Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime

Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length ofDaytimeAuthor(s): Alan C. Bowen and Bernard R. GoldsteinSource: Proceedings of the American Philosophical Society, Vol. 135, No. 2 (Jun., 1991), pp.233-254Published by: American Philosophical SocietyStable URL: http://www.jstor.org/stable/987033 .

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Hipparchus' Treatment of Early Greek Astronomy

The Case of Eudoxus and the Length of Daytime

ALAN C. BOWEN* and BERNARD R. GOLDSTEIN**

(ince Hipparchus (d. after -126:1 cf. Toomer 1978, 207-208) is an _ important source for Ptolemy's reports about the history of astron- J omy in the Almagest (ca. 150), it is worthwhile to investigate what

is known of Hipparchus independently of Ptolemy in order to determine if he is reliable as a witness or, to put it more constructively, to discover what kind of historical framework Hipparchus provided Ptolemy. Ideally, one should like to have all the works of Hipparchus that Ptolemy actually used, for this would enable one also to determine Ptolemy's role in the transmission of Hipparchus' remarks about his predecessors. Unfortunately, the only surviving complete treatise by Hipparchus is his extensive commentary on the works entitled Phaenomena by Aratus (ca. -314 to -238) and Eudoxus (-389 to -336: cf. de Santillana 1940), which Ptolemy neither cites nor, apparently, uses in the Almagest. Still, this commentary is sufficient, we think, to permit an assessment of Hipparchus' treatment of earlier Greek astronomy and to give a sense of the best that Ptolemy could do with Hipparchus' reports about others. Such an assessment would, of course, require a complete study of Hipparchus' commentary and thus lies beyond the scope of this paper. Accordingly, in what follows, we examine a test case - Hipparchus' ascription to Eudoxus of two different values, 5:3 and 12:7, for the ratio of the arc of the summer solstitial day-circle above the horizon to the arc below [In Arat. i 2.22, 3.5-10: see ? 6, below].

Our interest in this particular case, however, is not limited to the ques-

* Institute for Research in Classical Philosophy and Science, 1314 Browning Road, Pitts- burgh PA 15206; and Departments of Classics and of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA 15260.

** Department of History and Philosophy of Science, University of Pittsburgh, Pitts- burgh PA 15260; and Institute for Research in Classical Philosophy and Science, 1314 Browning Road, Pittsburgh PA 15206.

1 In this paper, we adopt the following standard convention: 1 BC is year 0; years prior to 1 BC are written as negative numbers and years AD, as positive numbers [see Neugebauer 1975, 1061-1062]. PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY, VOL. 135, NO. 2, 1991

233



234 BOWEN & GOLDSTEIN

tion of Hipparchus' reliability as a reporter. We are persuaded that the earliest examples of observations and precise measurements in Greek astronomy are those ascribed to Timocharis (beginning in -294) in Ptolemy's Almagest vii 3 and x 4.2 Previously [Bowen and Goldstein 1988], we have argued that the claims made on behalf of Meton that he made a precise astronomical measurement or observation are based on a misunderstanding of the ancient testimony. In this paper, we shall argue that similar claims made on behalf of Eudoxus are unsupported as well.3

Now the ratio of the arc of the summer solstitial day-circle above the horizon to the arc below is the ratio of longest daytime to shortest nighttime [see ? 6 ad [5], below].4 The two values for this ratio, 5:3 and 12:7, are close to one another but their relationship is not clear. The ratio 5:3 (or its equivalent, 15h:9h) was widely used in antiquity for the "region of Greece" [cf. Neugebauer 1975, 581, 711n26, 747]; but the ratio 12:7 is not found elsewhere for this purpose, so far as we know. In fact, Neugebauer points out that the ratio 12:7 would require that the day be divided into 19 parts, and that this is awkward, to say the least, if the day is divided into 24 hours or 3600; hence, he queries the text:

Hipparchus tells us that Eudoxus applied in the "Phaenomena" the ratio M:m = 12:7, presumably for a certain region in Greece, after having used 5:3 (i.e., M = 15h) in the "Enoptron" for Greece in general. The ratio M:m = 12:7 can hardly be correct, however, because it cannot be expressed in units of hours or degrees. It is tempting to emend the ratio to 11:7.... [Neugebauer 1975, 733n28]

Others avoid the problem Neugebauer discerns and take the Eudoxan values for the ratio of the arc on the summer solstitial day-circle above the horizon to the arc below, i.e., for the ratio of longest daytime to

2 See Goldstein and Bowen 1989, 1991. 3 We are aware of claims that other Greeks before Timocharis made precise observa-

tions, but we do not find the evidence convincing. For discussion of the observations attributed to Pytheas, see Goldstein 1983, 10-12; for those attributed to Eratosthenes, see Gold- stein 1983 and 1984. Many scholarly claims of precision in ancient observations do not bear close scrutiny. Thus, e.g., Aujac [1979, 12 and n2] states that Pytheas was known for the accuracy of his observations; and, citing Hipparchus [In Arat. i 4.1, not IV, 1], adds that Pytheas knew better than Eudoxus the exact position of the celestial north pole. But Hip- parchus simply affirms that Eudoxus was mistaken in thinking that there was a star at this pole; and then states that "there is an empty region, nearby which lie three stars with which the point at the pole contains a figure that is almost square, just as Pytheas of Marseilles also says." Hipparchus says nothing about why Eudoxus and Pytheas make the claims they do and he does not compare them as observers.

Nothing is known of Pytheas' life and only little more of his achievements. It is customary to put him in the generation after the death of Aristotle (-321) because, though Aris- totle did not know of him, Dicaearchus, Aristotle's pupil, and Eratosthenes (ca. -274 to -193) did [Strabo, Geog. ii 4.1-2].

4 Daytime (poppa) is the period from sunrise to sunset; nighttime (vi5E,), the period from sunset to sunrise. For the texts discussed in this paper, a day (p~topa) is the period from one sunset to the next sunset or from one sunrise to the next sunrise, or the period of one revolution of the celestial sphere.



HIPPARCHUS' TREATMENT OF GREEK ASTRONOMY 235

shortest nighttime, to be the results of precise measurements or observations [cf., e.g., Huxley 1963; Dicks 1970, 154-155].5

In this paper, we will argue against both alternatives. Our procedure is as follows. After a brief account [? 1] of the various ratios of longest daytime to shortest nighttime used by the ancients to specify terrestrial location, we demonstrate [? 2] how the ratios 12:7 and 5:3 may have been derived using only arithmetical techniques available at the time of Eudoxus. This demonstration is our answer to the question, How else might Eudoxus have found these ratios, if not by observation and measurement? Next [? 3], we turn to the question, Should the ratio 12:7 be emended because it is not expressible in units of hours or degrees?, and show that the division of the day into equinoctial hours or into 3600 of time was unknown in Greece at the time of Eudoxus. Then, we focus [? 4] on the related issue of the division of the circle into 3600 of arc and address the question of its earliest occurrence in Greek astronomical texts. It is here that we examine Hipparchus' testimony and demonstrate that many of Hipparchus' reports and criticisms concerning Eudoxus and Aratus are, on occasion, anachronistic and even polemical. In short, on the strength of this test case, we propose that Hipparchus does not describe faithfully the state of astronomy before his time but "modern- ized" it, thereby providing Ptolemy with an astronomical history that was in some respects inaccurate and distorted. Next, we elaborate [? 5] our conclusion that 12:7 does not need to be emended and, moreover, that the ratios 5:3 and 12:7 can be obtained without the aid of precise measurements or observations. Finally, as a convenience to the reader, we append [? 6] an annotated translation of the relevant passage from Hip- parchus' commentary [In Arat. i 3.5-10].

? 1. VALUES FOR LONGEST DAYTIME IN BABYLONIAN AND EARLY GREEK SOURCES

In Papyrus Hibeh 27 (a Greek text composed ca. -300 in Egypt), longest daytime is given as 14 hours and shortest daytime is 10 hours; hence, we have the ratio of 14h: 10h or 7:5 presumably for Sais, which was southwest of Alexandria in the Nile delta. This ratio is also given for Alexandria in the Anaphoricus 63-66 by Hypsicles (mid-second century BC), who is roughly contemporary with Hipparchus [cf. Neugebauer 1975, 306n34, 715]; and in Ptolemy's Almagest ii 6 we find that 14 hours is the longest daytime for lower Egypt [see Toomer 1984, 85]. For the

5 Hipparchus may himself have understood these ratios to be based on observations: see In Arat. i 1.8. According to Strabo [Geog. ii 5.14], Posidonius (ca. -134 to ca. -50) mentions an observatory (CSKonrv) of Eudoxus in Cnidus, from which it is also said (by others) that Eudoxus observed Canopus. Strabo [Geog. xvii 1.30] also writes of the observatories (oKoncq) of Eudoxus in Heliopolis. Such claims are very hard to assess: for while it is possible that they are factually sound, it is more likely that they wrongly retroject the increasing Hellenistic interest in observation to an earlier age and combine this with a subtle apolo- getic for the importance and centrality of Egypt in intellectual matters.




Babylonians, the standard value for the ratio of longest daytime to shortest nighttime was 3:2 [cf. Neugebauer 1975, 366]. Moreover, in some Baby- lonian texts such as a report to the king dating perhaps from -648 [Reiner and Pingree 1974-1977],6 we find the ratio 2:1.7

Associated with (a) the length of longest daytime (given either absolutely or in relation to the shortest nighttime or daytime) are two other methods for indicating geographical latitude that are mentioned as early as Hipparchus, namely, (b) the ratio of gnomon to shadow-length at equinox or solstice, and (c) the clima (i.e., the inclination of the celestial north pole to the local horizon) measured in degrees.8 To establish equivalent values in the three systems requires mathematical argument. The problem is solved in the Almagest by spherical trigonometry using the theorem of Menelaus who was active ca. 100. It has been argued that Hipparchus may well have treated such problems by means of an ana- lemma construction; if he did, he was very probably the first-there is no evidence that this problem had been solved by anyone before him [see Goldstein 1983, 6; Neugebauer 1975, 747n5: cf. ? 6 ad [6], below].

For the ratio of gnomon to equinoctial shadow-length, we have 4:3 for "the regions of Greece," and it is said to correspond to longest daytime of 143/5 equinoctial hours or to a lima of 370, almost exactly [Hipparchus, In Arat. i 3.6-7: see ? 6 ad [6], below; Neugebauer 1975, 746]. Other integer values for the ratio of gnomon to equinoctial shadow-length are to be found in Vitruvius, De arch. ix 7 (late first century BC) and Pliny, Hist. nat.

6 We have not consulted the Babylonian documents cited directly, but have relied on scholarly publications and translations of these texts.

7 See also the "astrolabe" texts that are discussed in Rochberg-Halton 1989. The same ratio appears in Mul.Apin, a text composed before -700 [Hunger and Pingree 1989, 9, 10-12] and roughly contemporary with the "astrolabes." In Mul.Apin, though this ratio is clearly a ratio of the weights of water used to measure out the lengths of one daywatch and one nightwatch respectively on the day of summer solstice, it is also taken as a ratio of the lengths of the watches themselves elsewhere in the same text [Hunger and Pingree 1989, 154]. Neugebauer [1947] proposed that the ratio 2:1 for the weights of water is a reasonable approximation of a ratio of 3:2 for the times so measured, if the water is released from the bottom of a cylindrical water-clock.

8 On clima, see ? 6 ad [5], below. The value of the lima is accordingly that of the geographical latitude. Yet, the lima and the latitude of a locality, though equivalent, are not identical concepts. Ptolemy [Alm. ii 6] indicates that the lima of a place is the same as the arc on the meridian from its zenith to the celestial equator and that this is in turn the same as the arc on a meridian from the terrestrial equator to the parallel circle (tcapiXXiiXoq) through the place. Note that in standard usage [cf., e.g., Strabo, Geog. ii 5.7, 5.16, 5.39-40], the distance to the parallel from the equator is given in stades, whereas for Ptolemy it is an arc given in degrees. In a work later than the Almagest, Ptolemy [Geog. i 6.4] actually uses the terms PtEKOE (longitude) and 7tkcioq (latitude) to designate, respectively, east-west and north-south distances on (a map of) the Earth [cf. Neugebauer 1975, 934], the same terms that he used in the Almagest for celestial coordinates.

Hipparchus not only uses degrees in his commentary to measure angular distance, he also uses the cubit (I tXuq) and the half-cubit (iltciur~lov). Further, in his (lost) book on geography, Adversus Eratosthenem [cf. Strabo, Geog. i 1.12], Hipparchus apparently used the cubit to measure the altitude of the Sun [cf. Strabo, Geog. ii 1.18]. According to Neugebauer [1975, 591-592], the cubit in this work is equivalent to 2?. In his translation of Hipparchus' commentary, Manitius renders mtXuq by "Mondbreite" ("Moonbreadth") which Neuge- bauer [1975, 592n20] finds perverse-the apparent breadth of the Moon is 1/20.




TABLE 1. Length of Daytime and Geographical Location

Ratio of Longest Daytime to Shortest Geographical

Nighttime Location Occurrences in Sample Greek Texts

1:1 Equator cf. Geminus, Int. ast. 5.6 (1st century AD)a

3:2 Babylon cf. Pliny, Hist. nat. vi 213 5:3 J Hellespontb cf. Hipparchus, In Arat. i 3.7

t Cyzicus, Dardanelles cf. Pliny, Hist. nat. vi 216 12:7 cf. Hipparchus, In Arat. i 3.7 7:4 Byzantium cf. Strabo, Geog. ii 5.41; note 9, above 2:1 Borysthenes cf. Strabo, Geog. ii 5.42; Pliny, Hist. nat. vi 216

a On Geminus' date, see Neugebauer 1975, 579-581. b See ? 6 ad [7], below.

ii 182, vi 211-218 [cf. Neugebauer 1975, 747f]. Such ratios, and the corresponding lengths of longest daytime in equinoctial hours, may well have been traditional values derived schematically rather than on the basis of precise measurements at the places associated with them.9

? 2. EUDOXUS AND THE RATIOS 5:3 AND 12:7

We claim that the ratios of longest daytime to shortest nighttime 2:1, 7:4, 12:7, 5:3, 3:2, 1:1-all of which are attested or implied in ancient texts [see Table 1]-lie in a single series and that they can be derived without recourse to precise measurements or observations. To establish this we depend on a theorem underlying Parmenides 154bl-d3 by Plato (-426 to -346) and proven in Collectio vii prop. 8 [Hultsch 1876-1878, ii 688-691: cf. Heath 1897, xc; Fowler 1987, 42-44] by Pappus (fourth century AD). According to this theorem, which was probably used by Eratosthenes, Aristarchus, and other early Greek astronomers [cf. Goldstein 1983, 8; 1984, 415],

if a:b > c:d, then a:b > (a+c):(b+d) > c:d.

If we begin with the ratios 2:1 (which is equivalent to the ratio of 16h:8h for longest daytime to shortest nighttime, assigned to some place far

9 Outside the commentary on the Phaenomena, there is a passage, Strabo, Geog. i 4.4 [cf. ii 5.8], where Hipparchus is cited in connection with the determination of the clima of Byzantium. Later in Strabo, Geog. ii 5.41, we are told that "at Byzantium and the regions thereabouts the longest daytime is 15 1/4 equinoctial hours and the ratio of the gnomon to shadow-length at summer solstice is that of 120 to 42 minus V5." It has been suggested [Goldstein 1983, 10-12] that this ratio for the gnomon to shadow-length has been derived (rather than observed) either from the value given for longest daytime, 15 1/4 hours, or from a ratio of 7:4 for longest to shortest daytime. In the latter case, where the value for longest daytime is 15;16,22h, Hipparchus may have rounded this value to the nearest quarter hour [for Hipparchus' use of a geographical scheme based on quarter-hour differences in longest daytime, cf. Neugebauer 1975, 731].




north of the equator)10 and 1:1 (which is equivalent to the ratio of 12h:12h for longest daytime to shortest nighttime, assigned to the equator) and apply the above theorem, we find the ratio 3:2 that lies between them. If we continue with the ratios 2:1 and 3:2, we find the intermediate ratio 5:3. Similarly, with the ratios 2:1 and 5:3, we are led to the intermediate ratio 7:4; and with the ratios 7:4 and 5:3, we find the intermediate ratio 12:7. Taking these ratios in ascending order, we have then

1:1 < 3:2 < 5:3 < 12:7 < 7:4 < 2:1,

a series that defines a sequence of locations beginning at the terrestrial equator and proceeding north [see Table 1]. Thus, it would have been possible for Eudoxus to obtain the ratios 5:3 and 12:7 without recourse to measurement or observation.

As Neugebauer remarks, the ratios 7:4 and 12:7 imply divisions of the day by the prime numbers 11 and 19, respectively, and for this reason were probably rejected by later Greek astronomers who used daytime schemes in equinoctial hours or time-degrees. The question now is whether the ratio 12:7 should be emended, that is, whether equinoctial hours or time-degrees had been introduced in Greece by the time of Eudoxus, since it is clear that before their introduction, the incon- venience of such divisions of the day by prime numbers would not have arisen.

? 3. THE EARLIEST OCCURRENCE OF EQUINOCTIAL HOURS AND TIME-DEGREES IN GREEK ASTRONOMY

To preface our remarks, we begin by distinguishing the relevant sorts of ancient evidence: in this case, there are three-(a) an author's quotations of passages mentioning a concept (such as that of an hour or a degree of time) that are from texts by others, (b) his testimonia or reports re-presenting the views of others that state or imply the use of this concept, and (c) passages in which an author presents his own views and introduces the concept either explicitly or implicitly. As a point of methodology, in determining the earliest occurrence of a given concept in Greek astronomy, we seek to discover the latest date possible for its first occurrence. In other words, we aim to answer the question, At what point is it absolutely necessary to admit the use of the concept or idea?, and thereby to avoid the excesses of conjecture and speculation that blemish most accounts of the history of Greek astronomy. Thus, in our arguments, the third kind of evidence mentioned above has the greatest weight in principle: for, if the date of a text is known, then so is the date

10 Strabo [Geog. ii 5.42], e.g., assigns it to Borysthenes. This ratio defines the northern- most of the canonical seven climate, a systematic division of the inhabited world probably first introduced in early Hellenistic times [cf. Neugebauer 1975, 333-335, 725-727, 735]. This new geographical system of the seven climate may have incorporated some values that were already traditional.




of use of the idea in question. Next in value are quotations, providing there is good reason for thinking that they are accurate. Last in value are the testimonia. This kind of evidence is very difficult to assess, especially when the works discussed are no longer extant-as is so often the case in the early history of astronomy-because ancient reports frequently recast the original in different terms for different purposes [see ? 4.1, below].

Further, in the interest of clarity we introduce some basic definitions. In later Greek astronomical texts, there is a distinction between seasonal and equinoctial hours: a seasonal hour (6cpa iccUp1cij) is 1/12 of daytime (i.e., the period from sunrise to sunset) or of nighttime (i.e., the period from sunset to sunrise); an equinoctial hour (copa io ltpivil) is 1/24 of a day [cf. n4, above], or /12 of the daytime or of the nighttime at equinox. Seasonal hours of daytime differ from those of nighttime on the same day (except at the equinoxes) and they vary in length throughout the year.

The earliest occurrence of equinoctial hours in an extant Greek text is in P. Hibeh 27 (ca. -300), a calendar that assigns 14h to the longest daytime of the year and 10h to nighttime on the same day. In this papyrus, however, there is a curious slip in stating the rule for the increase in the length of daytime: "(Phamenoth) Day 24: summer solstice; nighttime becomes longer by 45 of a twelfth (&co6086aTrj4poo) of daytime, that is, of an hour (60parq); nighttime becomes 10 [45] hours, daytime is 13 3 4 20 90 hours."1 [P. Hibeh 27 cols. 8.120-9.124]

The problem is the ambiguity of "hour." The rest of the passage and, indeed, of the entire calendar, makes it very clear that the daily incre- ment (or decrement) has to be 1/45 of /12 of daytime at equinox, i.e., 1/45 of an equinoctial hour (not a seasonal hour) or 1/1080 of a day [cf. P. Parisinus 1 col. 2; Blass 1887, 13 ad col. 2.51] Accordingly, there are two possibilities. Either the text is carelessly written here or it is evidence of a confusion

11 Reading e W' (5 40) as W' e' (45) (second occurrence). Following Neugebauer [1934, 111], we use fi for Vn and 3 for 23. There is an ambiguity

in the representation of fractions in the papyrus: e.g., i's'is to be read as 15 and not as 10 5. In general, the interpretation of the fractions in this papyrus is determined by the fact that the hours of nighttime and the hours of daytime sum to 24. This mathematical condition entails that the unit-fractions are to be understood in descending order of magnitude, though they are frequently not written this way. Editors usually assume that the unit-fractions were to be written in descending order as well, and use this "rule" to correct the text. But, since deviations from this scholarly "rule" occur as often as they do in P. Hibeh 27, we prefer the more conservative - and certainly more charitable - course, and suppose that the order in which the fractions were written down was of some indifference, rather than correct the documentary evidence on the ground that the scribe was not up to his task. Moreover, given that P. Hibeh 27 is an early document, correcting it in light of a rule inferred from much later documents may not be appropriate.

Phamenoth: month VII in the Egyptian civil calendar of 365 days (i.e., 12 months of 30 days each, and 5 epagomenal or supernumerary days).

Grenfell and Hunt [1906, 152, 155] understand Tin prtepay as a genitive of comparison with .tisfov, and thereby find yet more confusion in this sentence: after all, nighttime becomes 245 longer than daytime on Day 24. We prefer, however, to read Tin Prtepay with &I)6sKaTTi~topou, since the arithmetical sense of the passage and, indeed, of the entire text, requires this.




that involves taking 6opa (which elsewhere in this text is an equinoctial hour) to be a seasonal hour. Regarding the latter alternative, such confusion is, we think, plausible. In Greek texts of the fourth century BC and earlier (as well as later, of course), 6cpa usually means "a period of time," "duration," "time of year," "season," "proper or right time," "prime (of life)," and the like [cf. Liddell, Scott, and Jones 1940, s.v. 60pa];12 hence, one would anticipate that its initial or primary astronomical usage at the close of the fourth century and the beginning of the third would be to designate seasonal hours.

Though Babylonian documents testify to knowledge of equinoctial hours [see, e.g., Mul.Apin] and of seasonal hours [cf., e.g., Reiner and Pingree 1974-1977] before Hellenistic times,13 there is no unambiguous evidence in Greek texts [see n12, above] for the use of equinoctial hours until P. Hibeh 27, or for the use of seasonal hours until Timocharis [see Ptolemy, Alm. vii 3: cf. n2, above].14

As for time-degrees, their earliest occurrence in a Greek text is in Hypsicles' Anaphoricus 55-59. Hypsicles divides the day (i.e., the period of one revolution of the celestial sphere) into 360 equal sub-periods called ioltpat XpoviKwai (degrees of time). It seems, however, that these units of time were not yet in common usage. For, Hipparchus, who, as we have indicated, was roughly contemporary with Hypsicles, does not use time- degrees in his commentary: the least unit of time found there is 1/30 of an equinoctial hour [cf., e.g., toq X ugpoq 6cpaq at In Arat. iii 5.81.15 More- over, though there is evidence of the division of the day into time-degrees in Babylonian diaries dating from the sixth century BC [cf. Sachs and Hunger 1988, 46-53],16 there is, so far as we are aware, no evidence for

12 At Herodotus, Hist. ii 109.3, there is mention of the 12 parts of the day (T& 8cb8wcKa t1pscLa Tug; Kiapa;): the problem in interpreting this is that one has no way to decide whether a day (niitpp) is here to be understood as the period from sunrise to sunset or as that from one sunset to the next. In the latter case, the text would refer to the Babylonian beru or twelfths of the time it takes the heavens to revolve once (i.e., the so-called double hours); in the former, to the division of daytime into 12 seasonal hours. Both alternatives seem equally likely- one should not give preference to the first on the ground that the same passage seems also to mention sundials, at least until one establishes a satisfactory interpretation of tk6ov . . . Kai yvcwiova that accords with what is known of Babylonian sundials. In truth, this brief passage in Herodotus' narrative is peculiar, and sufficiently so that some have been moved to argue that it, or part of it, is an interpolation. Cf. Goldstein and Bowen 1983, 332n9; Neugebauer 1941, 15-16 and n8.

13 According to Rochberg-Halton [1989], simanu may be the Akkadian term used to designate a seasonal hour. In which case, there is remarkable coincidence in the semantic field of this term (i.e., season, period of time, time-interval, duration) and that of 6pa.

14 There is a report in Geminus, Int. ast. vi 9 suggesting that Pytheas used seasonal hours [cf. Rehm 1913; Neugebauer 1941, 15n8], but it is by no means clear that the quotation from Pytheas includes the expression in which seasonal hours appear: cf. Mette 1952, 28; Aujac 1966, 43. In Aristotle, Athen. resp. 30.6, sopa probably means "time (of day)" and not "hour."

15 Y30h corresponds to Y720 of a day or to Y2' of time: cf. n30, below. 16 It is worth noting that in the diaries, time-degrees are used only to specify the time

elapsed from the occurrence of one event to that of another; contrary to what one finds in Hypsicles' treatise, they are not given as part of a continuous count from a fixed starting- point in the day.




such a division of the day in Greek texts prior to Hypsicles nor even any reports attributing this division to a Greek who lived before the second century BC.

Accordingly, it is unlikely, in our view, that Eudoxus divided the day either into hours (whether equinoctial or seasonal) or into degrees of time. Thus, Hipparchus is probably right to represent Eudoxus' ratio of longest daytime to shortest nighttime consistently as a ratio of arcs on the Sun's day-circle at summer solstice rather than of hours.

? 4. THE DIVISION OF THE ECLIPTIC IN EARLY GREEK ASTRONOMY

Closely related to the division of the day into 360? of time is that of the ecliptic into 360? of arc. As a matter of history, division of the ecliptic into degrees is often (but not always) part of its division into 12ths. Thus, for example, in his commentary Hipparchus supposes that Eudoxus divided the ecliptic into 12 zodiacal signs-12 equal arcs that are named after the zodiacal constellations and further divided into 30? each -when he argues that Eudoxus put the solstices and equinoxes (the cardinal points) at the midpoints of their respective signs, i.e., at 5 15?, 18 15? and at mr 15?, 1 15?, respectively. Most historians of astronomy follow Hipparchus and in this, we think, they are mistaken: close examination of the evidence shows that Eudoxus did not divide the ecliptic into equal 12ths or into degrees of arc. This point is important because the question of how Eudoxus divided the ecliptic bears on that of how he divided the day and, thus, concerns the values for the length of longest daytime that are attributed to him by Hipparchus. At the same time, the issue is the reliability of Hipparchus' reports about earlier astronomy. Consequently, we will now review in detail Hipparchus' claim about how Eudoxus divided the ecliptic.

This review will proceed in two parts. In the first, we consider the evidence that Hipparchus adduces to support his claim that Eudoxus put the cardinal points at the midpoints of their respective zodiacal signs. In the second, we determine the earliest occurrence in Greek astronomical literature of the division of the ecliptic into degrees of arc.

? 4.1. EUDOXUS AND THE DIVISION OF THE ZODIACAL CIRCLE

To begin, a general observation about Hipparchus' commentary. This work is replete with quotations from the works of Aratus and Eudoxus and with testimonia about their views. Now, one should be aware that it is in general difficult to distinguish quotations and reports in ancient documents: the reasons for this are numerous, but not the least of them is the absence of conventions for writing quotations. Hipparchus' commentary, however, is exceptionally clear in making the distinction between quotations and reports. Moreover, given that the text of Aratus' Phaenomena is extant and so may serve for purposes of comparison, it is easily shown that Hipparchus is reliable in his quotations of Aratus'




poem. Unfortunately, there is no such means of checking Hipparchus' quotations of Eudoxus' works, since the two books by Eudoxus have sur- vived only in fragments and, judging from Lasserre's collection [1966], Hipparchus' commentary is the earliest extant source of these fragments. Still, there seems no reason to doubt that Hipparchus' quotations of Eudoxus are equally reliable.

Near the outset of book 2 of his commentary, Hipparchus posits that Aratus and Eudoxus located the solstices and equinoxes differently on the ecliptic:

First, let it be supposed in advance (npoO6iX1?00w) that Aratus has made the division of the zodiacal circle beginning from the tropic and the equinoctial points, so that these points are the beginnings of their zodiacal signs; but that Eudoxus has divided it so that the aforesaid points are the midpoints (ta gLaa) of Cancer and Capricorn and of Aries and Libra.17 [In Arat. ii 1.15]

Clearly, Hipparchus thinks that this supposition needs justification, for he argues it first for Aratus [In Arat. ii 1.16-19] and then for Eudoxus [In Arat. ii 1. 20-22]. Hipparchus' strategy, in either case, is to cite and interpret passages from their respective Phaenomena. So the question arises, Do the passages Hipparchus quotes actually support his contention that Aratus put the solstices and equinoxes at the beginnings of their respective signs and that Eudoxus put them at the midpoints?

Consider then Hipparchus' argument in defense of his claim about Aratus, who supposedly followed Eudoxus in most other matters:

Thus, Aratus becomes especially clear through these lines, when he locates the aforesaid points at the beginnings (apXaiq) of their dodecatemorial8 according to these (considerations). For, in speaking of the 4 circles-that is, of the 2 tropic (circles), the equinoctial (circle) and the zodiacal (circle)-he says that the [scil. first] 3 rise and set parallel to one another, and that each of them rises and sets at one and the same point (on the horizon), and that the zodiacal (circle) makes its risings and settings on such an arc (TotawT1) nspik0psFpF) of the horizon as it travels on it [scil. the horizon] from the rising (point) of Capri- corn to the rising (point) of Cancer. He says the following: And the (three circles) rise and19 forthwith set beneath all parallel, but one for each of them in order on each side is the descent and the ascent.20 But (the fourth) passes over as much water of Oceanus as rolls from Capricorn coming up to Cancer ascending. [Aratus, Phaen. 534-539] So it is clear from these (lines) that the solstitial points are located

17 zodiacal circle: scil. ecliptic. Editorial Conventions in Translated Passages: (a) parentheses enclose (i) Greek words or phrases underlying the translation, or (ii)

English words or phrases supplied to make the translation smoother; (b) square brackets enclose (i) the numbers of the chapter-sections in the edited Greek

text, or (ii) restored text; (c) italics are used to indicate passages quoted by Hipparchus. 18 dodecatemoria: see ? 4.2, below. 19 Manitius has Ts Kai; whereas Mair [1955, 248] has Kai. 20 Manitius has KaTiJuoi1l dvo866 Ts; whereas Mair [1955, 248] has KaTiiXUotii T'

6vo866 Ts.




in the beginnings, one in Cancer and the other in Capricorn: for when the zodiacal (circle) rises, it advances along the arc of the horizon that is cut off by these points at their risings. That he would seem to imply (&tpaiv~iv) this (follows) also from the (lines) he says in speaking of Leo: There are the hottest paths of the Sun where the fields appear empty of grain, when the Sun comes together with the first (parts) (ni 7rpc~ra) of Leo. [Aratus, Phaen. 149-151] For this happens especially around the rising of Sirius (Kuv6q) and the days of summer-heat (Ka6ctaTa). But this (rising) occurs almost 30 days after the summer solstice (Tponriq). Therefore, in his view, after so many days the Sun is in the beginning of Leo. Thus, in this solstice, it [scil. the Sun] occupies the beginning of Cancer.

And the zodiacal circle is divided in this way by almost all the ancient astronomers (taOUp~a-rKmCv) or by most (of them). [In Arat. ii 1.16-19]

At first glance, Hipparchus' interpretation looks pretty good: if the northern limit of the Sun's motion along the horizon is the point where Cancer begins to rise, then the summer solstice is at the beginning of Cancer. But does Aratus mean Cancer the zodiacal sign or Cancer the zodiacal constellation? Hipparchus assumes the former; in fact, however, the latter is correct. As any reader of Aratus' poem may verify, Aratus applies the name, "Cancer," and so on, only to constellations [cf., e.g., Phaen. 96-97 with 137-138 (Virgo), 167-171 (Taurus), 225-232 (Aries), 239-247 (Pisces), 301-307 (Sagittarius, Scorpio), 491-496 (Tropic of Cancer), 501-506 (Tropic of Capricorn)]. Thus, Aratus recognizes only the zodiacal constellations which he takes to occupy equal arcs of the zodiacal circle [cf. ? 4.2, below]. These arcs are neither named nor divided into degrees. In fact, at Phaen. 560 and 581, these 12ths of the zodiacal circle are called uoitpat (portions), the very term used later to signify degrees of arc and of time [cf., e.g., Hypsicles, Anaph. 55-59]. But, if Aratus' poem makes no mention of degrees and reserves the zodiacal names for constellations, it follows that Hipparchus is using later concepts and refinements in interpreting what Aratus has written.

Let us turn now to Hipparchus' argument that Eudoxus put the cardinal points at the midpoints of XYr, 5, A, and 3.

That he puts the tropic points at the midpoints of their zodiacal signs (Katr Auraa td c6ia) Eudoxus makes clear by these (remarks): Second is the circle in which [the] summer solstice (Ocpivai Tpoiraf) is; on this (circle) are the middle (parts) (rt ueca) of Cancer. And again he says: Third is the circle in which the equinoxes (ai iuigimgpiai) are: on this (circle) are the middle (parts) (i ,u peca) of Aries and of Libra. Fourth (is the circle) in which [the] winter solstice (EipcEpivai Tpo7ra[) is; on this (circle) are the middle (parts) (Ta& puca) of Capricorn. He makes it still more clear through these (remarks). For concerning the circles called colures that are described through the poles and through the tropic and equinoctial points, he says the following: Different (from the circles discussed so far) are the two (great) circles through the poles, that bisect one another at right angles. On them are these constellations (1cuTpa): first, the ever-visible pole of the cosmos, next, the middle (Tr pEuov) of Ursa Maior (taken) widthwise and the middle (TO pE5uov) of Cancer. And after a little, he says: The tail of Piscis Australis and the middle (Tr pgaov) of Capricorn. In what comes next, he says that both the middle (parts)




(ta pasta) of Libra2l (taken) widthwise and the back (parts) of Aries (taken) widthwise lie with the other (constellations) with which they are counted, on the other circle through the poles. [In Arat. ii 1.20-22]

There are two problems with this argument. First, the phrase t& Bi6oa (sing. to pitaov) is used in the case of Ursa Maior, a constellation that does not lie on the zodiacal circle. Accordingly, the phrase should signify only the interior of a constellation: it certainly need not indicate anything so precise as the midpoint. Second, from passages that Hipparchus quotes elsewhere in his commentary, it is clear that, like Aratus, Eudoxus too is concerned with zodiacal constellations and not zodiacal signs.

Consider the following:

Concerning the stars that move on the summer tropic (circle), the winter tropic circle, and, furthermore, on the equinoctial circle, Eudoxus says the following in the case of the summer tropic (circle): There are on this (circle) the middle parts (Ta , puca) of Cancer and the (parts) right through the body of Leo (taken) lengthwise, [the] slightly higher (parts) of Virgo, the neck of Serpens held fast, the right hand of Ingeniculus,22 the head of Ophiucus, the neck of Cygnus and the left wing, the feet of Equus,23 and further, the right hand of Andromeda and the (region) between her feet, the left shoulder of Perseus and his left (lower) leg, and further, the knees of Auriga and the heads of Gemini; then it joins up to the middle (parts) (Irpbq Ta pgua) of Cancer. [In Arat. i 2.18]

In the case of the winter tropic (circle), Eudoxus says the following: There are on it the middle (parts) (Tr puca) of Capricorn, the feet of Aquarius, the tail of Cetus, the bend of Flumen,24 Lepus, the feet and the tail of Canis, the stern and the mast of Puppis,25 the back and the breast of Centaur, Fera,26 and Scorpio's stinger; then it joins up through Sagittarius to the middle (parts) (irpbq rt piaa) of Capricorn. [In Arat. i 2.20]

Note that Eudoxus writes of the body of Leo, the slightly upper (parts) of Virgo, the heads of Gemini, the feet of Aquarius, Scorpio's stinger, the middle (parts) of Libra taken widthwise, and the back (parts) of Aries taken widthwise. None of this is consistent with the assumption that "Leo," "Virgo," and so on, denote zodiacal signs. In other words, Hip- parchus' efforts to argue that Eudoxus refers to the midpoints of zodiacal signs are unavailing: the evidence Hipparchus adduces does not support the claim that Eudoxus recognized zodiacal signs or that he divided them at their midpoints.

Moreover, Hipparchus' contention has consequences at odds with what Eudoxus writes, consequences which Hipparchus exploits. Thus, Hipparchus' assumption that Eudoxus put the tropic and equinoctial

21 Libra: Claws (scil. of Scorpio). 22 Ingeniculus: later supposed to be Heracles-cf. Mair 1955, 212na. 23 Equus: i.e., Pegasus. 24 Flumen: i.e., Eridanus. 25 Puppis: i.e., Argo. 26 Fera: i.e., Lupus.




points at the midpoints of their zodiacal signs is the basis for his criticism of the way that Eudoxus located certain constellations:

if in fact the aforesaid [points] lie at the midpoints of their zodiacal signs (KaTa pttaa d 4X6ia) as Eudoxus says, one should put not Leo but Virgo beneath the hindfeet of Ursa Maior [In Arat. i 5.11] For the last and brightest star of this (scil. Ursa Minor) lies at Pisces 18?; but, as Eudoxus divides the zodiacal circle, at Aries 3?. [In Arat. i 6.4]

In sum, we conclude that Hipparchus does not describe the state of astronomy at the time of Eudoxus. Contrary to what many historians suppose, then, one should not take for granted the historical accuracy of Hip- parchus' remarks about his predecessors. This does not mean, however, that Hipparchus is intentionally deceptive or that his recasting of the Phaenomena of Eudoxus and Aratus is unreasonable. Rather, it is a con- sequence of his effort to understand older texts in more modern terms and thereby to render them useful in dealing with current astronomical questions or problems. For Hipparchus, the works of Eudoxus and Aratus are not of antiquarian interest, but belong to living astronomy. Still, one does wonder why Hipparchus read Aratus and Eudoxus as he did on the particular question of the location of the cardinal points. One possibility is that he was influenced by Babylonian diaries dating from the fifth century BC that seem not to distinguish zodiacal signs and zodiacal constellations.27 Such conflation on Hipparchus' part would then indicate the transmission of Babylonian material after Aratus (d. -238), since there is no evidence for it in Aratus' poem or in any report of Eudoxus' writings.

Yet, for all that, one must also admit that Hipparchus' treatment of Eudoxus and Aratus is not devoid of polemic. For example, though Hip- parchus is well aware of the difference between zodiacal constellations (which cut off unequal arcs of the ecliptic) and zodiacal signs [In Arat. ii 1.7-8], he takes advantage of Aratus' assumption that the zodiacal constellations occupy equal arcs of the zodiacal circle, and proceeds to treat Aratus' 12 "portions" (40oipat) as signs and as constellations. Thus, at In Arat. ii 1-3, Hipparchus criticizes Aratus [cf. Phaen. 559-565] for thinking that, if the position of the Sun on the zodiacal circle is known, one can determine the hour of night by observing the zodiacal constellations- rather than the zodiacal signs. Even here, however, there may be yet further distortion: it is not at all clear that Aratus is concerned with timekeeping [cf. also ? 6 ad [5], below].

As we have shown, Hipparchus' commentary, the ultimate source of our knowledge of Eudoxus' Phaenomena and Enoptron, does not support the claim that Eudoxus divided the ecliptic into 12ths and each of these into 300 degrees of arc but even suggests that he did not. We will now assess the evidence for the earliest occurrence of this division in the corpus of surviving Greek texts in order to show that this claim on Eudoxus' behalf is implausible.

27 Professor Francesca Rochberg-Halton has kindly drawn these diaries to our attention.




? 4.2 THE EARLIEST OCCURRENCE OF DEGREES OF ARC IN GREEK ASTRONOMY

The earliest occurrence in a Greek text of the division of the ecliptic into 12 equal arcs (called 40oipai: portions-see Phaen. 560, 581) is in Aratus' Phaenomena [cf. 541-559]. These are defined by the zodiacal constellations -it is assumed that the constellations occupy equal arcs of the ecliptic -but they are not named after them. In Autolycus' De ortibus et occasibus [cf. ii props. 1-2],28 the ecliptic is also divided into 12 equal arcs (called 6o)8cit~rio6pta: twelfths); these arcs, however, seem to be designated by the names of the constellations.29 Similarly, in Euclid's Phaenomena [cf. props. 1, 8, 11-13],30 12 equal arcs are again defined on

28 See Schmidt 1949, for argument that book 2 of Autolycus' De ortibus is a revision of book 1.

29 Though it is common now to suppose that Autolycus lived from -359 to -289 [Aujac 1979, 8-10: cf., e.g., Mogenet 1950, 5-7, 8-9], his dates are really uncertain. The primary source, Diogenes Laertius [Vitae iv 29], claims that Arcesilaus, the founder of what was called the Middle Academy (third century BC), was his pupil in Pitane; but this may rest on nothing more secure than the fact that both Autolycus and Arcesilaus came from Pitane. The other source, Simplicius, in his commentary on Aristotle's De caelo [Heiberg 1894, 504.22-25], cites Sosigenes (second century AD) and writes "But none of them [scil. Cal- lippus and other Eudoxans [cf. ? 6 ad [10], below]] attempted, until in fact (wai) Autolycus of Pitane, to demonstrate this very (phenomenon) [scil. that the varying brightness of Venus and Mars gives the impression that their distance to the Earth varies], which is in fact mani- fest to the eye, by means of models (68t T6iv U'nofta?ov); and further not even Autolycus himself was successful (i8uvvOi9)-his dispute with Aristotheros shows (this)-. . .' Such testimony, however, gives no information that can be used to date Autolycus, since the date of Aristotheros is also disputed [cf. Aujac 1979, 9n2], and since there is no way to decide how much later than the Eudoxans Autolycus is supposed to have lived [cf. Mogenet 1950, 7-8]. Indeed, the assumption that Autolycus must have lived in the fourth century seems ultimately based on the dubious assumption that attention to simpler and less sophisticated theories generally belongs to a period prior to that of concern with more complex and sophisticated theories.

It is especially difficult to determine the relative dates of Autolycus and Euclid. The argument from internal evidence [cf. Heath 1921, i 348-353] that Autolycus is prior to Euclid is, as Neugebauer [1975, 750] points out, "singularly naive": there is no reason to dismiss the possibility that Autolycus and Euclid were contemporary. On Euclid's dates, see n30 below.

30 Euclid's dates are uncertain. Though it is customary to rely on Proclus, In Euc. prol. ii [Friedlein 1873, 68.6-20: cf., e.g., Heath 1956, i 1-2] and to interpret this passage to mean that Euclid flourished ca. -300, Proclus' report shows that he is merely guessing that Euclid lived during the time of Ptolemy I (reigned: -303 to -281) on the strength of the anecdote about Ptolemy and Euclid, and from Archimedes' (ca. -286 to -211) (trivial) citation of Euclid in De sph. et cyl. i prop. 2 and of a work entitled Elements in prop. 6. But such an inference is obviously very weak: the same anecdote is told of Menaechmus and Alexander [see Stobaeus, Ecl. ii 31.115] and so has no credibility of its own. Moreover, the citation of Euclid by Archimedes may in fact be an early interpolation [see Dijksterhuis 1956, 150nl]; and the reference to the Elements need not be to Euclid's Elements [cf. Proclus, In Euc. prol. ii: Friedlein 1873, 66.7-68.10]. Thus, at the very most, all one may know is that Euclid was either a predecessor or contemporary of Archimedes.

Unfortunately, it is also not entirely certain that Euclid even wrote the Phaenomena. As Menge [1916, xxxii] reports, Theodosius of Bithynia (first century BC) is the first to mention a book entitled Phaenomena -though, as Menge points out, 6cq V tIo4 k alvogtvol4 W&1wtat

(as is proven in the Phaenomena) may be an interpolation-and Galen (129-195) is the first to say that Euclid is its author. But, though Pappus as well as Marinus and Philoponus (both active in the sixth century AD) also speak of a Phaenomena by Euclid, the Phaenomena is not




the ecliptic and they too share the names of the zodiacal constellations. The problem in the latter two texts is that there is no basis for deciding whether Autolycus and Euclid divided these 12ths of the ecliptic further into 300 each.

The earliest occurrence of degrees of arc in a Greek literary text is found in Hypsicles' Anaphoricus 55-59.31 In this passage, Hypsicles divides the zodiacal circle into 360 equal arcs called 40oipac toitcat (degrees of arc). He also divides the zodiacal circle into 12 arcs of 30? each that bear the names of the zodiacal constellations [cf. Anaph. 67-70,139- 151]: we think these arcs are rightly viewed as zodiacal signs, though it must be admitted that Hypsicles does not state that the zodiacal constellations occupy unequal arcs of the ecliptic thereby distinguishing zodiacal signs and the arcs of the ecliptic that are cut off by the zodiacal constellations. For this distinction, one must turn to Hipparchus [see In Arat. ii 1.7-8, which maintains that the zodiacal constellations do not occupy equal arcs of the ecliptic], his near contemporary. Indeed, Hipparchus indisputably uses degrees and zodiacal signs in his commentary.

Given our analysis of the testimonia of Hipparchus about Eudoxus' and Aratus' use of zodiacal signs and degrees of arc, it would seem that there are no reliable reports that attribute their use to any Greek who lived before the close of the fourth century BC. There are, however, several such reports concerning the use of degrees (of arc) in the third. Thus, for example, Ptolemy reports observations of stellar declinations by Tim- ocharis (and Aristyllus) in degrees. These reports too, however, are based on information that comes from Hipparchus, and so do not constitute independent evidence; moreover, as we argue elsewhere [see n2, above], it is possible that Hipparchus has recast the original observational reports into degrees. In any case, note that in Alm. x 4 [Heiberg 1898-1907, ii 311.10-19], Ptolemy makes it clear that he has added zodiacal signs and degrees to the report by Timocharis that he cites. This means that even if one allows that Timocharis used degrees, the evidence suggests that he did not use zodiacal signs.

Thus, though there are Babylonian cuneiform texts showing the division of the ecliptic into zodiacal signs from the mid-fifth century BC [cf., e.g., Aaboe and Sachs 1969], the earliest occurrence of degrees and zodiacal signs in the corpus of extant Greek texts is not earlier than the third century-we here bypass the question of the dates of Autolycus and

listed as one of Euclid's works by Proclus (410-485) at In Euc. prol. ii [Friedlein 1873, 69.1-4] - however, neither are Euclid's Porisms and the Data, for example. On balance, then, it seems that there are no decisive arguments for doubting that Euclid was the author of this handbook of spherical astronomy.

31 There is a fragmentary astronomical inscription from Keskinto (on the Island of Rhodes) that divides the circle into 360 degrees (poipas) and 720 atiygai (points). Though epigraphic evidence suggests that the inscription was made around -100, this date is extremely insecure: cf. Neugebauer 1975, 698-699. Still, though the use of points and Moon- breadths is attested in Hellenistic times [cf. nn2, 8, above], there is nothing that suggests these units go back to Greek astronomy of the fourth century BC.




Euclid, because it does not affect our thesis. In short, it follows that the latest possible date for the introduction of the division of the ecliptic into degrees of arc in Greek astronomy comes well after the death of Alex- ander in -322, and that it is contemporary either with the division of the day into hours (third century) or with the division of the day into degrees of time (second century).

? 5. CONCLUDING REMARKS

It is not our contention that the series of ratios generated above in ? 2 (or any equivalent) was in fact used by Eudoxus or any other ancient astronomer. Our purpose is to show that (a) it was possible to generate a ratio for the relation of the arc above the horizon of the summer solstitial day-circle to the arc below (12:7) that is between 5:3 and 7:4, using an arithmetical procedure appropriate to the time of Eudoxus, and (b) that the ratio 12:7 posed no difficulty because it entails division of the day by a prime number, since the day was not divided into hours or time- degrees in the fourth century BC. This puts the burden of proof on those who would claim that the ratio 12:7 is textually corrupt. Further, those who maintain that this ratio was obtained by unattested observations or precise measurements have taken it for granted that no other means of derivation was available. The burden now lies on them to show that the method we propose was not possible.

Returning to Eudoxus, either the two values which Hipparchus reports for the ratio of the arc of the summer solstitial day-circle above the horizon to the arc below were intended to hold for the same place or for two different places. In the former case, it would be tempting to suppose that, since these values appear in different works, one value is a correction of the other. But, if so, given that there is no evidence of the place intended or of the means used to make this correction, nothing more can be said-except that it is highly unlikely that the correction involved observing the difference in the length of longest daytime (0;9,28h)32 using timekeeping devices of the period. In the latter case, it would follow that these two values (and the works they appear in) were written for different climata, the one assigned the ratio 12:7 being slightly north of the one assigned the ratio 5:3. Since Diogenes Laertius [Vitae viii 86-91] reports that Eudoxus is supposed to have visited Cyzicus prior to his (second) visit to Athens and his return to Cnidus, one might conjecture that Eudoxus associated Cyzicus with 12:7 and Athens with the ratio 5:3, thereby locating Cyzicus between Athens and Byzantium (assigned the ratio 7:4).33 Such conjecture is of course very weak for a variety of reasons, not the least of which is the fact that the ratios and the latitudes

32 5:3 corresponds to a value of 15h for longest daytime and 12:7, to 15;9,28h. 33 This is all that underlies the much repeated claim [cf. e.g., Dicks 1970, 154] that the

Phaenomena (which has the ratio 12:7 and is associated with Cyzicus) was written before the Enoptron (which has the ratio 5:3 and is associated with Athens).




TABLE 2. Comparison of Values for Geographical Latitude Ratio of Longest

Daytime to Shortest Computed Latitude Modern

Nighttime Latitudea Place following Plinyb Latitudec

- - Cnidus 37;360 36;400 5:3 40;410 Athens 37;180 380

12:7 42;30 Cyzicus 40;360 40;230 7:4 43;1? Byzantium 41;380 41;20

a An obliquity of 240 is assumed in all computations. According to Hipparchus [In Arat. i 3.7], the clima associated with the ratio 5:3 is "almost exactly 410."

b Values are computed from the gnomon to equinoctial shadow-length ratios listed in Pliny, Hist. nat. vi 211-218. Pliny states that the longest daytime at Byzantium is 15 1/9 hours; whereas Strabo [Geog. ii 5.41] puts it at 15 /4 hours (which is equivalent to a latitude of 42;500).

c See Times Atlas 1971.

of the places in question do not correlate very well, as Table 2 shows.34 The truth of the matter is, however, that the most we can do is state these alternatives, since the evidence is insufficient; in particular, Diogenes' account of Eudoxus' life does not by itself warrant any choice.

? 6. HIPPARCHUS, IN APRTi PHAENOMENA COMMENTARIA i 3.5-10 [MANITIUS 1894, 26.3-28.18]

Translation

[5] First, then, Aratus seems to me to be mistaken, since he thinks that in the regions of Greece the inclination (i'yKXt~ia) of the cosmos is such that the longest daytime (i'p6pav) has to the shortest the same ratio as 5 has to 3. For, in the case of the summer tropic, he says: when the (circle) has been measured out with the greatest care into 8 (parts), 5 revolve in the daytime, that is (Kai), above the Earth, and 3 in the (region) beneath (the Earth). [Aratus, Phaen. 497-499]

[6] Now, it is agreed that the gnomon has to the equinoctial shadow the ratio of 4 to 3 in the regions of Greece. So, there, the longest daytime is 14 and very nearly 3/5 equinoctial hours (WptWv iorllspiV0v), and the elevation (i'apia) of the pole is almost exactly 370 (4otp0v X4').

[7] But, where the longest daytime has to the shortest the ratio of 5 to 3, the longest daytime is 15 hours and the elevation of the pole is almost

3 Strabo, [Geog. ii 5.41] claims that longest daytime is 15 Y4 hours at Byzantium, which implies that the ratio of longest daytime to shortest nighttime is very nearly 7:4: see n9 above. According to Pliny [Hist. nat. vi 217], longest daytime in Byzantium is 15 Y9 hours, which implies a ratio of 17:10. Of course, neither Athens nor Cyzicus is at the geographical latitude corresponding to the ratios for longest to shortest daytime of 5:3 and 12:7, respectively: the error in each case is approximately 20. But, then, such inaccuracy is common in ancient specifications of geographical latitude [see Table 2] and is really a product of the different schemes used to locate places on the Earth in relation to one another.




exactly 410. So it is clear that it is not possible in the [regions] of Greece that the (ratio) just mentioned be the ratio of the longest daytime to the shortest; but rather (it is) in the regions of the Hellespont.

[8] And yet Aratus has not written offering a view (icpiat4) independently about such (matters), but following Eudoxus even concerning this. But if in fact he has written independently, since he has not made clear in what [regions] the aforesaid inclination (i"yji<Xolq) of the cosmos obtains, in this (point) at least he might perhaps evade criticism.

[9] Nevertheless, Attalus is admittedly mistaken when he says that the longest daytime has to the shortest the ratio of 5 to 3 in the regions of Greece. For, when he expounds the verses on the summer tropic, he adds the following: by this he [scil. Aratus] makes it clear that the entire work (irpdypaari) [scil. Phaenomena] has been written in the regions of Greece; for in those (regions) the longest daytime is to the shortest nighttime as 5 is to 3.

[10] Yet, one might wonder more how in the world (ncit bot&) he [scil. Attalus] does not pay attention when Eudoxus has set (the matter) out differently in his other work (ouvtcdyjait) and writes that the section of the tropic above the Earth has to the section beneath the Earth the ratio of 12 to 7; and when the Philippans (Xxwv iepi 4Diktnnov) and many others write in agreement with this (0'4oi(q . . . touq)), except that they have arranged (uvte~taxaoi) the co-risings and co-settings of the stars on the supposition that they have been observed (sac, 'tltpil1u4vwv) in the regions of Greece but have erred in the clima (Eyji<Xta) of these regions.

ad [5] * 'YKXia TOO cogou (inclination of the cosmos): cf. In Arat. i 2.22 (Kiga TOo

K6ogoou), 3.8 (?yKX1G1q TOO K6ogou). This inclination of the cosmos or clima at a location is the elevation of the (north) celestial pole (9ctapga tO6 n6koU) above the local horizon [cf., e.g., In Arat. i 3.6, 3.12], which is also the distance of the circle of ever-visible stars from the pole [cf. In Arat. i 4.8, 7.21]. (The circle of ever-visible stars is the circle that is tangent to the horizon at the north-point and thus contains all the stars that are circumpolar-i.e., do not rise or set-at that lima. In the latitude of Greece (i.e., Athens), this circle is just slightly larger than the outer limits of Ursa Maior, the apKT64, which is, therefore, circumpolar: hence, this circle came to be called the arctic circle [cf., e.g., Strabo, Geog. i 1.21 et passimi.) Szab6 and Maula [1986, 15; cf. 14, 155-156] translate clima as latitude terrestre (geog- raphique).

Aratus, Phaen. 497-499: (a) with the greatest care (6oove Ts paioTa), cf. Aeschylus, Prom. vinct. 524 (6oov gicitota) and Herodotus, Hist. i 185 (boa ?6t8varo gidktota); (b) Mani- tius has 9v~ta Otp9,06at Kai 65cgpTepa yafqq, whereas Mair [1955, 2461 has MvMia Otp&CTat Ka0' 65,tp-rpa yaiin (revolves in the daytime above the Earth). Note that "Earth" here stands for "the observer's horizon": the literary figure is synecdoche.

* One gets more of the flavor of Hipparchus' polemical tone when one recalls that Aratus does not give any geographical specifics for the lima of the celestial sphere he describes. Hipparchus admits this in [81 and, at the same time, reveals that Eudoxus too did not supply such geographical specifics. In fact, of those mentioned, the only one who seems to have done this is Attalus (see [9]). Moreover, as for Hipparchus' methodology, note that Aratus simply divides the day-circle for the Sun at summer solstice into two arcs of different size: five eighths above the horizon and three eighths below. Hipparchus, however, takes him to mean a ratio for longest to shortest daytime of 5:3, and so introduces the common assumption that the shortest daytime and the shortest nighttime are equal (cf. Neugebauer 1969, 158nl, which explains that atmospheric conditions make the shortest




ACKNOWLEDGMENT

The authors thank H. C. Avery, John P. Britton, David Fowler, Fran- cesca Rochberg-Halton, and Noel M. Swerdlow for their comments on an earlier draft of this paper. This study was supported by a research grant from the National Endowment for the Humanities.

daytime longer than the shortest nighttime). Attalus, on the other hand, takes Aratus to mean a ratio of longest daytime to shortest nighttime: cf. [9]. ad [61 * Hipparchus' computation of the length of daytime given the ratio of gnomon to equinoctial shadow-length is consistent with the assumption that the obliquity (?) of the ecliptic is 240.

* If the ratio of the gnomon to the shadow-length at equinox is 4:3, the elevation of the pole (4, which is also the geographical latitude) is 36;52?.

* Vitruvius [De arch. ix 7.11 assigns to Athens the gnomon to equinoctial shadow- length ratio of 4:3. Pliny [Hist. nat. vi 215], on the other hand, assigns to Athens the just slightly smaller ratio of 21:16 and adds that the longest daytime is 14 Y3 hours, a correlation that is again consistent with the assumption that ? = 24? [see Table 2]. On the arithmetical character of Pliny's tables, see Neugebauer 1975, 747-748.

* Hipparchus gives the results of his computations but does not indicate his procedure for finding the longest daytime and the clima from the ratio of the gnomon to equinoctial shadow-length. Though Szab6 and Maula [1986, 17-20, 221-2221 aim, in effect, to attribute to Eudoxus and his contemporaries the same trigonometric techniques more usually reserved for Hipparchus, and support this by drawing attention to the phrase au>)coveftat 68 (now, it is agreed), they do admit, however, that the sense of this passage is that only the ratio 4:3 for Greece is well known: toivuv (so) [Manitius 1894, 26.13] is in- ferential and it may suggest that the techniques of computation, as well as (perhaps) the correlations themselves, were not common knowledge. ad [71 If ? = 24?, X = 40;41? = 41?.

* The Hellespont is the strait at the final exit of the waters of the Pontus Euxinus (Black Sea) and the Propontis (Sea of Marmara) into the Aegean Sea, i.e., the Dardanelles. For Pliny [Hist. nat. vi 216-2171, the length of longest daytime in the Dardanelles and Cyzicus is 15 hours [see Table 2]. ad [9] * As Hipparchus [In Arat. i 1.3] says, Attalus was an astronomer (1LaOllartlc6) and his contemporary. ad [101 * Eudoxus' "other work" is the Phaenomena: Hipparchus [In Arat. i 2.22] claims that the ratio 5:3 was found in the Enoptron. Cf. Neugebauer 1975, 733n28.

* The Philippans (rov ntpi Vitnnov): a phrase such as oi ntpi CIXtIctov (literally, "those around Philip") is difficult. Though it is normally translated "the school of Philip," such a rendering probably suggests too much. The usual alternative, "the followers of Philip," is preferable, though perhaps not as good as "the Philippans." In any event, one should bear in mind that the phrase may mean nothing more than "those who share certain assumptions or procedures with Philip," whether they are contemporary with Philip or sub- sequent to him. Moreover, the modern historian should resist assuming that what is attributed to the Philippans also holds of Philip: such an identification requires more evidence than the locution ofi ntpi NiXtIctov, since there are many cases in which the "followers" departed dramatically from their "hero." Such is the case with Plato and his "followers,' for example. Cf. Toomer 1984, 137n19.

It has been suggested [cf., e.g., Neugebauer 1975, 574, 740n12] that the Philip in question is Philip of Opus, otherwise known as the editor of Plato's Laws and (perhaps) as the author of the dialogue, Epinomis, attributed to Plato [see Tarin 19751. Still, this identification is hardly secure, given that "Philip" is a fairly common name among Greeks of the Hellenistic period. There is also a Philip mentioned in the second Milesian parapegma [cf. Neugebauer 1975, 588, 6171 and, perhaps, he is the same as the Philip cited in Ptolemy's Phaseis. What we have in the present passage concerning the Philippans is consistent with these two occurrences.

* Note that the Philippans are not credited with making observations of the various co-risings and co-settings, but only with arranging (ouvrerdXaml) them. The same verb is used by Ptolemy [Alm. ix 2: Heiberg 1898-1907, i.2 210.8-19], when he says that Hipparchus arranged (ouv-rd~at) the planetary observations in a more useful way.




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Documents

Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime