauer [11] and Neugebauer Vorl., chap. iii ? 4. The theory set forth by ThureauDangin SS (English version Thureau-Dangin [1]) does not account for the place-value notation, which is the most essential feature of the whole system. 49Cf. Datta-Singh pp. 266 if. 5o E.g., Weissbach HHM I and Neugebauer BM, pp. 50-51. [12], omy and which we shall find again in early Greek astronomy; we must once more emphasize that elements from such schemes cannot be used for modern calculations, since this would assume quantitative accuracy where only qualitative results had been intended. The calendaric interest of these problems is obvious. The same is true of the 51 Weidner [1], pp. 198 if. 96. 52Weissbach BM, pp. 50-51; Weidner [1], pp. 19553 Schaumberger 54 Neugebauer Erg., p. 377. [19]. 55Thureau-Dangin [2] and Neugebauer MKT, I, pp. 173 ff. 56 This schematic year of 360 days, of course, does not indicate that one assumed 360 days as the correct length of the solar year. A lunar calendar makes correct predictions of a future date very difficult. The schematic calendar is in practice therefore very convenient for giving future dates which must, at any rate, be adjusted later.

JOURNAL OF NEAR EASTERN STUDIES oldest preserved astronomical documents from Mesopotamia, the so-called "astrolabes."57 These astrolabes are clay tablets inscribed with a figure of three concentric circles, divided into twelve sections by twelve radii. In each of the thirty-six fields thus obtained we find the name of a constellation and simple numbers whose significance is not yet clear. But it seems evident that the whole text constitutes some kind of schematic celestial map which represents three regions on the sky, each divided into twelve parts, and attributing characteristic numbers to each constellation. These numbers increase and decrease in arithmetic progression and are undoubtedly connected with the corresponding month ot the schematic twelve-month calendar. It is clear that we have here some kind of simple astronomical calendar parallel (not in detail, but in purpose) to the "diagonal calendars" in Egypt. In both cases these calendars are of great interest to us as a source for determining the relative positions and the earliest names of various constellations. But here, too, the strongest simplifications are adopted in order to obtain symmetric arrangements, and much remains to be done before we can answer such questions as the origin of the "zodiac." 11. Few statements are more deeply rooted in the public mind or more often repeated than the assertion that the origin of astronomy is to be found in astrology. Not only is historical evidence lacking for this statement but all well-documented facts are in sharp contradiction to it. All the above-mentioned facts from Egypt and Babylonia (and, as we shall presently see, also from Greece) show that calendaric problems directed the first steps of 57This name is rather misleading and is merely due to the circular arrangement. Schott [1], p. 311, introduced the more appropriate name "twelve-timesthree." Such texts are published in CT 33, Pls. 11 and 12. Cf. also Weidner Hdb., pp. 62 if. and Schott [1]. astronomy. Determination of the season, measurement of time, lunar festivalsthese are the problems which shaped astronomical development for many centuries; and we have seen that even the last phase of Mesopotamian astronomy, characterized by the mathematical ephemerids, was mainly devoted to problems of the lunar calendar. It is therefore one of the most difficult problems in the history of ancient astronomy to uncover the real roots of astrology and to establish their relation to astronomy. Very little has been done in this direction, mainly because of the prejudice in favor of accepting without question the priority of astrology. Before going into this problem in greater detail, we must clarify our terminology. The modern reader usually thinks in terms of that concept of astrology which consists in the prediction of the fate of a person determined by the constellation of the planets, the sun, and the moon at the moment of his birth. It is well known, however, that this form of astrology is comparatively late and was preceded by another form of much more general character (frequently called "judicial" astrology in contrast to the "genethlialogical" or "horoscopic" astrology just described). In judicial astrology, celestial phenomena are used to predict the imminent future of the country or its government, particularly the king. From halos of the moon, the approach or invisibility of planets, eclipses, etc., conclusions are drawn as to the invasion of an enemy from the east or west, the condition of the coming harvest, floods and storms, etc.; but we never find anything like the "horoscope" based on the constellation at the moment of birth of an individual. In other words, Mesopotamian "astrology" can be much better compared with weather prediction from phenomena observed in the skies than with astrology in the modern sense of the

HISTORY OF ANCIENT ASTRONOMY: word. Historically, astrology in Mesopotamia is merely one form of predicting future events; as such, it belongs to the enormous field of omen literature which is so familiar to every student of Babylonian civilization.58 Indeed, it can hardly be doubted that astrology emerged from the general practice of prognosticating through omens, which was based on the concept that irregularities in nature of any type (e.g., in the appearance of newborn animals or in the structure of the liver or other internal parts of a sheep) are indicative of other disturbances to come. Once the idea of fundamental parallelism between various phenomena in nature and human life is accepted, its use and development can be understood as consistent; established relations between observed irregularities and following events, constantly amplified by new experiences, thus lead to some sort of empirical science, which seems strange to us but was by no means illogical and bare of good sense to the minds of people who had no insight into the physical laws which determined the observed facts. Though the preceding remarks certainly describe the general situation adequately, the historical details are very much in the dark. One of the main difficulties lies in the character of our sources. We have at our disposal large parts of collections of astrological omens arranged in great "series" comprising hundreds of tablets. But the preserved canonical series come mainly from comparatively late collections (of the Assyrian period) and were thus undoubtedly subject to countless modifications. We must, moreover, probably assume that the collection of astrological omina goes back to the Cassite period (before 1200 B.C.)-a period about which our 58A comprehensive study of the development of the astrological omina literature by E. F. Weidner is in course of publication (Weidner [2]). PROBLEMS AND METHODS 15 general information is pretty flimsy. From the Old Babylonian period only one isolated text is preserved59which contains omina familiar from the later astrology. Predictions derived from observations of Venus made during the reign of Ammisaduqa (ca. 1600 B.C.) are preserved only in copies written almost a thousand years later60 and clearly subjected to several changes during this long time. We are thus again left in the dark as to the actual date of the composition of these documents except for the fact that it seems fairly safe to say that no astrological ideas appear before the end of the Old Babylonian period. Needless to say, there are no astrological documents of Sumerian origin. The period of the ever increasing importance of astrology (always, of course, of the above-mentioned type of "judicial" astrology) is that beginning with the Late Assyrian empire. The "reports" mentioned previously, preserved in the archives of the Assyrian kings, are our witnesses. But here, again, a completely unsolved problem must be mentioned: we do not know how the "horoscopic" astrology of the Hellenistic period originated from the totally different omen type of astrology of the preceding millennium. It is, indeed, an entirely unexpected turn to make the constellation of the planets at a single moment responsible for the whole future of an individual, instead of observing the ever shifting phenomena on the sky and thus establishing short-term consequences for the country in general (even if represented in the person of the king). It seems to me by no means self-evident that this radical shift of the character of astrology actually originated in Babylonia. We shall see in the next section that the horoscopic practice flourished especially in Egypt. It might therefore very well be that the new tendency originated in Hellenistic times 59Sileiko [1]. 60 Langdon

JOURNAL OF NEAR EASTERN STUDIES outside Mesopotamia and was reintroduced there in its modified form. It might be significant that only seven horoscopes are preserved from Mesopotamia, all of which were written in the Seleucid period,61a ridiculously small number as compared with the enormous amount of textual material dealing with the older "judicial" astrology. It must be admitted, however, that the oldest horoscopes known are of Babylonian origin. On the other hand, at no specific place can all the elements be found which are characteristic for astrology from Hellenistic times onward. Neither Babylonian astrology nor Egyptian cosmology furnishes the base for the fundamental assumption of horoscopic astrology, namely, that the position of the planets in the zodiac decides the future. And, finally, it must be emphasized that the problem of determining the date and place of origin of horoscopic astrology is intimately related to the problem of the date and origin of mathematical astrondmy. Horoscopes could not be cast before the existence of methods to determine the position of the celestial bodies for a period of at least a few decades. Even complete lists of observations would not be satisfactory because the positions of the planets in the zodiac are required regardless of their visibility at the specific hour. This shows how closely interwoven are the history of astrology and the history of planetary theories. IV. THE HELLENISTIC PERIOD 12. Before beginning the discussion of the Hellenistic period, we must briefly describe the preceding development in 61 Two are published by Kugler SSB, II, 554 ff., and refer to the years 258 and 142 B.C., respectively. One (probably 233 B.C.) is published in Thompson AB 251. Among four unpublished horoscopes, discovered by Dr. A. Sachs, two are very small fragments, one can be dated 235 B.C., and the last was cast for the year 263 B.c.; the last is the oldest horoscope in the world. Greece. Our direct sources of information about astronomy and mathematics before Alexander are extremely meagre. The dominating influence of Euclid's Elements succeeded in destroying almost all references to pre-Euclidean writings, and essentially the same effect was produced by Ptolemy's works. Original documents are, of course, not preserved-one must not forget that even our oldest manuscripts of Greek mathematical and astronomical literature were written many centuries after the originals.62It is therefore not surprising that our present-day knowledge of early Greek science is much more incomplete and subject to conjecture than the history of Mesopotamian or even Egyptian achievements where original documents are at our disposal. One point, however, can be established beyond any doubt: early Greek astronomy shows very strong parallelism with the early phases of Egyptian and Babylonian astronomy, with respect to scope as well as primitiveness. The astronomical writings of Autolycus63 and Euclid64struggle in a very crude way with the problem of the rising and setting of stars, making very strong simplifications which were forced upon them by the lack of adequate methods in spherical geometry. The final goal is again to establish relations between the celestial phenomena and the seasons of the years; the problem is thus of essentially calendaric interest. In addition to these simple treatises, however, we do find one work of outstanding character: the planetary theory of Eudoxos, Plato's famous contemporary. He made an attempt to explain the peculiarities of a planetary movement known as retrogra62 The oldest preserved manuscript of Euclid's Elements was written about twelve hundred years after Euclid (cf., e.g., Heath Euclid, I, p. 47). 63 Autolycus, ed. Hultsch (Leipzig, 1885). 64 Euclidis opera omnia, Vol. VIII, ed. Menge (Leipzig, 1916).

HISTORY OF ANCIENT ASTRONOMY: dation by the assumption of the superposition of the rotation of two concentric spheres around inclined axes and in opposite directions. In this way he reached a satisfactory explanation of the general type of planetary movement and thereby inaugurated a new period in the history of astronomy which was marked by attempts to explain the movements of the planetary system by mechanical models. It contains the nucleus for all planetary theories of the following two thousand years, namely, the assumption that irregularities in the apparent orbits can be explained as the result of superposed circular movements. It is only since Galileo and Newton that we know that the circular orbits do not play an exceptional role and that the great successes of the Greek theory were merely due to the accidental distribution of masses in our planetary system. It is, nevertheless, of great historical interest to see how a plausible initial hypothesis can for many centuries determine the line of attack on a problem, simultaneously barring all other possibilities. Such possibilities were actually contained in the approach developed by the Babylonian astronomers in the idea of superposing linear or quadratic periodic functions. These arithmetical methods were, however, almost completely abandoned by the Greek astronomers (at least so far as we know) and survived only in the treatment of certain smaller problems. One of these smaller problems is again related to calendaric questions but also to a basic problem of mathematical geography: the determination of the geographical latitude by means of the ratio of the longest to the shortest day. We have already mentioned the Babylonian methods of describing the change in the length of the days by means of simple sequences. These "linear" methods reappear in Greek literature and can be followed far PROBLEMS AND METHODS 17 into the early Middle Ages65in spite of the invention of much more accurate methods.66The term "linear" does not refer so much to the fact that the sequences in question form arithmetic progressions of the first order but is intended to emphasize the contrast with the "trigonometric" method applied to the same problem and explained in the first book of the Almagest. Here the exact solution of the problem by the use of spherical trigonometry is given. In contrast thereto, the linear methods yield only approximate results, but with an accuracy which was certainly sufficient in practice, especially when one takes into account the inaccuracy of the ancient instruments used in measuring time. Historically, however, the main interest lies much less in the perfection of the results than in the method employed and in its influence on the further development. A close investigation of early Greek astronomy and mathematics67 reveals an interesting fact. The determination of the time for the rising and setting of given arcs of the ecliptic, which lies at the heart of the question of the changing length of day and night, appears to be the most decisive problem in the development of spherical geometry. It is typical for the whole situation that a Greek "mathematical" work, the Sphaerics of Theodosius (ca. 200 B.C.), does not contain a single astronomical remark. The structure and contents of the main theorems, however, are determined by the astronomical problem in question; the methods applied constitute a very interesting link between the Babylonian linear methods and the final trigonometrical methods. Trigonometry undoubtedly has a very 65Neugebauer [13] and [18]. 66 Almagest 7 and II, 8. Cf. also Tetrabiblos I, 20 (ed. Robbins, p. 94), 21 (ed. Boll-Boer, pp. 46, 47 if.). 67 This investigation has been carried out by Olaf Schmidt (doctoral thesis, Brown Univ., 1943 [unpublished]).

JOURNAL OF NEAR EASTERN STUDIES long history. We find the basic relations necessary in order to appreciate the conbetween the chord and diameter of a circle tributions made by the Hindu-Arabic asalready in use in Old Babylonian texts tronomers which eventually led to the which employ the so-called "Thales" and modern form of spherical trigonometry. 13. It is of great interest to see that the "Pythagorean" theorems.68In sharp contrast to the Greek models for the move- very same problem-the determination of ment of the celestial bodies, which oper- rising times-leads to still other methods ate with circles and therefore necessarily which are now. known partly as "nomogrequire trigonometrical functions, we find raphy," partly as "descriptive geometry." no applications of trigonometry in the We have a small treatise, written by cuneiform astronomical texts of the Seleu- Ptolemy, called the Analemma.71He first cid period which are exclusively based on introduces in a very systematic way three arithmetical methods described above. different sets of spherical coordinates, each So far as we know, spherical trigonome- of which determines the position of a try appears for the first time in the Sphae- point on the celestial sphere. Then these ric of Menelaos69 (ca. A.D. 100). The as- coordinates are projected on different tronomical background of this work is planes, and these planes are turned into much more outspoken than in Theodosius, the plane of construction, just as we do but here, too, much is left to the reader, today in descriptive geometry. Finally, who must be familiar with the methods of certain scales are used to find graphically ancient astronomy to understand all the the relations between different coordiastronomical implications. The modern nates, again following principles which we scholar faces an additional difficulty, now use in nomography. The Arabs used namely, the modification of the Greek and developed these methods in connectext by the Arabic editors. The Greek tion with the construction of sundials.72 original is lost, and what we possess is only Another method of projection, today the Arabic version made almost a thou- called "stereographic," is given in Ptolesand years later. In this interval falls the my's Planisphaerium. The theory of pergradual transformation of Greek trigo- spective drawing in the Renaissance is dinometry, operating with chords, to the rectly connected with this work.73 modern treatment, which uses the sine The practical importance of the deterfunction. It is well known that this change mination of the rising times or the length goes back to Hindu astronomy, where the of the days is not restricted to the theory chords subtended by an angle were re- of sundials. The length of the longest day placed by the length of the half-chord of increases with the geographical latitude, the half-angle,70 i.e., our "sin a." It is, thus giving us the means to determine the however, a much more involved question latitude of a place from the ratio of the to separate these new methods from those 71 Ptolemy, Opera II, pp. 187-223. No complete used originally by Menelaos; this ques- translation of this badly preserved text has yet been tion must be answered if we wish to un- published, but an excellent commentary has been given by Luckey [1]. These methods, descripderstand the development of ancient tive geometry, are of an older date, as isusing evident from are already mentioned by Vitruvius spherical astronomy. This, in turn, is the fact thatofthey our era). Cf. Neugebauer [14] and Luck(beginning 68 Cf. Neugebauer-Struve [1], pp. 90-91; Neugebauer MKT, I, p. 180; and Neugebauer-Sachs MCT, Problem-Text A. 69Krause Men. 70Cf., e.g., Braunmiihl GT, chap. 3. ey [2]. 72 Cf., e.g., Garbers ES and Luckey [2]. 73 Ptolemy, Opera II, pp. 225-59, translated in Drecker [1]; cf. also Loria in M. Cantor, Geschichte der Mathematik, IV, p. 582.

HISTORY OF ANCIENT ASTRONOMY: shortest to the longest day. The ratio 3 : 2 accepted by Babylonian astronomers for the ratio of the longest to shortest daylight led the Greek geographers to determine erroneously the latitude of Babylon as 35? (instead of 321?). This error seriously affected the shape of the eastern part of the ancient map of the world.74 The precise relationship can only be established by using spherical trigonometry, but here, too, the "linear" methods were applied to various values of the basic ratio in order to give the law for the changing length of the days for the corresponding latitude. It must be remarked, however, that at this stage of affairs the concept "latitude" does not yet actually appear, but the ratio of the longest to the shortest day itself was used to characterize the location of a place. Zones of the same ratio were considered as belonging to the same "clima," a concept which plays a great role in ancient and medieval geography. The difference in character and behavior of nations living in different climates furnished one of the main arguments for the influence of astronomical phenomena on human life.75 The second geographical coordinatethe longitude-caused more trouble. The difference in longitude between two places on the earth is essentially equivalent to the difference in local time. But there existed no clocks or signals to compare the local time at far-distant places. Only one phenomenon could be used as a time signal, namely, records of simultaneous observations of a lunar eclipse from two different places. If each observer took note of the local time at which he observed the beginning and end of a lunar eclipse, a 74For the determination of the size of the earth by Eratosthenes Marinus of Tyre (about 250 B.c.), (about A.D. 100), and Ptolemy (about A.D. 150), see Mzik EGM, pp. 96 ff., and, in general, Heidel GM, chap. xi. Cf. also Honigmann SK and Neugebauer [13]. 75 E.g., Tetrabiblos II, 2. PROBLEMS AND METHODS 19 comparison of these records would then furnish the needed information. Hipparchus proposed the use of this method for an exact construction of the map of the world, but his program was never carried out. Only one pair of simultaneous observations seems to have been made, the eclipse of 331 B.C., September 20, recorded three hours earlier in Carthage than at Arbela.76Actually the difference in local time between these two localities is much smaller, and consequently the ancient map of the world suffers from a serious distortion in the direction from east to west. Here we see one of the most essential differences between ancient and modern science at work. Ancient science suffered most severely from the lack of scientific organization which is so familiar in our own times. In antiquity, generations passed before a new scientific idea found a follower able to use and develop methods handed down from a predecessor. The splendid isolation of the great scholars of antiquity can only be paralleled with the first beginnings of the new development in the European Renaissance. It seems to me beyond any doubt that even centers like Alexandria or Pergamon during their height would appear very poorly equipped if compared with a modern university of moderate size. And these centers themselves were few and practically isolated at any particular time; and at all times they were dependent upon the mood of some autocratic ruler. No wonder that the great achievements of antiquity are either the result of priestly castes of sufficiently stable tradition or of a few ingenious men who expended tremendous energyin restoring and enlarging the structure of a science known to them from the written legacy of their predecessors. One must not think 76 Ptolemy Geographia i. 4. 2 (ed. Nobbe, Cf. also 1Mik-Hopfner PDE, p. 21, n. 3. For Hipparchus' program see Strabo Geography i. C. 7; also Berger GFH, pp. 12 ff.

JOURNAL OF NEAR EASTERN STUDIES that mathematics and astronomy, like the contains a very interesting theory of map popular philosophical systems or the art projection, whereas the remaining twelve of rhetoric, were taught in the same man- chapters constitute an enormous cataner from generation to generation. Three logue of localities from all over the then centuries separate Hipparchus from Ptol- known world and the corresponding emy, one Eudoxos from Euclid, Euclid values of longitude and latitude to be from Archimedes and Apollonius. To be plotted into the network which was to be sure, the literary tradition was never in- constructed according to the method exterrupted between these outstanding men, plained in the first chapter. This, again, but most of the intermediate literature at was not geography for the entertainment best merely preserved and commented. of the general reader. To satisfy popular This explains not only why ingenious tastes, there was another literature, repreideas were frequently lost (e.g., Archi- sented by works like Strabo's Geography.80 medes' methods of integration) but also These more pleasant writings furnished why it was so easy to destroy ancient sci- serious competition to the strictly scienence almost completely in a very short tific literature and determined to a large time. Astronomy alone had a slight ad- extent the character of the field in late vantage because of its practical usefulness antiquity and the Middle Ages. 14. For the modern historian of ancient in navigation, geography, and time-reckoning, supplemented by the fortunate ac- astronomy it is therefore of the greatest cident that the Easter festival followed value to have an additional source'of asthe lunar calendar of the Near East, thus tronomical literature in which the earlier sanctioning lunar theory when other secu- tradition was kept alive without interruption for a much longer period: the astrolar sciences fell into total desuetude. The extreme paucity of scientists at al- logical texts. We have already mentioned most any given time in antiquity gave rise that astrology in the modern use of the to another phenomenon in Greek litera- word appeared very late in antiquity. ture: the publication of commentaries The art of casting horoscopes can be said and popularizing works. A work like the to be a typical Hellenistic product, the reAlmagest, written in purely scientific sult of the close contact between Greek style, was certainly unintelligible to the and oriental cultures.81We possess Greek majority of people who needed or wanted papyri from Egypt from the beginning of to know a modest amount of astronomy. our era to the Arabian conquest showing Hence books were written which attempt- us the application of astronomical methed to explain Ptolemy's text sentence by ods in a great number of specific horosentence,77 or which gave abstracts ac- scopes and in minor astronomical treacompanied by explanations of the main tises.82 In addition, an enormous astroprinciples as far as this could be done logical literature is preserved, catalogued without mathematics.78 We can observe during the last fifty years in the twelve the same phenomenon in geography. The volumes of the Catalogusby Cumont and first chapter of Ptolemy's Geography79 80 Edited and translated in the "Loeb Classical 77The commentaries of Pappus and Theon of Alexandria (and presumably of Hypathia) are of this type. For these texts cf. Rome CPT. 78 Represented, e.g., by Theon of Smyrna (second cent. A.D.) or Proclus (fifth cent. A.D.). 79Edited by Nobbe (1843). The first chapter is excellently discussed by M2ik and Hopfner PDE. Library" by H. L. Jones (8 vols.; 1917-32). 81 Cf., e.g., Capelle [1], who shows that only weak traces of astrological ideas in Greek literature can be followed as far back as 400 B.C. 82 Concerning horoscopes, see above, n. 17. Examples of astronomical treatises are Pap. Ryl. 27, 464, 522/24, 527/28, or Curtis-Robbins [1].

HISTORY OF ANCIENT ASTRONOMY: his collaborators.83 Finally, Vettius Valens, who wrote shortly before Ptolemy,84 and Ptolemy himself as the author of the famous Tetrabiblos,must be mentioned.85 Modern scholars have not yet made full use of this vast material. The reason is only too clear: the amount of work to be done surpasses by far the power of a single individual, and the work itself is certainly not very pleasant. The astronomical part must be extracted from occasional remarks, short computations, and similar instances submerged beneath purely astrological matter of a very unappealing character. But this work must eventually be done and will give valuable results. As an example might be mentioned the question of discovering the principle according to which the equinox was placed in the zodiac. This question must be answered, for on it depend our calculations in the determination of constellations, chronology, etc. Moreover, systematic checking of astrological computations will frequently yield information about the character of the astronomical tables used at the time. We touch here upon a point of great importance for the modern attitude toward ancient astronomy. The usual treatment of ancient sciences as a homogeneous type of literature is very misleading. It is necessary to realize that very different levels of astronomy or mathematics were coexistent, almost without mutual contact or interference. One misses the essential points in the understanding of ancient astronomy if one naively considers various documents in their chronological order. Even works by the same person must sometimes be separated from one another. Ptolemy's Almagest is purely mathematical, the Tetrabiblos (written Cf. also Boll 83 CCAG. 84 Kroll [2]. VV. 85Ptolemy, Opera III, 1, and "Loeb Classical Library" (ed. F. E. Robbins). PROBLEMS AND METHODS 21 after the Almagest)86is purely astrological, and his Harmonics7 contains a chapter on the harmony of spheres employing concepts of the planetary movements which contains such strong simplification of the actual facts that one would try in vain to find similar assumptions in any of the other works of Ptolemy. In other words, it is necessary to evaluate each text in its proper surrounding and according to its traditional style. One cannot, for example, speak without qualification of the contact between Babylonian and Greek astronomy. Such a contact might even have worked in opposite directions in different fields. For instance, we have already referred to the possibility that Hellenistic astrology returned to Babylonia in the form acquired in Egypt or Syria, whereas observational material from Mesopotamia undoubtedly influenced Greek mathematical astronomy deeply. In general, it can be said that the growth of ancient sciences shows much more irregularity and stratification than modern scientists, accustomed to the fact of the uniform spread of modern ideas and methods, are prone to assume. The lack of uniformity in the whole field of ancient astronomy in general necessarily interferes also with the investigation of any special problem. We have already mentioned the fact that astrology in the Assyrian age differed considerably from the horoscopic type which prevailed in late antiquity and the Middle Ages. But there exists a third type, standing between the omina type ("when this and this happens in the skies, then such and such a major event will be the consequence") and the individual birth horoscope, namely, the "general prognostication," explained in full detail in the first two books of the Tetrabiblos.This type of 86 This follows from the introduction biblos. 87 Diring HP and PPM. to the Tetra-

JOURNAL OF NEAR astrology is actually primitive cosmic physics built on a vast generalization of the evident influence of the position of the sun in the zodiac on the weather on earth. The influence of the moon is considered as of almost equal importance, and from this point of departure an intricate system of characterization of the parts of the zodiac, the nature of the planets, and their mutual relations is developed.88 This whole astronomical meteorology is, to be sure, based on utterly naive analogies and generalizations, but it is certainly no more naive and plays no more with words than the most admired philosophical systems of antiquity. It would be of great interest for the understanding of ancient physics and science in general to know where and when this system was developed. The question arises whether this is a Greek invention, replacing the Babylonian omen literature, which must at any rate have lost most of its interest with the end of independent Mesopotamian rule, whether it precedes the invention of the horoscopic art for individuals or merely represents an attempt to rationalize the latter on more general principles.89 Thus we see that even in a single field of ancient astronomical thought the most heterogeneous influences are at work; the analysis of these influences has repercussions on almost every aspect of the study of ancient civilizations.90 15. The same branching-off into very different lines of thought must also be recognized in the development of Greek mathematics. The line of development characterized by the names of Eudoxus, Euclid, Archimedes, and Apollonius is to be separated sharply from writings like 88 For the whole complex of the ancient justifications of astrology, see Duhem, SM, II, 274 ff. 89 This is the assumption of Kroll [1], p. 216, for the tendency exhibited in Ptolemy's Tetrabiblos. 90Cf. the excellent survey of this situation in Boll [21. EASTERN STUDIES Heron91and Diophantus92or the Arithmetic of Nicomachus of Gerasa.93Here, again, the question of oriental influence cannot be discussed as one common phenomenon. Egyptian calculation technique and mensuration were certainly continued in similar works in Hellenistic Egypt and found their way into Roman and medieval practices. At the same time, Babylonian numerical methods influenced Alexandrian astronomy. How Babylonian algebraic concepts eventually reached Greek writers like Diophantus is still completely unknown, but that it did is supported by the strong parallelism in methods and problems.94Equally lacking is detailed information as to the revival of these methods in Moslem literature.95 On the other hand, the problems which emerged from the discovery of the irrational numbers are undoubtedly of Greek origin. It is, however, not correct to consider writings of the same person as equally representative of "Greek" mathematics. Those parts of Euclid's Elements (the majority of the work) which deal more or less directly with the problem of irrational numbers are, as we said before, Greek. Most likely of equally Greek origin is Euclid's astronomical treatise called Phenomena,96 which is written on so elementary a level that nobody would attribute it to the author of the Elements if the authorship were not so firmly established. And, finally, Euclid's Data97 contains the treatment of purely algebraical problems by geometrical means-which can be interpreted as the direct geometrical transla91 First century A.D.; cf. for this date Neugebauer [14], pp. 21 ff. 92 Usually dated about A.D. 300; cf., however, Klein [1], p. 133, n. 23. 93Greek text ed. Hoche (Leipzig, 1866); English translation: D'Ooge-Robbins-Karpinski Nic. 94 Vogel [2]; Gandz [3]. 95Gandz [1], [2], [3]. 96 Opera VIII; cf. above, p. 16. 97 Opera VI.

HISTORY OF ANCIENT ASTRONOMY: tion of methods well known to Babylonian mathematics.98 These methods of "geometrical algebra" in turn determine the whole structure of Apollonius' theory of conic sections.9 Greek mathematics is by far the bestinvestigated field of ancient science (and of the history of science in general) ;"1 the situation with respect to the source material is very good?"'-except where only Arabic manuscripts are preserved.102But one must not forget that also this tradition suffers from severe gaps. This is due not only to the destruction of manuscripts over a period of two thousand years but also to the effect of literary influence. I refer not only to the above-mentioned elimination of older treatises by the overshadowing of the great works of the Hellenistic period. The Greeks themselves contributed to the distortion of the picture of the actual development by inventing seemingly plausible stories where the real records were already lost. The oft-repeated stories about Thales, Pythagoras, and other heroes are the result.103We should now realize that we know next to nothing about earlier Greek mathematics and astronomy in general and about the contact with the Near East and its influence in particular. The method which involves the use of a few obscure citations'04 from 98 Neugebauer [15]. 99Zeuthen KA and Neugebauer [16]. 100Best and exposition: Heath GM MGM and Euclid. A selection of texts is given in Thomas GM W. 101Most of the texts are edited in the Teubneriana collection. 102 Menelaos alone is now edited (Krause Men.), but Books v, vi, and vii of Apollonius' Conic Sections are still unavailable in a modern edition. Archimedes' construction of the heptagon is published in a free translation of the Arabic version in Schoy TLAB, pp. 74-91; cf. also Tropfke [1]. 103 As an example might be mentioned the criticism of the story of the Thales eclipse by Pannekoek [3], p. 955; Dreyer HPS, p. 12, n. 2; Neugebauer [9], pp. 295-96. Cf. also Frank, Plato, or Heidel [1]. 104 The fragments collected by Diels VS not only give an extremely incomplete picture of the lost writings but were certainly very much distorted by the PROBLEMS AND METHODS 23 late authors for the restoration of the history of science during the course of centuries seems to me doomed to failure. This amounts to little more than an attempt to understand the history of modern science from a few corrupt quotations from Kant, Goethe, Shakespeare, and Dante. 16. Undoubtedly the most spectacular advances in the history of astronomy until very recent times were scored in the theory of the planets. The catch-words "Ptolemaic" and "Copernican" refer to different assumptions as to the mechanism of the planetary movement. This is not the place to underline the fact that the Copernican theory is by no means so different from or so superior to the Ptolemaic theory as is customarily asserted in anniversary celebrations,"05but we must briefly analyze Ptolemy's own claims to having been the first one who was able to give a consistent planetary theory."10This claim seems to contradict not only the existence of pre-Ptolemaic planetary tables in Roman Egypt as well as in Mesopotamia but also Ptolemy's own reference to such texts. What Ptolemy means, however, becomes clear if one reads the details of the introduction to his own theory. He requires an explanation of the planetary movement by rheans of a combination of uniform circular movements which refrains from simplifications like the assumption of an invariable amount for the retrograde arc and similar deviations from the actual observations. Indeed, in order to remain in close agreement with the observations, Ptolemy had to overcome difficulties which Hipparchus was not able to authors from whose works they are taken. One needs only to look at the picture of oriental writings obtained from Greek tradition as compared with the originals. 105The correct estimate can be found in Thorndike H M, Vol. V, chap. xviii. 106 Almagest

JOURNALOF NEAR EASTERN STUDIES master and which led Ptolemy to a model my.112 We do not know how these tables which is very close to Kepler's final solu- were computed, and their occurrence in tion of the problem, by assuming not only Greek as well as in Demotic leaves us in an eccentric position of the earth but also doubt as to their origin-showing us only an eccentric point around which the the degree of interrelation we can expect movement of the planetary eccenter ap- in Hellenistic times. The most interesting question would, of pears to be uniform. The resulting orbit is of almost elliptical shape with these two course, be to learn more about Hipparpoints as foci.107 This whole theory is chus' astronomy: He is most famous as closely related in method to the explana- the discoverer of the precession of the tion of the 'evection" of the moon (a pe- equinoxes. Though this fact cannot be riodic perturbation of the moon's orbit doubted,1l3 underlining its importance discovered by Ptolemy) by a combination lays the wrong emphasis on a phenomeof eccentric and epicyclic movements. non which gained its importance only Both theories are real masterpieces of from Newton's theory, which showed ancient mathematical astronomy which that precession depends on the shape of far surpassed all previous results. the earth and thus opened the way to It is not surprising that Ptolemy's re- test the theory of general gravitation by sults overshadowed all previous works. All direct measurements on the earth. For that we know about his forerunners comes ancient astronomy, however, precession mainly from the Almagest itself. We hear played a very small role, requiring nothing that Hipparchus used eccenters and epi- more than sufficiently remote and sufficycles for the explanation of the anomalies ciently reliable records of observations of in the movement of the sun and the positions of fixed stars. The change in moon,108and we learn about theorems for positions must then eventually become such movements proved by Apollonius.109 evident; and little difficulty was enThis brings us to the very period (about countered in incorporating this slow 200 B.C.)from which the oldest cuneiform movement into the adopted model of celestial mechanics. What we actually planetary texts are preserved-computed, however, on entirely different principles. need to appreciate in Hipparchus' contriThese cuneiform texts cover the two cen- bution must be derived from a careful turies down to the time of Caesar. A direct study of all relevant sections of the Almacontinuation, chronologically speaking, gest, not by the schematic method of obbut of still another type, are planetary taining "fragments" from direct quotatables from Egypt, written in Demotic or tions but by a comparison of Ptolemy's Greek.110These tables give the dates at methods and the older procedures which which the planets enter or leave the signs he frequently mentions. That such an apof the zodiac. Such tables were known to proach can lead to well-defined results Cicero"' and are most likely the "eternal has recently been shown in the theory of tables" quoted with contempt by Ptole- eclipses.114 107 Cf. Schumacher 17. One of the most important prob[1] for the Ptolemaic theory of Venus and For the Greek Mercury. in general, see Herz GB I. 108 Almagest 109 Almagest III, XII, planetary theory 112 4. 1 (=Apollonius, ed. Heiberg, II, 137). 110Neugebauer [3]. Cf. above, p. 5. M Cicero De divinatione Almagest IX, 13 Schnabel's ii. 6, 17; cf. also ii. 71. 146. 2. attempts (Schnabel [1]) to prove that precession was taken into consideration in the cuneiform texts are, to say the least, inconclusive and in part based on mere scribal errors. 114 Schmidt

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS lems in connection with Hipparchus is, of course, the problem of the dependence of Hipparchus (and Greek astronomy in general) on Babylonian results and methods. Whatever the conclusions derived from a deeper knowledge of Hipparchus' astronomy may turn out to be, one thing is clear: the century between Alexander's conquest of the Near East and Hipparchus' time is the critical period for the origin of Babylonian mathematical astronomy as well as for its contact with Greek astronomy. Since Kugler's discoveries, which showed the exact coincidence between numerical relations in cuneiform tablets and in Hipparchus' theory,115no one has doubted Babylonian priority. It is an undeniable fact that the Babylonian theory is based on mathematical methods known already in Old Babylonian times and does not show any trace of methods considered to be characteristically Greek. The problem remains, however, to answer the question: What caused the sudden outburst of scientific astronomy in Mesopotamia after many centuries of a tradition of another sort? On what background can we understand, for example, the report16 that the "Chaldaean" Seleucus from Seleucia on the Tigris117completed the heliocentric theory, previously proposed as a hypothesis by Hipparchus? Greek influence on late Babylonian astronomy must not be denied or asserted on aprioristic grounds, if we really want to understand a phenomenon of great historical significance. These remarks are not intended to make Greek influence alone responsible for the new developments in Mesopotamia. As a matter of fact, this answer would only raise the equally unsolved 11,Kugler BMR, p. 40. Plat. quaest. vii. 1. 1006 C (ed. BerAS, pp. nardakis, Moralia, VI, 138). Cf. also Heath, 305 if. and Duhem SM, I, 423 ff. 116 Plutarch 117 Strabo 150 B.C. xvi. 739. Seleucus may have lived about 25 question why Greek astronomy suddenly emerged from many centuries of primitiveness to a scientific system. The alternative, Greek or Babylonian, might even exclude the right answer from the very beginning. It also seems possible that the rise of mathematical astronomy in Hellenistic times resulted from the suddenly intensified contact between several types of civilization, in some respects to be paralleled with the origin of modern science in the Renaissance. In other words, neither the Greeks nor the Orientals might have been alone responsible for the new development but rather the enormous widening of the horizon of all members of the culture of the Hellenistic age. One result of this process was probably the new attitude toward the relationship between the individual and the cosmos, expressed in the new form of horoscopic astrology. In this case it is quite evident that Egypt and Greece-and perhaps Syria as wellcontributed about equally much to the refinement and spread of this new creed. It is equally possible that the contact between Greek scholars, trained to think in geometrical terms which Greek mathematics had developed in the fifth century, and Babylonian astronomers, equipped with superior numerical methods and observational records, brought into simultaneous existence two closely related types of mathematical astronomy: the treatment by arithmetical means in Babylonia and the model based on circular movements in the Greek centers of learning in the eastern Mediterranean. It may well be that competition, not borrowing, was the chief contributor to the initial impetus."1 At any rate, it is clear that each detail in the development of Hellenistic astronomy which we will be able to understand better will reveal a new aspect in the fascinating process of the Neugebauer [17], pp. 30-31.

JOURNAL OF NEAR EASTERN STUDIES creation of the new world which was destined to become the foundation of the Roman and medieval civilizations. The unique role of the Hellenistic period in the field of sciences, as in other fields, can be described as the destruction of a cultural tradition which dominated the Near East and the Mediterranean countries for many centuries, but also the founding of a new tradition which held following generations in its spell. The history of astronomy in the Hellenistic age is especially well suited to demonstrate that the great energies liberated by the disintegration of an old cultural tradition are very soon transformed into stabilizing forces of a new tradition, which includes about as many elements of development as of stagnation. V. SPECIAL PROBLEMS 18. Every research program in a complex field will face the need of constant modification and adjustment to unforeseen complications and new ramifications. Problems can arise and results be obtained without having been anticipated in the original question. The context of a mathematical text, for example, can determine with absolute certainty the meaning of a word otherwise only vaguely defined; sign-forms in a papyrus which is exactly dated by astronomical means may furnish valuable information for purely paleographical problems. From dates and positions given in Demotic astronomical texts, it follows that the Alexandrian calendar introduced by Augustus was used by Egyptian scribes only a few years after the reform,19very much in contrast to the common opinion that the Egyptians were especially conservative in general and in calendaric matters in particular. In short, from few, but solidly established, facts we can learn more than from all general speculations. 119Neugebauer [6], p. 119. One of the problems which at first sight lies very much outside the history of ancient astronomy is the study of social and economic conditions of the ancient civilizations. There are, however, several points of contact between these studies and astronomy. We are indebted to Cumont for a masterly investigation of the information contained in the astrological literature from Hellenistic Egypt.l20 His results are not only of interest for the history of ancient civilization but also illustrate very well the background of the men who used and transmitted the astronomical material known to us from the planetary tables or from Vettius Valens. It turns out that the soil in which these practices were rooted was essentially Egyptian, in spite of the use of the Greek language in the documents. This is in perfect harmony with the close parallelism between Greek and Demotic planetary texts mentioned above and shows the constant interaction of Greek and native influences in Hellenistic Egypt. It also shows how dangerous it is to decide the authorship of Hellenistic doctrines or methods simply on the basis of such superficial grounds as the language used. The analogous question for Babylonia seems to be easier to answer. The Mesopotamian origin of the astrological omina cannot be doubted. We would, however, like to know more about the background of the astronomers of the latest period. It is well known that the names of three Babylonian astronomers appear in Greek literature121and that two of them actually were found on astronomical tablets, though in an unclear context. For one particular place, the famous city of Uruk in South Babylonia, we can go much further. It can be shown that the scribes and owners of our texts belong to one of two 120 Cumont EA. 121 Cumont See also Kroll

HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS 27 "families," or perhaps "guilds," of scribes are of great value because they contain who frequently call themselves scribes of numerous examples which give detailed the omen-series "Enuma Anu Enlil."122 solutions of problems in which metrologiWe can follow the work of these scribes cal relations play a major role. The consevery closely for almost a hundred years quences of such relations, established with until the school of Uruk ceased to exist, absolute certainty, are manifold. For exprobably because of the Parthian invasion ample, we now know from Old Babylonian of Babylonia in 141 B.C. In contrast there- mathematical texts the measurements of to, the school of Babylon survived the several types of bricks123as well as the pecollapse of the Greek regime, as is proved culiar notation used in counting bricks. It by a continuous series of astronomical is evident that such information is of imtexts down to 30 B.C.This is an interesting portance for the understanding of conresult in comparison with the assumption temporary economic texts dealing with that Babylon practically ceased to exist the delivery of bricks for buildings, thus after the Parthian occupation. The group- leading to purely archeological questions. ing of our texts according to well-defined Metrological relations are also needed if schools is also of interest from another we wish to gain an insight into wages and point of view. It can be shown that two prices.124Returning to our subject, it must different systems of computation existed be said that metrology is of great imporside by side for a long time. Competing tance not only for the history of the ecoschools of this sort constitute a phenome- nomics of Mesopotamia but also for purenon which is usually considered charac- ly astronomical problems. Distances on the celestial sphere are measured in asteristic for Greek culture. 19. Countless thousands of business tronomical texts by units borrowed from documents are preserved from all periods terrestrial metrology. The comparison beof Mesopotamian history. For the urgent- tween ancient observation and modern ly needed investigation of ancient eco- computations thus requires a knowledge nomics, a precise knowledge of the metro- of the ancient relations between the varilogical systems is of the greatest impor- ous units. This problem is by no means tance. Unfortunately, the scientific study simple because our astronomical material of Babylonian measures has been sadly belongs to relatively late periods, Assyrian neglected. Fantastic ideas about the level and Neo-Babylonian, and the metrologiand importance of astronomy in the earli- cal system of these times is much more est periods of Babylonian history led to involved than the Old Babylonian. Maththeories which brought measures of time ematical texts would certainly be of great and space in close relationship with al- help, here too, but the few tablets from leged astronomical discoveries. We know this period are so badly preserved that today that all these assumptions of the they present us with at least as many new early days of Assyriology must be aban- questions as they answer. Neo-Babylonidoned and that Babylonian metrology an economic texts will therefore furnish must be studied from economic and re- the main point of departure for the study lated texts clearly separated according to 123 MCT, Problem-Text O and period and region. For the determination Sachs Neugebauer-Sachs [1]. of Old Babylonian relations between 124 Waschow [1, p. 277, found, in discussing mathevarious measures, the mathematical texts matical texts, that the value of the area-measure "se" 122 For this series cf. Boll-Bezold-Gundel 2 if., and Weidner [2]. SS, pp. must be changed by a factor 60 against older assumptions. It is obvious how such facts influence the interpretation of economic texts.

JOURNAL OF NEAR EASTERN STUDIES of the latest phase of Mesopotamian metrology and its astronomical applications. It might be mentioned, in this connection, that theories about direct relationship between early Mesopotamian metrology and astronomy also gave rise to the rather unfortunate concept of high accuracy in the determination of weights, measures of length, etc. It is of great importance to realize that the absolute values of all metrological units are subject to great margins of inaccuracy and local and temporal variations. The first step in a historical investigation of Mesopotamian metrology must therefore be to establish from economic and mathematical texts the ratios between the units; these ratios have an incomparably better chance of showing unformity than the absolute values deduced from accidental archeological finds. 20. Closely related to metrological problems is the question of the accurate identification of ancient star configurations. Much work remains to be done before it will be possible to give a reliable history of the topography of the celestial sphere in general, or even of the zodiacal constellations.125In spite of attempts to make Egypt responsible for many forms,'26the predominant influence of Babylonian concepts on the grouping of stars into pictures must be maintained. But neither Babylonian nor Egyptian developments are known in detail. The identification of Egyptian constellations is especially difficult, mainly because it must be based on relations between the times of rising and setting and therefore depends on elements which are grossly schematized in the texts at our disposal. The situation in Mesopotamia is slightly better because we have actual observations in addition to the 125 The best summary is given by the Boll-Gundel article, "Sternbilder," in Roscher GRM, Vol. VI (1937), cols. 867-1072. schematic lists, at least for the later periods which are of special importance for the Hellenistic forms of the constellations. For the period following the publication of the Almagest, we must take into account the possibility of still other complications. We know from explicit remarks in the Almagest that Ptolemy's star catalogue introduced deviations from older catalogues.127 Astrological works, however, may very well have maintained prePtolemy standards both with respect to the boundaries of constellation and the counting of angles in the zodiac. We have already mentioned the stubborn adherence of astrological writers to methods of computation which were made obsolete by the development of spherical trigonometry.128For the modern historian it is therefore of importance to establish the specific standard according to which a given document was written, especially when chronological problems are involved. 21. While metrology is a much-needed implement for economic history and the understanding of ancient astronomy, astronomy itself serves general history in chronological problems. Chronology is the necessary skeleton of history and owes its most important fixed points to astronomical facts. We need not emphasize the use of reports of eclipses, especially solar eclipses, for the determination of accurate dates to form the framework into which the results of relative chronology must be fitted. It must be underlined, however, that the available material is by no means exhausted. A better understanding and reinvestigation of the reports of the Assyrian astronomers will certainly furnish new information of chronological value. It must be stated, on the other 127Almagest VII, 4 (ed. Heiberg, p. 37). 126 Cf. esp. Gundel DD and HT and the criticism of Schott [2]. 128 Cf., 95). e.g., Tetrabiblos I, 20 (ed. Robbins,

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS hand, that not too much is to be expected from older material. In order to make ancient observations accessible to modern computation, a certain degree of accuracy must be granted; this accuracy seems to be missing in the earlier phases of the development of astronomy. This, for instance, makes the older Egyptian material so ill suited for chronological purposes. For later periods, however, Egypt has furnished and will furnish much information from astrological documents. It is particularly calendaric questions, such as the use of eras and similar problems, which have been illuminated by the dating of horoscopes. The great variety of calendaric systems, local eras, and older methods of dating raises many difficulties in ancient chronology. This difficulty was clearly felt also by ancient astronomers and was the cause of the early use of consistent eras in Babylonian and Greek astronomy. The Babylonian texts always use the Seleucid Era, whereas Ptolemy reduces all dates to the Nabonassar Era but uses the Old Egyptian years of constant length. This crossing of Egyptian and Babylonian influences is paralleled by the subdivision of the day into hours. The Egyptians divided the day into twelve parts from sunrise to sunset, thus obtaining hours whose length depended on the season. The Babylonian astronomers used six subdivisions of day and night, but these units were of constant length. Combining the Egyptian division into 24 hours with the Babylonian constancy of length, the Hellenistic astronomers used "equinoctial" hours for their computations and solved the problem of finding the relationship between seasonal and equinoctial hours by spherical trigonometry.129 One sees here again what a multitude of relations, problems, and methods contributed to shape concepts such as a continuous era or the 24129 Almagest II. 9, 29 hour day which are so familiar to us today. Ancient chronology and the accurate analysis of ancient reports have turned out to be of interest even to a modern astronomical problem. In 1693 Halley discovered the fact'30 that the moon's position appeared to be advanced compared with the expected position as computed from positions recorded by Ptolemy. This "acceleration" can be explained by a slow increase in the length of the solar day or by a decrease in the rotational velocity of the earth. Such a decrease is caused by tidal forces,l31and it is of great interest to determine the amount as accurately as possible. For this purpose, accurate positions of the moon in remote times are of great value, and such positions can, indeed, be derived from records in cuneiform texts.132 Modern measurements of high precision can thus be supplemented by observations in antiquity. 22. Not only are Hellenistic astronomy and Hellenistic astrology the determining factors for the astronomy and astrology of the Middle Ages in Europe, but its influence is equally important for the development of astronomical methods and concepts in the Middle and Far East. We must therefore at least mention an enor-, mous field which still awaits systematic research: Hindu science. This does not mean that there is not an extensive literature on this subject; indeed, even a small number of original texts are published.'33 The main trouble lies, however, in the tendency of the majority of publications by Hindu authors to claim priority for Hindu discoveries and to deny foreign in130Edm. Halley, "Emendationes ac notae Abatenii observationes astronomicas, cum restitutione tabularum lunisolarum ejusdem authoris," Philosophical 17 (1693), No. 204, pp. 913-21. Transactions, 131Cf., e.g., Jeffreys [1]. 132 p. V. [1]. Neugebauer 133 For the literature until 1899, see Thibaut AA M. The best discussion of Hindu astronomy is still Burgess SS (1860).

JOURNAL OF NEAR fluence, as well as in the opposite tendency of some European scholars. This tendency has been especially strong so far as Hindu mathematics is concerned,134 and it is aggravated by the inadequate publication of the original documents, from which usually only scattered fragments are cited in order to prove some specific statement. As a result, there is no means today to obtain an independent judgment from the study of the original texts which are preserved in enormous number, though of relatively late date for the most part. The situation with respect to Hindu astronomy is not much better. There can be little doubt that the original impetus came from Hellenistic astronomy; the use of the eccentric-epicyclic model alone would be sufficient proof even if we did not also find direct witness in the use of Greek terminology.135This fact is interesting in itself, but it may very well be that the period of reception lies between Hipparchus and Ptolemy; systematic study might therefore reveal information about pre-Ptolemaic Greek astronomy no longer preserved in available Greek sources. Hindu astronomy would in this case constitute one of the most important missing links between late Babylonian astronomy and the fully developed stage of Greek astronomy represented by the Almagest. The fundamental difficulty in the study of Hindu astronomy lies in the character of the preserved textual material. The published and commented texts consist exclusively of cryptically formulated verses giving the rules for computing certain phenomena, making it extremely difficult to understand the actual 134 Cf., e.g., Datta-Singh HHM (reviewed in Neugebauer [12]). 135Thibaut AA M, pp. 43 ff. The Babylonian ratio 3 : 2 for the ratio between the longest and shortest days of the year also occurs in India (Thibaut AA M, pp. 26-27; Kugler BMR, pp. 82 and 195), though it would be suitable only for the latitude of the northern corner of India. For the planetary theory, see Kugler BB, p. 120; Schnabel [2], p. 112; Schnabel [1], p. 60. EASTERN STUDIES process to be followed. It is evident, on the other hand, that no astronomy of an advanced level can exist without actually computed ephemerids. It must therefore be the first task of the historian of Hindu astronomy to look for texts which contain actual computations. Such texts are, indeed, preserved in great number, though actually written in very late periods. Poleman's catalogue136of Sanskrit manuscripts in American collections lists about a hundred such manuscripts in the D. E. Smith collection in Columbia University in New York. In their general arrangement, these texts are reminiscent of the cuneiform ephemerids from Seleucid times and must reveal many details of the Hindu theory of the planetary movement if attacked by the same methods which have proved so successful in the case of the Babylonian material. The complete publication of this material is an urgent desideratum in the exploration of oriental astronomy. As mentioned above, the texts in the D. E. Smith collection are of very recent origin, only a few centuries old. This does not mean that the methods used are not of very much earlier date. This is shown by the investigation of one of these texts,l37 which deals with the problem of the varying length of the days during the year. Though written about 1500, the computations are based on methods going back to a much older period. Analogous results can be expected in the remaining material, and there is no reason to assume that the D. E. Smith collection exhausts all the preserved material. 23. In the preceding sections we have frequently touched on methodological questions. In closing, I wish to underline a few principles in a more general way. As is only natural, the study of the development of ancient science began under the 136Poleman CIM, pp. 231 ff. See also Emeneau PIT, pp. 318 ff. 137 Schmidt

THE HISTORYOF ANCIENTASTRONOMY:PROBLEMSAND METHODS influence of the ancient tradition. Herodotus, Diodorus, the commentators of Plato, etc., were the sources which determined the picture of the early stages of Greek and oriental mathematics and astronomy. But while students of political history, art, economics, and law learned in the early days of systematic archeological research to consider this literary tradition about the ancient Orient as nothing more than a supplementary source to be checked by the original documents, the majority of historians of the exact sciences have remained in a stage of naive innocence, repeating without criticism the nursery stories of ancient popular writers. This is all the more surprising because many of these stories should have revealed their purely fictitious character from the very beginning. Every invention considered of basic importance is attributed to a definite person or nation: Thales "discovered" that a diameter divides the area of a circle into two equal parts, Anaximandes and several others are credited with the discovery of the obliquity of the ecliptic, the Egyptians discovered geometry, the Phoenicians arithmetic-and so on, according to an obvious pattern of naive restoration of facts the origins of which had been totally forgotten. Modern authors then add stories of their own, such as the idea that the construction of the pyramids required mathematics, the assumption of supposedly marvelous skies of Mesopotamia,138 and the notion of Egyptian Stone Age astronomers industriously determining the heliacal rising of Sirius or carrying out a geodetic survey of the Nile Valley. It is clear that the replacement of the traditional stories by statements based exclusively on results obtainable from the original sources will not be very appealing. This is the inevitable result in the devel138 For the poor conditions of actual observation cf. Koldewey WB, p. 192; Vogt [11, pp. 38-39; cf. also Boll [1], pp. 48 and 157. 31 opment of every science; for increased knowledge means giving up simple pictures. In the history of science, an additional element must be added to the steady increase of complexity resulting from a better understanding of our sources. Not only do we learn to interpret our material more accurately but we also learn to see everywhere the immense gaps in our preserved sources. We will more and more be forced to admit that many, and essential, steps in the development of science are hopelessly destroyed; that we, at best, are able to sketch mere outlines of the history of science during certain sharply limited periods; and that many of the driving forces might actually have been quite different from those which we customarily restore on the analogy of later periods. One consequence of this situation seems to me to be evident: unless the history of science now enters the stage of specialization, it will lose all value in the framework of historical research. It must be clearly understood that the history of science must work with methods and must consider its problems from viewpoints which correspond to the methods and standards of other branches of historical research. The idea must definitely be abandoned that the history of science must adapt its level to the alleged requirements of the teaching of the modern fields of science. The intrinsic value of this research must be seen in its contribution to our understanding of the historical processes which shaped human civilization, and it must be made clear that such an understanding cannot be reached without the closest contact with the other historical fields. The call for specialization is not very popular. I am convinced, however, that a well-founded insight into the details of a single essential step in the development is at present of higher value and more fascinating than any attempt at general syn-

JOURNAL OF NEAR EASTERN STUDIES thesis. It is ridiculous to believe that we are anywhere able to reach "final" results in the study of the development of human civilization. But the overwhelming richness of all phases of human history can be appreciated only if we occupy ourselves with the real facts as accurately as possible and do not attempt to hide their manifold aspects under the veil of hazy generalizations or let our judgment be guided by the naive idea of human "progress." Every synthesis written fifty years ago is now completely antiquated and at best enjoyable for its literary style; the careful study of the original works of the ancients, however, will reveal to everyone and at any time the development of their achievements.139 The call for specialization must not be misunderstood as a plea for the disregard of the general outlines of the historical conditions. On the contrary, specialized work can be accomplished successfully only if the points of attack are selected under constant consideration of possible interference from other problems and other fields. It is indeed the most gratifying result of detailed research on a welldefined problem that it necessarily uncovers relationships which are of primary importance for the understanding of larger 139An excellent example is Delambre HA A, published in 1817 and still not surpassed or even equaled because of its direct contact with the original sources. historical processes. The actual working program, however, needs restriction and minute detail work. The most essential task is that of making the original sources accessible as easily as possible in their best available form. By the indefatigable work of Heiberg, Hultsch, Tannery, and many others, we possess today a great part of the extant writings of the Greek scientists in excellent editions. We owe to Sir Thomas Little Heath many brilliant commentaries and translations of Greek mathematicians.140 To make Greek and oriental source material more generally accessible, supplemented, of course, by modern translations and commentaries, will be the foremost problem of the future. The extension of this program to include medieval material, on the one hand, and Middle Eastern documents, on the other, appears as a logical consequence, worthy of the serious efforts of all scholars who wish to contribute to the understanding of the past of our own culture. BROWN UNIVERSITY 140 On the other hand, much remains to be done to repair the harm caused by classical philologists who made their editions inaccessible to modern scientists by translating them into Latin instead of a modern language. Great opportunities have been spoiled by this absurd attitude. It has fortunately never occurred to Orientalists to translate their texts into Hebrew. It should be mentioned, however, that the Arabic version of Euclid's Elements was published in Latin(!) translation by Besthorn, Heiberg, and others (Copenhagen, 1897-1932). BIBLIOGRAPHY AJP AJSL Almagest AN American Journal of Philology. American Journal of Semitic Languages and Literatures. See Ptolemy. AstronomischeNachrichten. Baillet [1] J. "Le Papyrus mathematique d'Akhmim," Mem. publ. par les BAILLET, membresde la mission arch. franc. au Caire, Vol. 9, Fasc. 1 (1892). BASOR Bulletin of the American Schools of Oriental Research. Berger GFH BERGER, H. Die geographischenFragmentedes Hipparch. Leipzig, 1869. BOLL. F. Sphaera. Leipzig, 1903. Boll, Sphaera Boll [1] BOLL, F. "Antike Beobachtungen farbiger Sterne," Abh. K. Bayerischen Akad. d. Wiss., Philos.-philol. u. histor. Kl. 30, No. 1 (1918). Boll [2] BOLL, F. "Die Erforschung der antiken Astrologie," Neue Jahrbiicherfur das klassische Altertum, 21 (1908), 103-26. Boll-Bezold-Gun- BOLL, F.; BEZOLD, C.; and GUNDEL, W. Sternglaube und Sterndeutung. 4th del SS ed. Leipzig, 1931.

THE HISTORYOF ANCIENTASTRONOMY:PROBLEMSAND METHODS Bouriant, P. [1] P. "Fragment d'une manuscripte copte de basse epoque ayant BOURIANT, contenu les principes astronomiques des arabes [cf. n. 12]." Journal asiatique, 10. ser., 4 (1904), 117-23. Bouriant, U.-Ven- BOURIANT, U., and VENTRE BEY. "Sur trois tables horaires coptes," Metre Bey [1] moires presentesa l'institut egyptien, 3 (1900), 575-604. Braunmiihl GT BRAUNMtHL, A. v. Vorlesungen iiber Geschichteder Trigonometrie. Leipzig, 1900. H. Thesaurus inscriptionum aegyptiacarum, I. Astronomischeand Brugsch, Thes. 1 BRUGSCH, astrologischeInschriften. Leipzig, 1883. Burgess SS BURGESS,E. "Translation of the Surya-Siddhanta," JAOS, 6 (1860), 141498. Reprinted, Calcutta, 1935 [with introduction (45 pp.) by P. C. Sengupta]. W. "Alteste Spuren der Astrologie bei den Griechen," Hermes, 60 Capelle [1] CAPELLE, (1925), 373-95. CCAG et. al. Catalogus codicum astrologorumGraecorum. Edited by BOLL,CUMONT, 12 vols. Bruxelles, 1898-1936. Chace RMP Cicero De div. Crum CO CT Cumont EA Cumont [1] Curtis-Robbins CHACE, A. B.; MANNING, H. P.; and ARCHIBALD,R. C. The Rhind Mathematical Papyrus. 2 vols. Oberlin, 1927-29. "Loeb CICERO,M. T. De divinatione. (English trans. by W. A. FALCONER, Classical Library.") CRUM,W. E. Coptic Ostraca. London, 1902. Cuneiform texts from Babylonian tablets, etc., in the British Museum. F. L'Egypte des astrologues.Bruxelles, 1937. CUMONT, F. Comment les grecs connurent les tables lunaires des chaldeens: CUMONT, Florilegium ... dedies a M. le marquis Melchior de Vogue ..., pp. 159-65. Paris, 1910. [1] CURTIS, H. D., and ROBBINS, F. E. "An Ephemeris of 467 A.D." Publ. of the Observatoryof the Univ. of Michigan, 6 (1935), 77-100. Daressy [1] DARESSY,G. "Notes et remarques 181," Rec. trav., 23 (1901), 126-27. Datta-Singh HHM DATTA,B., and SINGH,A. N. History of Hindu Mathematics. 2 vols. Lahore, 1935-38. Delambre HAA J. B. J. Histoire de l'astronomieancienne. 2 vols. Paris, 1817. DELAMBRE, D'Ooge-Robbins- D'OOGE,M. L.; ROBBINS,F. E.; and KARPINSKI,L. C. Nicomachus of Gerasa. "Univ. of Michigan Studies: Humanistic Series," No. 16. Ann Karpinski Nic. Arbor, 1926. Drecker [1] J. "Das Planisphaerium des Claudius Ptolemaeus," Isis, 9 (1927), DRECKER, 255-78. DREYER,J. L. E. History of the Planetary Systems from Thales to Kepler. Dreyer HPS Cambridge, 1906. Diiring HP DtRING, I. Die Harmonielehredes Klaudios Ptolemaios. "Goteborgs Hogskolas Arsskrift," No. 36 (1930). Diuring PPM DURING,I. Ptolemaios und Porphyrios iiber die Musik. "G6teborgs H6gskolas Arsskrift," No. 40 (1934). Duhem SM DUHEM,P. Le Systeme du monde;histoire des doctrinescosmologiquesde Platon a Copernic. 5 vols. Paris, 1913-17. Emeneau PIT EMENEAU, M. B. A Union List of Printed Indic Texts and Translations in American Libraries. "Amer. Oriental Series," Vol. 7, 1935. AB Epping EPPING,J. Astronomischesaus Babylon. "Stimmen aus Maria-Laach, Erganzungsheft," No. 44. Freiburg, 1889. Euclid Euclidis opera omnia, ed. J, L. HEIBERGand H. MENGE.8 vols. Leipzig, Frank, Plato FRANK,E. Plato und die sogenanntenPythagoreer. Halle, 1923.

Frankfort CSA Gandz [1] Gandz [2] Gandz [3] Garbers ES Ginzel Chron. Gundel DD Gundel HT Harper Letters Haskins MS Heath AS Heath Euclid Heath GM Heath MGM Heidel GM Heidel [1] Herz GB Honigmann SK JAOS Jeffreys [1] JNES Klein [1] Koldewey WB Krause Men. Kroll [1] Kroll VV Kugler BB Kugler BMR Kugler MP Kugler SSB JOURNAL OF NEAR EASTERN STUDIES H. The Cenotaphof Seti I at Abydos. 2 vols. "Egypt Explor. Soc., FRANKFORT, Memoir," No. 39 (1933). GANDZ,S. "The Sources of al-Khowarizmi's Algebra," Osiris, 1 (1936), 263-77. GANDZ,S. "The Algebra of Inheritance," Osiris, 5 (1938), 319-91. GANDZ,S. "The Origin and Development of the Quadratic Equations in Babylonian, Greek, and Early Arabic Algebra," Osiris, 3 (1937), 405-557. GARBERS, K. "Ein Werk Tabit b. Qurra's iiber ebene Sonnenuhren," QS, A, 4 (1936). GINZEL,F. K. Handbuch der mathematischenund technischen Chronologie. 3 vols. Leipzig, 1906-14. GUNDEL,W. Dekane und Dekansternbilder. "Studien d. Bibl. Warburg," No. 19 (1936). GUNDEL,W. "Neue astrologische Texte des Hermes Trismegistos," Abh. d. Bayerischen Akad. d. Wiss. Phil.-hist. Abt. (N.F.), 12 (1936). HARPER,R. F. Assyrian and Babylonian Letters Belonging to the Kouyunjik Collectionof the British Museum. 14 vols. Chicago, 1892-1914. HASKINS,C. H. Studies in theHistory of Mediaeval Science. 2d ed. Cambridge: Harvard University Press, 1927. HEATH,T. L. Aristarchus of Samos. Oxford, 1913. HEATH,T. L. The Thirteen Books of Euclid's Elements. 3 vols. 2d ed. Cambridge, 1926. HEATH,T. L. A History of GreekMathematics. 2 vols. Oxford, 1921. HEATH,T. L. A Manual of GreekMathematics. Oxford, 1931. HEIDEL,W. A. The Frame of the Ancient GreekMaps: With a Discussion of the Discovery of the Sphericity of the Earth. New York, 1937 (=Am. Geographical Soc., Research Series No. 20). HEIDEL,W. A. "The Pythagoreans and Greek Mathematics," AJP, 61 (1940), 1-33. HERZ,N. Geschichteder Bahnbestimmungvon Planeten und Kometen, 1: Die Theorien des Altertums. Leipzig, 1887. HONIGMANN,E. Die sieben Klimata und die Poleis episemoi. Heidelberg, 1929. Journal of the American Oriental Society H. "The Chief Cause of the Lunar Secular Acceleration," MN, 80 JEFFREYS, (1920), 309-17. Journal of Near Eastern Studies. KLEIN,J. Die griechische Logistik und die Entstehung der Algebra," QS, B, 3 (1934-36), 18-105, 122-235. R. Das wiedererstehendeBabylon. 4th ed. Leipzig, 1925. KOLDEWEY, KRAUSE,K. "Die Spharik von Menelaos aus Alexandrien in der Verbesserung von Abi Nasr Mansur b. Ali b. Iraq," Abh. Ges. d. Wiss. zu Gittingen, Phil.-hist. Kl., 3. Folge, No. 17 (1936). KROLL,W. "Kulturhistorisches aus astrologischen Texten," Klio, 18 (1923), 213-25. KROLL,W. Vettii Valentis anthologiarumlibri. Berlin, 1908. KUGLER,F. X. Im Bannkreis Babels. Miinster, 1910. KUGLER,F. X. Die babylonischeMondrechnung.Freiburg, 1900. KUGLER,F. X. Von Moses bis Paulus. Miinster, 1922. KUGLER,F. X. Sternkunde und Sterndienst in Babel. 2 vols. Miinster, 1907-24. Erganzungen, in three parts (Part III by J. SCHAUMBERGER).Miinster,

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS Langdon VT Lange-Neugebauer [1] L'H6te LE Luckey [1] Luckey [2] 35 J. K.; and SCHOCH,C. The Venus Tablets of LANGDON,S.; FOTHERINGHAM, Ammizaduga, Oxford, 1928. O. "Papyrus Carlsberg No. 1," Kongl. LANGE,H. 0., and NEUGEBAUER, Danske Vidensk. Selskab, Hist.-fil. Skrifter, Vol. 1, No. 2 (1940). L'HOTE, NESTOR. Lettres ecrites d'Egypte en 1838 et 1839 .... Paris, 1840. LUCKEY, P. "Das Analemma von Ptolemaus," AN, 230 (1927), 18-46. P. "Tabit b. Qurra's Buch iiber die ebenen Sonnenuhren," QS, B, 4 LUCKEY, (1937), 95-148. MN Monthly Notices of the Royal Astronomical Society. Mfik EGM M1IK, H. v. Erdmessung, Grad, Meile und Stadion nach den altarmenischen Quellen (= Studien zur armenischen Geschichte No. 6). Wien, 1933. [Reprinted from Handes Amsorya, Z. f. armenische Philologie, 47 (1933), cols. 283-305, 432-59, 559-82.] MZIK,H. v., and HOPFNER,F. Des Klaudius Ptolemaios Einfiihrung in die Mzik-Hopfner PDE darstellendeErdkunde, 1. Wien, 1938 (=Klotho 5). Neugebauer ACT NEUGEBAUER,O. Astronomical Cuneiform Texts. (In preparation.) Neugebauer MKT NEUGEBAUER, 0. MathematischeKeilschrift-Texte. 3 vols. (QS, A, 1 [193538].) Neugebauer Vorl. NEUGEBAUER, 0. Vorlesungen iiber Geschichteder antiken mathematischen Wissenschaften, 1: VorgriechischeMathematik. Berlin, 1934. Neugebauer [1] NEUGEBAUER, O. "Arithmetik und Rechentechnik der Agypter," QS, B, 1 (1930), 301-80. NEUGEBAUER,0. "Die Geometrie der agyptischen mathematischen Texte," Neugebauer [2] QS, B, 1 (1931), 413-51. O. "Egyptian Planetary Texts," Trans. of the Amer. Philos. Neugebauer [3] NEUGEBAUER, Soc., 32 (new ser., 1942), 209-50. Neugebauer [4] NEUGEBAUER, O. "Die Bedeutungslosigkeit der 'Sothisperiode' fir die alteste agyptische Chronologie," Acta orientalia, 17 (1938), 169-95. Neugebauer [5] NEUGEBAUER,O. "The Origin of the Egyptian Calendar," JNES, 1 (1942), 396-403. NEUGEBAUER,O. "Demotic Horoscopes," JAOS, 63 (1943), 115-26. Neugebauer [6] Neugebauer [7] NEUGEBAUER, O. Review of Kugler SSB Erg. (Schaumberger), QS, B, 3 (1935), 271-86. NEUGEBAUER,O. "Untersuchungen zur antiken Astronomie, II: Datierung Neugebauer [8] und Rekonstruktionn on Texten des Systems II der Mondtheorie," QS, B, 4 (1937),. 34-91. Neugebauer [9] NEUGEBAUER,O. "Untersuchungen zur antiken Astronomie, III: Die babylonische Theorie der Breitenbewegung des Mondes," QS, B, 4 (1938), 193-346. Neugebauer [10] NEUGEBAUER,O. "Jahreszeiten und Tageslangen in der babylonischen Astronomie," Osiris, 2 (1936), 517-50. Neugebauer [11] NEUGEBAUER,O. "Zur Entstehung des Sexagesimalsystems," Abh. d. Ges. d. Wissenschaften zu G6ttingen,Math.-phys. Kl. (N.F.), 13, No. 1 (1927). Neugebauer [12] NEUGEBAUER,O. Review of Datta-Singh HHM I (QS, B, 3 [1936], 263-71). Neugebauer [13] NEUGEBAUER,O. "On Some Astronomical Papyri and Related Problems of Ancient Geography," Trans. of the Amer. Philos. Soc., 32 (new ser., 1942), 251-63. Neugebauer [14] NEUGEBAUER,O. "Uber eine Methode zur Distanzbestimmung AlexandriaRom bei Heron," Kongl. Danske Vidensk. Selskab, Hist.-fil. Meddel., Vol. 26, No. 2 (1938). Neugebauer [15] NEUGEBAUER, O. Zur geometrischenAlgebra ("Studien zur Geschichte der antiken Algebra," III) (QS, B, 3 [1935], 245-59).

JOURNALOF NEAR EASTERN STUDIES O. Apollonius-Studien ("Studien zur Geschichte der antiken NEUGEBAUER, Algebra," II) (QS, B, 2 [1932], 215-54). [17] Neugebauer NEUGEBAUER,O. "Exact Science in Antiquity," University of Pennsylvania Bicentennial Conference:Studies in Civilization, pp. 23-31. Philadelphia, 1941. O. "Some Fundamental Concepts in Ancient Astronomy," Neugebauer [18] NEUGEBAUER, University of Pennsylvania Bicentennial Conference:Studies in the History of Science, pp. 13-29. Philadelphia, 1941. O. "The Water-Clock in Babylonian Astronomy" (to be Neugebauer [19] NEUGEBAUER, published in Isis in 1945). Neugebauer-Sachs NEUGEBAUER, 0., and SACHS,A. Mathematical CuneiformTexts (to be published in "Amer. Oriental Series," New Haven, 1945). MCT 0., and STRUVE,W. "Uber die Geometrie des Kreises in NeugebauerNEUGEBAUER, Struve [1] Babylonien," QS, B, 1 (1929), 81-92. Neugebauer-Vol- NEUGEBAUER, 0., and VOLTEN,A. "Untersuchungen zur antiken Astronoten [1] mie, IV: Ein demotischer astronomischer Papyrus (Pap. Carlsberg 9)," QS, B, 4 (1938), 383-406. P. V. "Eine Konjunktion von Mond und Venus aus dem Neugebauer, NEUGEBAUER, P. V. [1] Jahre -418 und die Akzeleration von Sonne und Mond," AN, 244 (1932), cols. 305-8. Pannekoek [1] A. "Calculation of Dates in the Babylonian Tables of Planets," PANNEKOEK, Koninkl. Akad. van Wetensch. te Amsterdam, Proceedings, 19 (1916), 684-703. Pannekoek [2] A. "Some Remarks on the Moon's Diameter and the Eclipse PANNEKOEK, Tables in Babylonian Astronomy," Eudemus, 1 (1941), 9-22. Pannekoek [3] A. "The Origin of the Saros," Koninkl. Akad. van Wetensch.te PANNEKOEK, Amsterdam, Proceedings, 20 (1917), 943-55. B. P., and HUNT,A. S. The OxyrhynchusPapyri. London: Egypt Pap. Oxyrh. GRENFELL, Exploration Fund, 1898 ff. C. H. Catalogueof the Greekand Latin Papyri in Pap. Ryl. HUNT,A. S., and ROBERTS, the John Rylands Library, Manchester. 3 vols. 1911-38. Peet RMP PEET,T. E. The Rhind Mathematical Papyrus. Liverpool, 1923. Pfeiffer SLA PFEIFFER,R. H. State Letters of Assyria. "Amer. Oriental Series," No. 6. New Haven, 1935. Pococke DE R. A Description of the East and Some OtherCountries.2 vols. LonPOCOCKE, don, 1743-45. POGo,A. "Calendars on Coffin Lids from Asyut," Isis, 17 (1932), 6-24. Pogo [1] POGO,A. "The Astronomical Inscriptions on the Coffins of Heny," Isis, 8 Pogo [2] (1932), 7-13. POGO,A. "Three Unpublished Calendars from Asyut," Osiris, 1 (1935), Pogo [3] 500-509. POGO,A. "Der Kalender auf dem Sargdeckel des Idy in Tiibingen," in Pogo [4] Gundel DD, pp. 22-26. POGO,A. "The Astronomical Ceiling-Decoration in the Tomb of Senmut," Pogo [5] Isis, 14 (1930), 301-25. Poleman CIM H. I. A Census of Indic Manuscripts in the United States and POLEMAN, Canada. "Amer. Oriental Series," Vol. 12. 1938. Porter-Moss TB PORTER,B., and Moss, R. L. B. Topographical Bibliography of Ancient Egyptian Hieroglyphic Texts, Reliefs and Paintings. 6 vols. Oxford, 1927-39. Neugebauer [16] Ptolemy CL. PTOLEMAEUS,Opera. I. Syntaxis mathematica, ed. J. L. HEIBERG.2 vols. Leipzig, 1898-1903. German translation by K. MANITIUS.2 vols. Leipzig, 1912-13.

THE HISTORY OF ANCIENT ASTRONOMY: PROBLEMS AND METHODS Rec. trav. Revillout [1] Robbins [1] Rome CPT Roscher GRM Sachs [1] Schaumberger Erg. Schmidt [1] Schmidt [2] Schnabel Ber. Schnabel [1] Schnabel [2] Schott [1] Schott [2] Schoy TLAB Schumacher [1] Sethe ZAA Sethe ZZ Sileiko [1] Struve MPM Thibaut AAM 37 II. Opera astronomica minora, ed. J. L. HEIBERG.Leipzig, 1907. III, 1. Apotelesmatica, ed.'F. BOLLand AE. BOER.Leipzig, 1940. Tetrabiblos, ed. and English trans. by F. E. ROBBINS("Loeb Classical Library" [1940]), and Opera, III, 1. Geographia,ed. Nobbe. Leipzig, 1843. Harmonics. See During. Quellen und Studien zur Geschichteder Mathematik, Astronomie und Physik. Revue d'assyriologie. Recueil de travauxrelatifs a la philologie et a V'archeologie egyptienneset assyriennes. REVILLOUT,E. Melanges sur la metrologie,l'economiepolitique et l'histoire de l'ancienne Egypte avec de nombreuxtextes demotiques,hieroglyphiques,hieratiques ou Grecs inedits ou anterieurmentmal publies. Paris, 1895. ROBBINS,F. E. "A Greco-Egyptian Mathematical -Papyrus," Classical Philol., 18 (1923), 328-33. ROME,A. Commentairesde Pappus et de Theon d'Alexandrie sur l'Almagest. 2 vols. (="Studi e testi," Vols. 54 and 72). Roma, 1931-36. W. H. Ausfihrliches Lexikon d. griechischenu. romischenMytholoROSCHER, gie. Leipzig, 1884-1937. SACHS,A. J. "Some Metrological Problems in Old-Babylonian Mathelmatical Texts" BASOR, 96 (1944), 29-39. See Kugler SSB. 0. "Bestemmelsen af Epoken for Maanens Middelbevaegelse i Bredde hos Hipparch og Ptolemseus," Matematisk Tidsskrift, B (1937), pp. 27-32. SCHMIDT,O. "The Computation of the Length of Daylight in Hindu Astronomy" (to b3 published in Isis in 1945). SCHNABEL, P. Berossos und die babylonisch-hellenistischeLiteratur. Leipzig, 1923. SCHNABEL, P. "Kidenas, Hipparch und die Entdeckung der Prazession," ZA, 37 (1927), 1-60. SCHNABEL,P. "Neue babylonische Planetentafeln," ZA, 35 (1924), 99-112. A. "Das Werden der babylonisch-assyrischen Positionsastronomie SCHOTT, und einige seiner Bedingungen," ZDMG, 88 (1934), 302-37. SCHOTT,A. Review of Gundel HT, in QS, B, 4 (1937), 167-78. SCHOY, C. Die trigonometrischenLehren des persischen Astronomen AbuDlRai.hm Muh. ibn Ahmad al-Btruni. Hannover, 1927. SCHUMACHER, C. J. Untersuchungeniber die ptolemdischeTheorie der unteren Planeten. Mutnster, 1917. SETHE, K. "Die Zeitrechnung der alten Aegypter," Nachr. Ges. Wiss. zu Gittingen, Phil.-hist. Kl., 1919, pp. 287-330; 1920, pp. 28-55, 97-141. SETHE, K. Von Zahlen und Zahlwortenbei den alten Agyptern (= Schriften der Wissenschaftlichen Gesellschaft Strassburg, No. 25). Strassburg, 1916. SILEIKO, V. "Mondlaufprognosen aus der Zeit der ersten babylonischen Dynastie," Comptes-Rendusde l'Academie des Sciences de I'URSS, 1927, B, pp. 125-28. STRUVE,W. W. "Mathematischer Papyrus des staatlichen Museums der sch6nen Kiinste in Moskau," QS, A, 1 (1930). THIBAUT, G. "Astronomie, Astrologie und Mathematik" (art.) in Grundriss d. Indo-Arischen Philologie und Altertumskunde,III, 9 (1899). SCHMIDT,

Thomas GMW Thompson AB Thompson Rep. Thorndike HM Thureau-Dangin SS Thureau-Dangin TMB Thureau-Dangin [1] Thureau-Dangin [2] Tropfke [1] van der Waerden [1] van der Waerden [2] Vettius Valens Virolleaud ACh. Vogel [1] Vogel [2] Vogt [1] Waterman RC Weidner Hdb. Weidner [1] Weidner [2] Weissbach BM Winlock [1] Winlock [2] Winlock EDEB ZA ZDMG Zeuthen KA JOURNAL OF NEAR EASTERN STUDIES THOMAS,I. Selections Illustrating the History of Greek Mathematics. 2 vols. 1939-41. ("Loeb Classical Library.") THOMPSON,R. C. A Catalogueof Late Babylonian Tablets in the Bodleian Library, Oxford. London, 1927. THOMPSON,R. C. The Reports of the Magicians and Astrologers of Niniveh and Babylon. 2 vols. London, 1900. THORNDIKE,L. A History of Magic and Experimental Science. 6 vols. New York, 1923-41. THUREAU-DANGIN, F. Esquisse d'une histoire du systeme sexagesimal. Paris, 1932. THUREAU-DANGIN, F. Textes mathematiquesbabyloniens. Leiden, 1938. THUREAU-DANGIN, F. "Sketch of a History of the Sexagesimal System," Osiris, 7 (1939), 95-141. THUREAU-DANGIN,F. "La Clepsydre chez les Babyloniens," RA, 29 (1932), 133-36. TROPFKE, J. "Archimedes und die Trigonometrie," Archivf. Gesch.d. Math., d. Naturwiss. und d. Technik, 10 (1928), 432-63. VAN DER WAERDEN, B. L. "Die Voraussage von Finsternissen bei den Babyloniern," Berichte d. math. phys. Kl. d. sdchs. Akad. d. Wiss. zu Leipzig, 92 (1940), 107-14. VANDERWAERDEN,B. L. "Zur babylonischen Planetenrechnung," Eudemus, 1 (1941), 23-48. See Kroll VV. VIROLLEAUD,CH. L'Astrologie chaldeenne. 4 vols. Paris, 1908-12. VOGEL, K. "Beitrage zur griechischen Logistik," Sitzungsber. d. Bayerischen Akad. d. Wiss., Math.-nat. Abt., 1936, pp. 357-472. VOGEL, K. "Bemerkungen zu den quadratischen Gleichungen der babylonischen Mathematik," Osiris, 1 (1936), 703-17. VOGT,H. "Der Kalender des Claudius Ptolemaus" (=F. BOLL,Griechische Kalender, V), Sitzungsber. d. HeidelbergerAkad. d. Wiss. Philos.-hist. Kl., 1920, p. 15. WATERMAN,L. Royal Correspondenceof the Assyrian Empire. 4 vols. "Univ. of Michigan Studies: Humanistic Series," Vols. 17-20. Ann Arbor, 1930-36. WEIDNER, E. F. Handbuch der babylonischenAstronomie, I: Der babylonische Fixsternhimmel (=Assyriologische Bibliothek, 23). Leipzig, 1915. [Only 146 pages are published; pp. 147-80 were printed but not published.] WEIDNER, E. F. "Ein babylonisches Kompendium der Himmelskunde," AJSLL, 40 (1924), 186-208. WEIDNER, E. F. "Die astrologische Serie Enuima Anu Enlil," Archiv fiir Orientforschung,14 (1942), 172-95 [to be continued]. WEISSBACH,F. H. Babylonische Miscellen. Leipzig, 1903 (= Wiss. Veroffentl. d. Deutschen Orient-Ges.,4). WINLOCK,H. E. "The Origin of the Egyptian Calendar," Proc. of the Amer. Philos. Soc., 83 (1940), 447-64. WINLOCK, H. E. The Egyptian Expedition, 1925-1927. Section II of the Bulletin of the Metropolitan Museum of Art, 1928, pp. 3-58. WINLOCK,H. E. Excavations at Deir el Bahri. New York, 1942. Zeitschriftfir Assyriologie. Zeitschrift der Deutschen MorgenldndischenGesellschaft. ZEUTHEN, H. G. Die Lehre von den Kegelschnittenim Altertum. Copenhagen,