On some aspects of early greek astronomy

26sa el NSCS BAKES, GC: 2 On Some Aspects of Early Greek Astronomy Author(s): O. Neugebauer Reviewed work(s): Source: Proceedings of the American Philosophical Society, Vol. 116, No. 3 (Jun. 9, 1972), pp. 243-251 Published by: American Philosophical Socicty Stable URL: http://www.jstor.org/stablc/9861 18 Accessed: 21/01/2012 14:30 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www. jstor.org/page/info/about/policies/terms.jsp 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. American Philosophical Society is collaborating with Proceedings of the American Philosophical Society. http://www.jstor.org JSTOR to digitize, preserve and extend access to

Reviewed work(s): Source: Proceedings of the American Philosophical Society, Vol. 116, No. 3 (Jun. 9, 1972), pp. 243-251 Published by: American Philosophical Society Stable URL: http://www.jstor.org/stable/986118 . Accessed: 21/01/2012 14:30 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp 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. American Philosophical Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the American Philosophical Society. http://www.jstor.org

Professor Emeritus of the History of Mathematics, Brown University (Read November 11, 1971) SOMEEIGHTYEARS ago a "Symposium on Cuneiform Studies and the History of Civilization" was held at a meeting of this Society. As one of the speakers on that occasion I presented a paper on "The Survival of Babylonian Methods in the Exact Sciences of Antiquity and Middle Ages" 1 in which I tried to distinguish as far as possible between those areas of ancient astronomy in which Babylonian influence was decisive and those which represent an independent development. In the present paper some sections of this earlier study will be amplified. Its main purpose, however, is methodological. The progress of modern astronomy since Brahe and Kepler is inextricably connected with the ever-increasing accuracy and range of observational techniques and it therefore has seemed plausible to assume that a similar trend existed also in the first phase of astronomical development in the Greek world, that is, in the period from the beginnings in the fifth century B.C. to the crowning achievement, Ptolemy's "Almagest" in the second century A.D. I think that this retrojection of conditions prevailing during the last five centuries into a fundamentally different milieu, two millennia earlier, has resulted in a severe distortion of the actual situation and has deprived us of a better insight into the origin of scientific methods that are difficult enough to reconstruct from our fragmentary sources. Furthermore, since most of the sources in question were made accessible through the industry and philological competence of the classical scholars of the nineteenth century we also have inherited much of their basic attitudes. Classicists during this period were still undisturbed by fields concerned with "Ueberresten von gemischter Art"2 (e.g., Archaeology or Papyrology-not to mention oriental material), and so they could act sicut Deus, scientes bonum et malurn. Thus it was simply taken for granted that "progress" from Eudoxus and Aristotle to Aristarchus and Hipparchus could be measured by the increasing agreement with modern data and methods. Wilamowitz (in 1897) did not hesitate to declare that around 240 B.C.(in the reign of Ptolemy III Euergetes) "man arbeitete auf der Sternwarte Alexandreias an einem Fixsternkataloge"-although there exists no trace of organized observational activity before the Abbasid period. There is no need to multiply such examples of baseless anachronisms; they would easily fill another paper. Instead, I shall make an attempt to describe a drastically different aspect that emerges from fragments of early Greek astronomy, i.e., from the period from Eudoxus (early fourth century B.C.) to Archimedes and Apollonius (i.e., to about I think it is essential for our under- 200 B.C.). standing of this early period to realize, first that its approach to fundamental problems of astronomy is in many respects totally different from what we customarily consider to be "Greek" astronomy, and, secondly, that Greek mathematics and Greek astronomy progressed in quite distinct levels, a distinction which left its effects until deep into the Renaissance. Two more introductory remarks. I shall abstain from giving the bibliographical references and the discussions of details which would be necessary to support statements made in this paper; I hope to do this elsewhere within a wider framework. Secondly, it is not my intention to present a complete picture of what I think we do know about early Greek astronomy. I have only selected certain topics which seem to me particularly revealing for the situation in the formative period of Greek astronomy. But I shall far transgress the traditional chronological and geographical framework of early Greek science simply because I am convinced that many a medieval and non-Greek source gives us important information about hellenistic origins. 1. MEASUREMENT OF TIME The division of the into 24 hours-itself day 1Proc. Amer.Philos.Soc. 107 (1963): p. 528-535. 2 Fr. Aug. Wolf, Museum der Alterthums-Wissenschaft the outcome of a complicated mixture of Egyptian, Babylonian, and Greek components-presents 1 (1807): p. 77. PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY, VOL. 116, NO. 3, JUNE

TABLE 1 hora Month Iet XII II XI III X IV IX V VIII VI VII 3 6 9 17 15 13 11 9 7 11 9 7 5 3 2 17 15 13 11 9 7 itself in two different forms: one, of greater popular appeal in the Mediterranean world, is the division of the time of daylight and night separately into 12 ("seasonal") hours each; the other operates with 24 hours of constant length which agree with the seasonal hours at the equinoxes, therefore called "equinoctial"hours and commonly used in astronomical contexts, though the separation seasonal/equinoctial is by no means equivalent to popular versus scientific usage. Even the same text may have both types simultaneously. The "shadow tables" to be discussed presently concern 12 seasonal hours for each month but they also give the length of daylight for the same months in equinoctial hours-without any distinction in the terminology. It is obviously equinoctial hours that are meant when the calendar page for the month of June in the "Tres belles heures de Notre Dame" says, "Les heurs de la nuit 6 et duiour 18." This "Book of Hours" of the Duke of Berry (about 1400) presents us with a simple pattern for the variation of length of daylight from month to month during the year: the maximum M = 18h in June, the minimum m = 6h in December and a fixed increase or decrease of 2h per month, i.e., a strictly linear variation between nmand M. However, not only does the abrupt change from increase to decrease stand in obvious contradiction to the most elementary experience, but a maximum of 18h corresponding to a geographical latitude of almost 58? (as tabulated, e.g., in the Almagest), correct about half-way between Copenhagen and Stockholm, but surely not in Paris or Bourges. How these data got into a Book of Hours I cannot say. Only the antiquity of the pattern is evident: "linear zig-zag functions" are a characteristic feature in cuneiform astronomical texts of which we have a great variety from the centuries between Artaxerxes and Caesar. But this is by no means the earliest evidence for linear calendaric schemes: a hieratic papyrus, known as [PROC. AMER. PHIL. SOC. the "Cairo Calendar," written in the Ramesside period (twelfth century B.C.) shows exactly the same scheme as the Book of Hours of 1400 A.D., i.e. the same extrema of 18h end 6h and a linear variation with 2h each month. I can only once more admit my inability to explain the origin of the basic parameters. Fortunately arithmetical schemes constitute the leading principle also in a group of texts, the socalled "shadow tables," where we can reach a fairly clear historical understanding. To begin with the European medieval tradition, we have, e.g., a "Horalogium Horarum" of the ninth or tenth century which gives in six pairs information of the following type: "Januarius et december, hora 3 et 9 pedes 17, hora 6 pedes 11." Here "hora" must mean seasonal hour, the 6th always representing noon. The "pedes" measure the length of the shadow of a man standing upright and using his own feet as units of length. The resulting scheme is very simple (table 1). The last noon shadow of 2 feet is an obvious arithmetical error, both with respect to the sequence of the noon shadows and the shadows 3 hours before or after noon that are always 6 feet (= 1 + 2 + 3) longer than the noon shadow. Many more shadow tables are preserved in Greek from the Byzantine period (thirteenth and fourteenth centuries). They lead us to distinguish two major types and to recognize a systematic error in many tables, e.g., in the above-given Latin example (table 1). The arrangement in six pairs of months is wrong: there should be seven entries, one for each extremum alone (e.g., December and June) and five pairs of equidistant months (e.g., I and XI or V and VII). This holds for both types of tables: the one which uses names of months for the entries, the other which does not depend on calendaric conventions but uses the solar positions in the signs of the zodiac. For this second type one has one single entry for Capricorn and Cancer each and five pairs of signs symmetric to the solstices. Another solid group of shadow tables is preserved in Ethiopic codices, all of very recent date (e.g., seventeenth and eighteenth centuries) but undoubtedly copied from much older (presumably Coptic) sources. Except for the adaptation to the Ethiopic calendar and many scribal errors, these tables are closely parallel to the Byzantine ones. This parallelism is further emphasized by two peculiarities. Several tables mention for each month the length of daylight, and these numbers again form a linear zigzag function, now always

VOL. 116, NO. 3, 1972] with M = 15h, m = 9h. Since according to ancient geography M = 15h is characteristic for the "clima" of the Hellespont, and since Byzantine relations with Ethiopia are well attested for the early Middle Ages (e.g., sixth century), a transmission of Byzantine astronomy to Ethiopia seems evident here. A second element of parallelism is found in the textual preambles to the tables, which address a "King Philip." In astronomical context one will think, of course, of Philip Arrhidaeus whose regnal years constitute the basis for the "Era Philip," used, e.g., in the famous "Handy Tables" of Ptolemy and Theon. I have no doubt that both conclusions are wrong. The preserved material amply suffices to restore the original pattern of the shadow tables (cf. table 2). Both the lengths of the noon shadows and the lengths of daylight form arithmetical sequences with difference 1. Since a daylight of 12h is necessary for the equinoxes, such a sequence leads automatically to M = 15h and m = 9h. In other words, these extrema are the consequence of a primitive arithmetical pattern, not the result of observations which we could then utilize to determine the underlying geographical location, e.g., Hellespont or Byzantium. In fact we have good evidence from Hipparchus and from Geminus that the ratio 15:9 was considered by Eudoxus, Aratus, and Attalus as representative for Greece in general. Thus our pattern does not belong to the Byzantine period but originated in early Greek astronomy in Greece, presumably at Athens. This conclusion is supported by another consideration. Shadow tables have nothing to do with a chronological era but they naturally belong to the "parapegmata," i.e. texts which associate the risings and settings of fixed stars in the course of the year with weather conditions-much in the way that our Farmer's Almanacs still do. Ptolemy, among others, wrote a whole treaties on these "Phaseis," and he cites his authorities for all predictions, e.g., "unwholesome air and turbulence according to Callippus, Euctemon and Philip; rain and thunder TABLE 2 IN FEET FOR THE HOURS OF DAYAT SUNRISE LIGHT BEGINNING (NOON = 6h), DEPENDING ON THE SOLAR POSITIONS BETWEEN AND SUMMER WINTER SOLSTICE (6) LENGTH OF SHADOW SOLSTICE(F / 28 18 14 11 9 8 9 11 14 18 28 X T F s9 X 26 16 12 9 7 6 7 9 12 16 26 25 15 11 8 6 5 6 8 11 15 25 24 14 10 7 5 4 5 7 10 14 24 23 13 9 6 4 3 4 6 9 13 23 ; )t hour 1 2 3 4 5 6 7 8 9 10 Ii ) 27 17 13 10 8 7 8 10 13 17 27 22 12 8 5 3 2 3 5 8 12 22 ength of daylight: 15 14 13 12 11 10 9 hours noon shadow: 2 3 4 5 6 7 8 feet Thus it seems fairly certain that the arithmetical patterns of table 2 for the shadow lengths and lengths of daylight originated in early Greek astronomy, and this conclusion is supported by the remark that the zodiacal patterns (which, in view of the Greek lunar calendars, must be the original form) presuppose the Eudoxan norm which places solstices and equinoxes in the middle of the zodiacal signs. Again, Athens seems to be the plausible center for this development. A lucky accident allows us to show that shadow tables of the type discussed here appear already in the Ptolemaic period in Egypt. A papyrus fragment of an astronomical treatise (now in Vienna) has preserved a little corner of such a table, just enough to demonstrate the identity with the pattern of our table 2. Having once established the basic structure of these tables it is no longer difficult to recognize their survival in more or less significant variations all around the Mediterranean medieval world: in an Armenian treatise, in Syriac, in Coptic, in Nubia (a Greek inscription in a temple at Taphis), in North Africa and Spain (in "Anwac "tables), according to Eudoxus. . . ." Finally, Ptolemy gives the list of his authorities from Eudoxus to and in monastic manuscripts of France and EngCaesar and mentions the regions where they ob- land. Incidentally we can now say that the aptained their climatic experiences. Here "Philip" parent arithmetical error 2 in the last line of our is associated with the Peloponnesus, Locris, and table 1 is the only correct residue of the original Phocis, and there is little doubt that we are deal- pattern, whereas all other numbers are adapted ing with Philip of Opus, who flourished in the to the faulty six-pair pattern that replaced the first half of the fourth century B.C. (well known original 1 + 2 5 + 1 scheme. It should be noted, because of his connection with Plato's "Epin- however, that this modification of the ancient Greek scheme is only a misguided arithmetical reomis").

arrangement and is by no means based on any empirical correction for different geographical situations. It is easy to show that the assumption of a linear variation of the length of daylight was a common feature in astronomical and calendaric treatises of the early Hellenistic period. The calendar of P. Hibeh 27 (about 300 B.C.) gives the length of daylight day by day, increasing linearly during 180 days, decreasing in the same fashion for 180 days, and simply kept constant for 3 days at M = 14h and for 2 days at .m = 10h (cf. fig. 1); the ratio 14:10 is the norm characteristic for "Lower Egypt" (Alexandria) in ancient geography. The multiples of the daily increment of 4h/180 = 1lh/45 are given for each day on the basis of the clumsy Egyptian rules for operations with unit fractions. Similar linear schemes are found throughout antiquity, e.g., with Porphyry (around A.D. 300) for M = 15h, ,m = 9h1. Strictly linear patterns are not the only arithmetical devices of early astronomy. The shadow tables, e.g., have for all months the same increase of shadow lengths per hour before or after noon (cf. table 2) 1, 2, 3, 4, 10( =1+2+3+4). [PROC. AMER. PHIL. SOC. Obviously this is an attempt to describe by a simple numerical pattern the rapid increase of shadow lengths toward sunrise and sunset. The pattern is a difference sequence of the second order (or nearly so) and it is clear that it is only arithemetical expediency that determined these numbers, not any set of actual measurements, however crude. Much more sophisticated arithmetical methods appear on the scene with closer contact with Babylonian astronomy during the Seleucid-Parthian period. About the details of this contact we know very little except for the evidence of fundamental Babylonian parameters in Hipparchus's lunar theory. This does not exclude earlier contacts, which are indeed suggested by the use of sexagesimal units about a century earlier (Eratosthenes). To the period of earlier borrowing probably also belong two types of arithmetical schemes (denoted as "System A" and "System B") which produce a quite satisfactory representation of the actual variation of the length of daylight during the year. Instead of assuming a linear variation of the length of daylight with its abrupt changes at the extrema, the underlying rising times of the subsequent zodiacal signs are brought into an arithmetical pattern, linear in "System A," with double difference at the equinoxes in System B. In this way the lengths of daylight become a kind of difference sequence of second order (as the sum of six consecutive rising times) with smooth approach to the extrema. Nevertheless all computations follow very simple patterns which are completely determined by the value of the longest daylight and the choice of the system, A or B. Transplanted to Alexandria, these patterns had TABLE

VOL. 116, NO. 3, 1972] a profound influence on Greek and medieval geography. Babylonian astronomy itself had never introduced (at least so far as we know) any element of geographical variation. The Babylonian schemes for the length of daylight are always based on the ratio 3:2, i.e., on the extrema M = 216? = 14;24h and m = 144? = 9;36h. This parameter appears also in Greek geography as characteristic for "Syria" (cf. table 3) but the same pattern is expanded to seven (why seven?) exactly similar schemes such that the determining parameters M form a linear sequence of constant difference 4? = 0;16h. For each "clima," as these geographical steps were called, the computational method follows "System B" (incidentally, also the system to which the Hipparchian lunar parameters belong). A similar sequence of seven climata, but now centered in Alexandria and based on System A is first attested in a little treatise by Hypsicles (around 150 B.C.) with M = 14h as the point of departure (cf. table 3, middle section). The constant difference is the same as before. Both sequences are frequently found in Greek astrological literature and in the famous work of the poet Manilius who had pretentions to astronomical competence but mixed data from Babylon and System A with elements for Rhodes and System B. A radical departure from these arithmetical patterns occurred after the invention of spherical trigonometry by Menelaos (around A.D. 100) which made it possible to compute from correct trigonometric relations the variation of the length of daylight for any geographical latitude. This astronomically correct theory is then again applied to a set of "seven climata," defined by a linear progression of longest daylight with constant difference 1/2h, beginning at M = 13h (Meroe in Nubia) and ending at M = 16h (Borysthenes = Dnjeper). The choice of M, instead of the geographical latitude, as characteristic parameter shows the strength of the arithmetical tradition. This does not mean that the concept of geographical latitude (or altitude of the pole) did not play a role in Greek astronomy or geography. On the contrary, one can even get the impression that at an early time the latitude of Rhodes, <p= 36?, played a special role-often explained by the prominence of Hipparchus or Posidonius. But again numerology undermines such arguments. One of the earliest symptoms of Babylonian influence seems to be a sexagesimal division of the circumference of the circle-not into 360 degrees of the later standard, but simply a division into 60 parts such that a quadrant contains 15 parts. 247 FIG. 2. Luckily for such speculations, the obliquity of the ecliptic can be described fairly enough as representing the side of a regular polygon of 15 sides (i.e., = 24?), hence covering exactly 4P. The remaining lip of the quadrant can then be divided into 5P plus 6P with the "arctic circle" as boundary (cf. fig. 2), producing a neat numerical pattern with 4P, 5P, 6P for each quadrant. This, however, assigns a fixed position to the arctic circle, that is to say, a distance of 6P = 36? away from the pole and hence creating a situation demanding p = 36?. Hence it seems at least conceivable that it is not a "school" or "observatory" which accounts for the prominence of this latitude but a cosmologic doctrine of "pythagorean" flavor. 2. COSMICDIMENSIONS From the period of early Greek astronomy we have three outstanding attempts at bringing cosmic motions and cosmic dimensions under the control of mathematical methods: the "homocentric spheres" of Eudoxus (and their modification by Callippus and Aristotle), Aristarchus's treatise on the sizes and distances of sun and moon, and some very strange schemes for planetary distances by Archimedes, as well as his "Sand-reckoner." That the last-mentioned work is a work of mathematics and not of astronomy is obvious: in it the "universe" is of interest only in so far as its enormous but definite size furnishes the concrete space which nevertheless can be measured in accurately defined numerical terms. I consider it of vital importance for our understanding also of the

other above-mentioned works to realize that their real goal is to demonstrate the power of the mathematical approach, not the solution of some specific astronomical problems. The conceptual beauty and mathematical elegance of the Eudoxan homocentric spheres are undeniable, but equally evident is their inability to explain obvious details in the observable planetary motions. I think it is in vain that modern scholars have tried to reconstruct numerical parameters to be substituted in order to make the Eudoxan model represent the planetary motions with some semblance of truth. I think such numerical data never existed. And I confess that I consider it quite possible that exactly the same thing could still be said with respect to the sophisticated investigations of epicyclic and eccentric motions by Apollonius some 150 years later. The same general attitude explains the much discussed disrespect of Aristarchus for observational data. We know from Archimedes that Aristarchus was aware of the fact that the apparent diameter of the sun and moon is about 1/2?; nevertheless, in his treatise he computes with a value of 2?, thus avoiding trigonometric difficulties with small angles. The basic mathematical idea to determine the ratio of lunar to solar distance is very neat (cf. fig. 3): if one knows the angle r/ of elongation between sun and moon when the latter appears exactly half illuminated, then one has only to solve one right triangle to obtain the desired ratio. Astronomically the method is totally impracticable: the moment of dichotomy cannot be determined with any accuracy, and the value of r is so near 90? that the difference falls below the limit of accuracy of visual observation-Aristarchus's value of 87? is purely fictitious. What really interests him in this treatise is, on the one hand, the trigonometry of the problem (at his time an undeveloped topic) and, secondly, an accurate mathematical investigation of a question which is again without any practical importance: how far is the terminator between light and darkness removed from the great circle which contains the radius moon-earth FIG. 3. [PROC. AMER. PHIL. SOC. 0 0 FIG. 4. (ME in fig. 3). The answer that a minute eccentricity of the terminator cannot materially affect the ratio of the distance is, of course, obvious from the very beginning. But it is the rigorous mathematical aspects of the problem and not observational techniques which concern Aristarchus.3 And his treatise ends without giving actual distances and sizes. To determine the distance of sun and moon Hipparchus eventually introduced a method which is based on geometric conditions that prevail at a lunar eclipse and which are well suited for observational refinements. None of these methods were applicable to planetary distances, and ancient astronomers remained entangled in philosophical doctrines which eventually led Ptolemy to the model of nested spheres that dominated Islamic and western astronomy during the Middle Ages. In a completely different direction, however, lies a system of planetary distances, proposed by Archimedes, that escapes our understanding so far as the underlying principle is concerned. It is a very peculiar accident which has preserved for us a record of this Archimedean theory. Hippolytus, in the first half of the third century, 3 The figure which belongs to the discussion of the terminator had been garbled already in antiquity, a fact not realized by Pappus (early fourth century A.D.) or by the modern editor, Sir Thomas Heath (1913). Actually two cases were condensed into one figure: one, the earth E at quadrature(cf. fig. 4), the other E0oin the axis moon-sun in order to find the smallest terminator TU, assuming equal apparent diameter of sun and moon. The problem is to find how big LG < GU appears from E. Pappus did not realize that EM must be perpendicular to EoM and he therefore discusses the apparent size of GU as seen from Eo, which is absurd because this arc is invisible from Eo.

Plato. We must, alas, admit that we do not understand how Archimedes, the greatest scientist of antiquity, came to these numbers. TABLE 4 B A From earth to 249 ASPECTS OF EARLY GREEK ASTRONOMY VOL. 116, NO. 3, 1972] Stadia taearth Moon Sun Venus Mercury 5544130 = la + 2d 55816195 = lla + 3d 76088260 = 15a + 4d 126904455 = 25a + 7d Mars Jupiter Saturn Zodiac 167448585 = 33a + 9d 187720650 = 37a + 10d 227992715 = 45a + lid 248264780 = 49a + 12d From to 3. TRIGONOMETRY Stadia Venus Mercury Sun 50815160 52688256 121604451 Mars Jupiter Saturn 132418581 202770646 222692711 a = 5-106 d = 272065 Moon to Mercury = Mercury to Zodiac = 121360325 = 24a + 5d Mercury to Saturn = Sun to Saturn = 101088260 was perhaps the last Greek-educated bishop of Rome. He was of an uncompromising fighting spirit, not only directed against his more worldly competitors for the episcopal throne but also against heresies about which he composed a still extant treatise. Though not a heresy by itself, astronomy is full of speculations about the structure of the world and is thus potentially a preparation for heretical ideas. As an example of such useless speculations he gives the list of two sets of distances assumed by Archimedes (cf. table 4), unfortunately without any further details. All that we can say is derived from the bare numbers. For no apparent reason we find two different arrangements: in A the sun is nearer to the earth than Venus and Mercury, in B it is located beyond these two planets. In the first scheme, Mercury is exactly at the midpoint between moon and zodiac (cf. fig. 5) and the distance Mercury-Saturn in A is exactly the same as the distance sun-Saturn in B. But the most surprising feature is the structure of the numbers in A. Each one of them is a linear combination of a multiple of a = 5000000 and of d = 272065 (cf. table 4). Why these numbers were chosen and why they were combined in this peculiar fashion is a complete mystery. All that the pious bishop has to say is that Archimedes should have taken numbers that satisfy harmonies postulated by Aristarchus's discussion of the solution of one right triangle demonstrates the absence of a systematic trigonometry in the time before Hipparchus. It is obviously nonsense that Pliny ascribes to Hipparchus a work on chords in 12 books. The total of ancient trigonometry, plane and spherical, with proofs, tables, and applications, requires only two of the 13 books of the whole Almagest. How Pliny's error originated I do not know; at any rate it is not worth the effort to rescue this passage in Pliny by emendations. It is important, however, not to be misled by Pliny into the assumption of a very developed stage of trigonometric tables in the time of Hipparchus, comparable, say, to the table of chords in the Almagest computed in steps of 1/2?. On the contrary, I think we have good evidence that suggests a rather crude set of tables as the core of Hipparchus's trigonometry. That Hipparchus had no spherical trigonometry in the proper sense is certain, simply because it did not become clear before Menelaus (in the first century A.D.) that spherical triangles must be made up exclusively of great circle arcs. This does not exclude, however, the possibility of solving certain problems of spherical astronomy by means of the so-called "Analemma" methods, to which also Ptolemy devoted an elegant treatise. The basic idea can best be described as using "descriptive geometry" for the transformation of three-dimensional configurations to two-dimensional problems which then only require plane trigonometry for their solution. These methods fortunately do not depend on the distinction between great circles and parallels which have such a natural place in spherical astronomy. The theory of sundials (as preserved, e.g., by Vitruvius) also makes use of the analemma methods. Though direct proof is lacking, it seems very likely that Hipparchus knew (if indeed he did not invent) these methods; it is certainly not accidental that analemma methods play a vital role also in Indian astronomy. A ?([ o 9 . iI I I. ... d' 4 l . . H- -I II II B *

TABLE 5 Hipparchus Ptolemy M a lAa 16h 9 cubits pO 18? 48;32? 17;37? 12 54;1 12;8 8 58 8;9 6 61 5;9 3 cub. = 6? 17 6 18 4 19 3 - E a a =-- Aa 5;30? 2 cub. = 4? 4 1 cub. = 2? 3 In dealing with problems of spherical astronomy Hipparchus was not restricted to the Analemma; we know from data transmitted by Strabo that he also made use of arithmetical patterns that are characteristic for Babylonian astronomy. The problem in question concerns the altitude the sun can reach at the winter solstice at given geographical latitudes. The correct solution has been given by Ptolemy and the results are tabulated in the Almagest (cf. table 5, right half). Hipparchus also answered the question for latitudes where the longest daylight is 16'hto 19h. The corresponding solar altitudes are given in "cubits," an angular measurement of 2? well attested as a Babylonian norm (cf. table 5, left). His answer agrees fairly well with the correct data, but the basis is obviously a simple sequence of second order with 1, 2, 3 as differences. What Hipparchus's plane trigonometry looked like is not to be judged from Pliny but from Indian astronomy whose dependence on Hellenistic (and thus Babylonian) prototypes has long been recognized. Our insight into this process of transmission has moved one important step forward with the proper interpretation of two Greek papyri concerned with the motion of the moon: P. Lund Inv. 35a from the time of Nero-Domitian (A.D. 60 to 84) and P. Ryl. 27 written around A.D. 250. The methods displayed in these papyri are based on Babylonian parameters and procedures; their exact counterparts had been found two centuries ago by LeGentil in South India where he attempted to observe the Venus transits of 1761 and 1769 from the French colony at Pondicherry. He missed the transits (first because of war and then because of clouds), but he succeeded in getting native Tamil scholars to compute for him the circumstances of a lunar eclipse using rules which Cassini eventually was able to explain and which are, as we now know, identical with the [PROC. AMER. PHIL. SOC. methods in the above-mentioned papyri. It is certainly no accident that a site near Pondicherry, Arikamedu, was a center of Roman trade in the early imperial period. No doubt the interest in astrology, a discipline brought to its highest development in hellenistic Egypt, was the vehicle that transmitted also the strictly astronomical techniques to India. It is not within the scope of this paper to describe how much we owe to Indian astronomy (e.g. through Varahamihira's Pancasiddhantika) for our knowledge of Babylonian and Greek astronomy of the hellenistic period. I shall only draw attention to some features of Indian trigonometry which, I think, reflect very accurately the type of Hipparchus's trigonometry. In the only work of Hipparchus that has come down to us, his Commentary on Aratus (more a sharp critique than a "Commentary"), he uses peculiar spherical coordinates- arcs on the ecliptic in combination with arcs on circles of declination,4 a system also used in Indian astronomy. As units he repeatedly uses "signs," i.e., twelfths of the circumference regardless of whether these arcs lie on the ecliptic or not. Again the same terminology is found in India. Another peculiar terminology is revealed to us by Theon's Commentary to the Handy Tables (fourth century A.D.) according to which 15?sections, i.e., 24ths of the circumference, are called "steps" (ftaOtko'), in particular in relation to lunar latitudes and solar declinations. Once alerted to this concept, one finds many instances of its application from Roman Egypt to the late Byzantine period. It is particularly significant in the present context that the above-mentioned P. Ryl. 27 uses this concept in relation to the lunar argument of latitude. These "steps" also appear in the astrological literature, e.g., in Vettius Valens (second century A.D.) and in relation to planetary latitudes. An apparently very old connection exists with meteorology (i.e., to the same background from which the parapegmata and shadow tables come) since the steps of the four quadrants are related to wind directions. Finally, we find in the geographical literature, e.g., in fragments from Eratosthenes, in Posidonius, and in Geminus, a 48-division of the circle, resulting in "parts" (;pp) of 7;30?. All these units are more or less interwoven in our sources 4 This peculiar system could well have been suggested by its convenience for stereographic projection which is fundamentalfor the "astrolabe";but explicit evidence for Hipparchus is still missing.

VOL. 116, NO. 3, 1972] and represent together a simple sequence of arcs: 30?, 15 , and 7;30?. We can now turn to Indian trigonometry. In India the great step was made of changing the Greek "chords" to "half-chords" i.e., to sines in modern terminology: 1 - crd a = sin tr . Applying this transformation to the previously mentioned units of signs, steps, and parts, we obtain arcs that are multiples of 3;45?: this is exactly the norm in the Indian tables.5 I have very little doubt that these tables are nothing but the transformation of the Hipparchian table of chords to a table of sines. Through the intermediary of Islamic astronomy these Indian tables did reach medieval western Europe, in particular Spain and England, under the name of "kardaga," a crude rendering of a Sanskrit term meaning "half-chord." In this circuitous way, Hipparchian trigonometry returned to a part of the world from which it had originated well over a millennium before. A concluding remark must be added concerning the practical use of tables for a trigonometric function, computed only for a few values of the argument, 3 ;45? apart for the sines, 7;30? for the chords (24 values in all). Here again the Tamil computers testify to the existence of a rational procedure. Their table of lunar latitudes progressing in steps of integer degrees of the argument of latitude is simply found by linear interpolation between the values computed for the multiples of 3;45? as argument. Once more we see at the very end of the "early" period of ancient astronomy the all-pervading convenience of the linear function. And it seems to me of methodological interest that Tamil astronomers of the eighteenth century can provide us indirectly with better information about early Greek astronomy than does Pliny's encyclopedia. 4. EPILOGUE That we could use sources from the Roman imperial period until deep into the Middle Ages is due to more than a lucky accident of preservation of antiquated material. The extreme simplicity of the arithmetical methods, their entanglement with 5 I know of no proof for the commonlyaccepted explanation that 3;45? = 225' was chosen because it is the largest arc for which sin a = a numerically. Obviously this implies additional assumptions concerning the length of the radius and the accuracy of the tables. 251 a priori speculations of numerological character (politely called "pythagorean"), the lack of observational accuracy, all this makes early Greek astronomy ideally suited to the mental climate of the Middle Ages. Our results imply a strong warning against periodization of cultural history. Contemporary with the haphazard steps of an elementary astronomy and mathematical geography are the most brilliant achievements of Greek abstract mathematical thought. The theory of irrational quantities by Theaetetus and Eudoxus, Aristotelian logic, Euclid, Archimedes's integrations, and Apollonius's conic sections are examples of mathematical structures whose significance was fully recovered only in modern times. Of the astronomy of the same period nothing remained that could be incorporated into the new kinematic astronomy that emerged from the work of Apollonius and Hipparchus in the second century B.C. While mathematics had clearly passed its peak, astronomy progressed to become an exact science of splendid methodology, including observational techniques, numerical and graphical methods, and theoretical optics. If one had only to rely on methodology, one would date Peurbach, Regiomontanus, Brahe, and Kepler in the century following Ptolemy. In fact it took two centuries to produce the competent mediocrity of Pappus and Theon who turned Ptolemy's work into a segment of higher scientific education, not much less sterile than the tradition followed by astrological practitioners who preserved for us so much of early Greek astronomy. In our discussion, we have repeatedly referred to the background formed by Babylonian astronomy, whose influence is evident in the use of the sexagesimal system or in the basic procedures of arithmetical methods. Nevertheless, I think that one has to concede to early Greek astronomy a good measure of independence, in particular so far as geometrical considerations are concerned and also with respect to the trend to numerological speculation. How far the Greeks ever reached a detailed understanding of the refined Babylonian methods for the computation of lunar and planetary ephemerides is difficult to say. We only know of the use of basic Babylonian parameters by Hipparchus, and one may also conjecture that the idea of the tabulation of numerical material is an important borrowing from Mesopotamia. But the roads soon parted when geometric models became the basis of Greek astronomy which made possible a physical interpretation some 1,500 years later.