ΑΚAΔΗΜΙΑ ΘΕΣΜΩΝ ΚΑΙ ΠΟΛΙΤΙΣΜΩΝ ACADEMY OF INSTITUTIONS AND CULTURES

ΑΚAΔΗΜΙΑ ΘΕΣΜΩΝ ΚΑΙ ΠΟΛΙΤΙΣΜΩΝ ACADEMY OF INSTITUTIONS AND CULTURES Στοά των Επιστημών-Επιστημονική Επιθεώρηση Stoa of Sciences-Scientific Review Τόμος 1 Volume 1 Ελληνιστική Αστρονομία και Μηχανική Hellenistic Astronomy and Mechanics Τεύχος Β Issue B Ο Μηχανισμός των Αντικυθήρων The Antikythera Mechanism Corpus operae Ι Διεύθυνση- Direction ΙΩΑΝΝΗΣ HUGH ΣΕΙΡΑΔΑΚΗΣ, Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης IOANNIS HUGH SEIRADAKIS, Aristotle University of Thessaloniki Μαγδαληνή Αναστασίου et al. Magdalene Anastasiou et al. ISSN: 2241-9993 : Ακαδημία Θεσμών και Πολιτισμών - Academy of Institutions and Cultures Έτος εκδόσεως - Υear of issue : 2016

JHA, xliv (2013) THE ASTRONOMICAL EVENTS OF THE PARAPEGMA OF THE ANTIKYTHERA MECHANISM MAGDALINI ANASTASIOU, Aristotle University, JOHN H. SEIRADAKIS, Aristotle University, JAMES EVANS, University of Puget Sound, STELLA DROUGOU, Aristotle University, and KYRIAKOS EFSTATHIOU, Aristotle University 1. introduction The Antikythera Mechanism, 1 found in an ancient shipwreck in a.d. 1901, is an extraordinary 2000-year-old astronomical computer that calculated the position of the Sun and the Moon in the sky, was used as a calendrical device, could predict eclipses, and displayed the year when ancient panhellenic games were held. Extended astronomical and technical inscriptions covered its front and back doors and plates. The front side of the Antikythera Mechanism is mainly preserved in Fragment C where small parts of two concentric rings are found: the outer one displays the Egyptian calendar (with month names ΠΑΧΩΝ, ΠΑΥΝΙ, etc.) and the inner one the signs of the zodiac (ΠΑΡΘΕΝΟΣ, ΧΗΛΑΙ, etc). 2 On the zodiac dial, above some of the scale divisions of each zodiac sign, small capital letters (A, B, Γ,...) are inscribed, that follow the sequence of the letters of the Greek alphabet. 3 Attached to Fragment C, there are also two pieces (C1-a and C1-b 4 ) of flat plate, inscribed with lines of text describing astronomical phenomena. At the beginning of each line is a separate letter. These index letters almost certainly correspond to the small capital letters found on the zodiac dial, thus forming a parapegma. 5 Parapegmata calendars of astronomical and meteorological events were widely used in ancient Greece. 6 Incomplete lines of parapegma text have been found in four other fragments (Fragments 9, 20, 22 and 28). On the largest of the two pieces of flat plate (C1-a) attached to Fragment C, the most extensive parapegma inscription, with 9 lines of text (see Figure 1), can be read. From the lines bearing the index letters Λ, M and N, only the last words are visible. The rest of the letters were visible duringalbert Rehm s investigation (early twentieth century) but nowadays they are not, as part of the inscribed plate has broken away since then. These inscriptions were reported by Price, 7 who based his reconstruction on unpublished notes of Albert Rehm. The events mentioned on plate C1-a are typical of events found in parapegmata and are categorized as follows: (a) There are 6 events that refer to the rising and setting of stars. The verb used is either EΠΙΤΕΛΛΕΙ (rises) or ΔΥΝΕΙ (sets), indicating that the star is in the east or the west, respectively. It is followed by the adjective ΕΩΙΟΣ (in the morning) or ΕΣΠΕΡΙΟΣ (in the evening) indicating that the Sun is in the east or the west, 0021-8286/13/4402-0173/$10.00 2013 Science History Publications Ltd

174 Magdalini anastasiou and colleagues Fig. 1. Parapegma inscription from plate C1-a. Left: The Greek text. Underlined letters can no longer be read today. Right: Translation in English. respectively. The subject is always a star, a star cluster or a constellation. Combining the star and Sun positions, four star events are formed (morning rising (mr), evening rising (er), morning setting (ms) and evening setting (es)). These four events are sometimes called star phases or heliacal risings and settings by modern scholars. (b) There are 2 events that refer to the signs of the zodiac (henceforward called zodiac statements). The subject is always a zodiac constellation/sign, followed by the phrase ΑΡΧΕΤΑΙ/ΑΡΧΟΝΤΑΙ ΑΝΑΤΕΛΛΕΙΝ/ΕΠΙΤΕΛΛΕΙΝ (begins to rise), indicating that the constellation/sign is in the east. The ancients used the same names to indicate the irregularly shaped and sized zodiac constellations and the uniformlysized, 30 -long, zodiac signs. The fact that there is always an index letter at the first scale division of each sign of the zodiac on the zodiac dial (for all three signs with their first scale division preserved (ΧΗΛΑΙ, ΣΚΟΡΠΙΟΣ, ΤΟΞΟΤΗΣ)) indicates that these events most probably mark the crossing of the Sun from one zodiac sign to the next. 8 This is somewhat unusual among Greek parapegmata. Parapegmata, not uncommonly, indicate the entry dates in some manner but this is the only parapegma that uses the verb rise to designate the Sun s entry into zodiac signs. More typically we see a notice that a zodiac constellation begins to rise on a day that is not necessarily the first day that the Sun is in the zodiac sign of the same name. For example, in the Geminos parapegma, we read that, according to Callippus, the constellation Leo begins to rise on the 30 th day that the Sun is in the sign of Cancer; or that, again according to Callippus, the constellation Aries begins to rise on the 3 rd day that the Sun is in the sign of Aries. (Such begins to... statements are also used for other large (not zodiac) constellations.) Star risings and settings were differentiated into true ones (ἀληθινοί) and visible ones (φαινόμενοι). The true ones occur when the Sun and the star have an altitude of 0 (are simultaneously on the ideal horizon) and can be readily calculated using basic spherical trigonometry. The true risings and settings are unobservable as the star is obscured by the glare of the Sun or the sky brightness. The first day that a star becomes visible above the eastern horizon in the morning twilight, after a period when it was below the horizon or when it was just above the horizon but fainter than the sky background, is called a visible morning rising. The visible risings and

the astronomical events of the parapegma 175 settings are observable events, of practical interest to farmers and sailors, and this is why their list in chronological order served as a supplement to the rather chaotic civil calendars. 9 The computation of these events is complicated as the depression of the Sun below the horizon and the altitude of the star in order to be visible are not known a priori. 10 The basic spherical trigonometric equations for the computation of risings and settings are functions of the geographic latitude of the observer. The geographic dependence of the star events inscribed on the parapegma of the Antikythera Mechanism is investigated in this article. In order to calculate the dates of a star s visible risings and settings, atmospheric extinction and atmospheric refraction were taken into account. As the extinction coefficient k varies with location, three different cases with k = 0.17, 0.23, 0.30 covering reasonable values for ancient observations were examined. The visibility of a star was determined by comparing its apparent magnitude with the sky background brightness. Two independent studies of sky background brightness measurements (Nawar 11 and Koomen et al. 12 ) were used. The mathematical analysis, as well the formulas for the atmospheric refraction and the atmospheric extinction, can be seen in Appendices A and B respectively of the online edition. 2. stars and constellations used We calculated the star events of plate C1-a for 150 b.c., as the construction of the Mechanism has been placed around 150 100 b.c. 13 Because the dates of star phases shift only slowly with time, the results would be essentially the same for any date between 100 and 200 b.c. The equatorial coordinates of the Sun and the stars were calculated using the procedures described in Duffett-Smith, 14 taking into account the precession of the equinoxes, the decrease in the obliquity of the ecliptic, nutation and aberration, as well as, for the stars, the effects of proper motion. Constellations were identified with their brightest star as the star events of a constellation are always the ones of its brightest star unless a certain star of the constellation is explicitly specified. Thus, Lyra was identified with Vega, Aquila with Altair and the Hyades with Aldebaran (in Ptolemy s Almagest, 15 Aldebaran (Λαμπαδίας) is mentioned as the brightest star of the Hyades). For the Pleiades, a total apparent magnitude and mean equatorial coordinates 16 were calculated from the ten brightest stars of the cluster. Results hardly differ if one uses the modern Pleiades total cluster magnitude. The relation of the signs of the zodiac to the days of the year was determined from the smaller plate C1-b. On this plate we read that the autumn equinox takes place when the Sun enters the sign of ΧΗΛΑΙ [Libra], i.e. at A ΧΗΛΑΙ as we have seen above regarding the zodiac statements. 17 (This is an important piece of information, (a) as some early Greek astronomers may have placed the equinoxes and solstices at the midpoints of their signs and (b) as a number of Roman writers used a Babylonian convention, according to which the equinoxes and solstices occurred when the Sun reached the 8 th degree.) According to the calculated equatorial coordinates of

176 Magdalini anastasiou and colleagues Table 1. The duration of the Sun s passage through the signs of the zodiac, as given in the parapegma of Geminos (see Evans et al., Geminos s (ref. 18)). Zodiac sign Dates in the Number Zodiac sign Dates in the Number Julian calendar of days Julian calendar of days Libra 27/9 26/10 30 Aries 24/3 23/4 31 Scorpio 27/10 25/11 30 Taurus 24/4 25/5 32 Sagittarius 26/11 24/12 29 Gemini 26/5 26/6 32 Capricorn 25/12 22/1 29 Cancer 27/6 27/7 31 Aquarius 23/1 21/2 30 Leo 28/7 27/8 31 Pisces 22/2 23/3 30 Virgo 28/8 26/9 30 the Sun for 150 b.c., the autumn equinox took place on September 27 (27/9). Thus, 27/9 marks the beginning of the sign of Libra. The duration of the crossing of the Sun through each of the zodiac signs, used in our calculations (see Table 1), is that mentioned in the parapegma of Geminos. 18 3. results 3.1. The Sequence of Star Events The time order of two star events can reverse as one moves from lower to higher latitude. As we see in Figure 2, the largest effects are shown by Vega and Arcturus, Fig. 2. Dates of the stars visible risings and settings v. geographic latitude for the events on plate C1-a using (above) the Nawar ( Sky (ref. 11)) and (below) the Koomen ( Measurements (ref. 12)) sky brightness values. k = 0.23.

the astronomical events of the parapegma 177 Fig. 3. The total sequence error (e s solid curves) and the total zodiac error (e z dashed curves) v. geographic latitude for the events on plate C1-a using the Nawar (above) and the Koomen (below) sky brightness values. k = 0.17, 0.23, 0.30. for the dates of their events change by a whole month as we move from 25 N to 45 N. By contrast, the dates of the events for stars near the ecliptic depend only weakly on latitude. The sequence of the six star events inscribed on plate C1-a was compared to the calculated sequence, for the year 150 b.c., for latitudes between 25 N and 45 N, with k = 0.17, 0.23, 0.30. Sequence errors were determined by placing the calculated events in the order mentioned on the parapegma and by assigning errors as follows: when two events in the parapegma are listed in the time order that agrees with the actual phenomena an error of 0 days is assigned, while in the opposite case the assigned error equals the number of days by which the first event deviates from the following event. So each star event is compared just with the star events that come immediately before and after it on the parapegma. Figure 3 shows the sum of the sequence errors (hereafter: total sequence error (e s )) for each latitude (φ). For k = 0.17, we have a perfect matching of the sequence of the parapegma events at 34.0 N for Nawar s sky brightness values and at 34.5 N for Koomen s. An extinction coefficient of 0.17 corresponds to a very clear sky at an elevation a several hundred metres above sea level. This is the lowest plausible value that could have been enjoyed by the ancient observer. At k = 0.30, the sky is not quite hazy, but is on the verge of becoming so. Two results apparent in Figure 3 are that a hazier sky somewhat broadens the range

178 Magdalini anastasiou and colleagues Table 2. Latitudes of best fit, according to the sequence of star events and the zodiac signs, for both cases of sky brightness values and all three values of k. Sequence of k = 0.17 k = 0.23 k = 0.30 star events e s 5 days e s 7.5 days e s 10 days Nawar 32.3 37.0 N 32.2 37.2 N 32.0 39.0 N Koomen 33.3 38.0 N 33.2 38.6 N 32.9 39.2 N Nawar-Koomen 33.3 37.0 N 33.2 37.2 N 32.9 39.0 N Zodiac k = 0.17 k = 0.23 k = 0.30 signs e z 10 days e z 12.5 days e z 15 days Nawar 25.8 38.5 N 25.8 39.1 N 27.0 40.0 N Koomen 26.8 39.5 N 27.3 39.7 N 30.3 40.5 N Nawar-Koomen 26.8 38.5 N 27.3 39.1 N 30.3 40.0 N of possible latitudes and shifts its centre to the north. For k = 0.23, we still have well-defined minima, with only slightly elevated total error. For k = 0.30, the total error becomes large enough that we may reasonably set this case aside. Table 2 summarizes the latitudes of best fit for each case of sky brightness values and k. The dates of the events calculated with the two sky brightness measurements (by Nawar or Koomen) can differ from 0 to 5 days, so we take as latitudes of best fit those lying within a range of about 5 days from the minimum of the curves. When all cases of brightness values and k are considered, the parapegma events are, in terms of their sequence, best fitted for latitudes in the range 33.3 37.0 N. This is basically the range established for k = 0.17 and k = 0.23. For k = 0.17, the best fitted latitudes are in the range 33.3 37.0 N. As we have mentioned, an extinction coefficient as low as 0.17 could suggest that the observations were made at an elevation of a few hundred metres above sea level. This is consistent with ancient mentions of astronomical and astro-meteorological observations being made on hills or mountains. 19 But if one deems a slightly higher value for the extinction coefficient more plausible, the k = 0.23 curve implies practically the same latitude, in the range 33.2 37.2 N. The results do not really allow us to settle on the most likely extinction coefficient, and any value between 0.17 and 0.23 seems acceptable. Again, with the latitude range defined by 5 days error above the minimum of the curve, these two curves give nearly the same range. 3.2. The Zodiac Signs Using the zodiac statements of plate C1-a, zodiac errors for each of the six star events were estimated. When the event takes place within the boundaries of the correct (as mentioned on plate C1-a) zodiac sign, we assign an error of 0 days while in the opposite case the assigned error is set equal to the number of days by which the event deviates from the correct sign. Figure 3 shows the sum of the zodiac errors (hereafter: total zodiac error (e z )) for each latitude (φ). All curves of this zodiac analysis have a similar structure with an almost constant e z except for the northern latitudes where e z increases. e z reaches the minimum of 5 days for k = 0.17.

the astronomical events of the parapegma 179 Table 2 summarizes the latitudes of best fit. All cases of brightness values and k considered, the parapegma events are, according to the zodiac signs, best fitted between 30.3 N and 38.5 N. This is a rather large range of latitudes that leads us to conclude that the basic sequence of star events in time provides a more useful tool than does the zodiac sign in which an event is said to occur. However it contains the range mentioned in 3.1. 4. discussion 4.1. The Geographic Dependence Considering the analyses of both the sequences and the zodiac signs in 3, the parapegma events on plate C1-a of the Antikythera Mechanism are best fitted for latitudes 33.3 N 37.0 N. Several ancient Greek cities lie within or immediately adjacent to this zone. Among these cities, Rhodes (36.4 N) and Syracuse (37.1 N) have often been suggested as places of origin and/or use of the Antikythera Mechanism. 20 The northern cities in Epirus, Corcyra (latitude 39.6 N), Dodona (39.5 N), and Bouthrotos (39.7 N), whose month names well match the luni-solar calendar of the Mechanism, 21 seem not to be favoured. Corinth (37.9 N) as well as Tauromenion in Sicily (37.8 N) lie just outside the most likely zone. Of course, we cannot give a sharp cutoff, but the likelihood decreases progressively as one moves north of 37.0 N. The preserved events of the parapegma are too small in number to decisively establish the latitude for which the Antikythera parapegma was composed. And of course, we do not really know whether the parapegma was created from direct observations or under the influence of an already existing body of knowledge (which must ultimately have rested on observations of some kind) or even from rules of thumb or schematic patterns or questionable mathematical calculations. However, the basic order in time of events involving stars that were not only conspicuous in the sky but also prominent in the parapegma tradition may give us at least a strong indication. When we consider the care that went into other parts of the Mechanism, it seems unlikely that the designer would overlook accuracy and would get wrong the basic time sequence of easily observed events involving such prominent stars. The fact that the Antikythera Mechanism data fit reasonably well with our numerical solutions of the real events strongly supports that (a) the parapegma was accurate and (b) its data very likely originated from real observations. 4.2. Star Detection Probability Tousey and Koomen s threshold stellar apparent magnitude values 22 apply to an approximate 98% probability of seeing the star against a given sky brightness. This is a good detection probability to use as the observation of a star gets dubious for a level of probability less than 0.90. For example, Blackwell 23 reports that experimental observers did not feel confident of having seen a stimulus (a point of light)

180 Magdalini anastasiou and colleagues unless the level of probability of detection was greater than 0.90. (Moreover, these probability levels for visual detection do not include the effects of any stray wisps of clouds: that is, the probabilities are more likely to overstate than to understate the actual likelihood of detection in a realistic situation.) However, results were also obtained using values of threshold stellar apparent magnitude for a 50% probability of detection. The parapegma events were, in this case, best fitted between 34.2 and 38.2 N. These results basically show a shift of the range of the best fitted latitudes towards northern latitudes, which is rather expected as going to 50% is more or less equivalent to accepting greater atmospheric extinction. Using the 50% probability criterion, the northern cities of Epirus would still be comfortably excluded, while Corinth and Tauromenion would be allowed. 4.3. Visible or True Phenomena? A diagram of the total error v. geographic latitude was also calculated on the assumption that the events in the parapegma represent true risings and settings 24 rather than visible ones (see Appendix C of the online edition). This results in substantially elevated errors (for both the sequence and the zodiac graphs) and much less welldefined minima. This seems to confirm what one would expect on a priori grounds, that the parapegma of the Antikythera Mechanism records visible risings and settings. But it also provides a good test of the discriminatory power of our method. 4.4. Ptolemy s Parapegma One might wonder whether our method of analysis can be tested in order to confirm that it returns reasonable results. For this purpose, we examined Ptolemy s parapegma, as recorded in his Phaseis. 25 In the Phaseis, Ptolemy reported a very detailed parapegma, using the visible risings and settings of 30 bright stars, for five different geographic latitudes. We investigated the star events referring to the clima of 14 hours. In the tables of ascensions in the Almagest, Ptolemy identifies the clima of 14 hours with lower Egypt and a latitude of 30 22. Using a more accurate value for the obliquity of the ecliptic in Ptolemy s day gives 30 33 26 for the latitude where the longest day is 14 hours. And the actual latitude of Alexandria, where Ptolemy lived and worked, is 31 13. Table 4 in Appendix D of the online edition shows the stars as cited in Phaseis and the stars that they were identified with. In Ptolemy s parapegma, the star events are dated according to the Alexandrian month in which they occur (instead of the zodiac constellations that we see in the case of the Antikythera Mechanism). (For more details on this, see Appendix D.) So, the statistical method of the zodiac signs was slightly modified. Instead of zodiac errors, we calculated month errors, on the basis, though, of the same concept: when the event takes place within the boundaries of the correct Alexandrian month, we assign an error of 0 days, while in the opposite case the assigned error is set equal to the number of days by which the event deviates from the correct Alexandrian month. So the total month error (e m ) is essentially equivalent to the total zodiac error (e z ).

the astronomical events of the parapegma 181 The calculations were done using star places for a.d. 130. Table 5 in Appendix D shows the boundaries of the months of the Alexandrian calendar, for a.d. 130. The mr of Canopus that occurs on the 2 nd of the 5 epagomenal days was not considered for the month analysis, as the Epagomenae days do not belong to any month, but to a very short interval of 5 (or 6) days in a year. Naturally, sequence errors were also calculated, using the method described in 3.1. For the star events that occur on the same day, the sequence errors displayed in Figure 4 are the ones calculated with these events in the order that they are mentioned by Ptolemy. Interchanging their order was tested but the results were practically similar in each case. The total number of star events mentioned by Ptolemy for the clima of 14 hours is 110, so it seems that Ptolemy made no reports for 10 star events 27 (we expected to have a total of 30 stars 4 events = 120 star events). For the events of the low declination and faint stars α Sagittarii and θ Eridani, the software calculated no dates with the Koomen brightness values for most cases of k. This is due to the high apparent magnitudes of these stars, which are comparable to the smallest sky background brightness values in Koomen s study. The dates of these star events calculated with the Nawar brightness values are also problematic as they give unstable results and are therefore not trustworthy. 28 Consequently, all events of α Sagittarii and θ Eridani 29 were excluded from the analysis and the total number of events taken into account in our calculations is 110 8 = 102 events. Figure 4 shows the total sequence (e s ) and month (e m ) errors. The rather strange peaks that can be seen in the range 33 38 N are due to the events of the very low declination stars Canopus and α Centauri. These stars become invisible (i.e., remain permanently below the horizon) at the higher geographic latitudes and so the program calculates no dates (and consequently no errors) for these regions, while it calculated Fig. 4. The total sequence (e s solid curves) and month (e m dashed curves) error v. geographic latitude (φ), for k = 0.23, for the star events reported in Ptolemy s Phaseis for the clima of 14 hours, omitting the events of α Sagittarii and θ Eridani. Both the Nawar and the Koomen sky brightness values have been used. The apparent sharp drop-offs occur at latitudes where two stars (Canopus and α Centauri) cease to have risings and settings because of their very low declination. Thus the significant minima are in the vicinity of 30.

182 Magdalini anastasiou and colleagues errors for the southern latitudes. When one removes the star events of Canopus and α Centauri, Ptolemy s resulting curves lose the strange peaks and get quite smooth. However, although Canopus and α Centauri may cause disturbances in the graph towards the northern latitudes, their role in the southern ones is substantial and they cannot be ignored in our calculations. Figure 7 in Appendix D displays the total sequence (e s solid curves) and month (e m dashed curves) errors per event. (That is, we divide the total error by the number of star events.) This facilitates comparison with the errors of the Antikythera Mechanism parapegma. The error values of Figure 4 for Ptolemy s parapegma that may have seemed quite large are due to the large number of events and, as Figure 7 shows, when these errors are calculated per event, they turn out to be quite small error values, comparable to the ones of the Antikythera Mechanism. Ptolemy s resulting curves are plotted from 25 N to 33 N. The zodiac error per event for the Antikythera Mechanism is comparable to the month error per event for Ptolemy s parapegma for the clima of 14 hours, which again suggests that the Antikythera Mechanism parapegma is based on actual observations for a single latitude. The sequence error per event for Ptolemy s parapegma is about four times higher than the sequence error per event for the Antikythera Mechanism. However, this is probably only a density effect. In the case of the Antikythera Mechanism parapegma, we have 6 useable events distributed over three signs of the zodiac a density of about 2 events per sign. In the case of Ptolemy s parapegma, we have 102 useable events distributed over 12 months a density of about 8.5 events per month. Having more events per month multiplies the opportunities for crossings and thus leads to larger sequence error values. As we have seen, the sequence error is the more sensitive guide for determining latitude. However, the zodiac error provides the more directly useful tool for comparing accuracies of two different parapegmata, as it does not need to be corrected for the density effect. Figure 5 displays the average of the errors obtained using Koomen and Nawar sky brightness values, for all 102 events, for latitudes up to 33. Assuming that both Ptolemy s (102) and the Antikythera Mechanism (6) parapegma events are independent and therefore follow a normal distribution, the range of 5 days from the minimum of the curves taken for the Antikythera Mechanism parapegma corresponds to a range 101 5 days 22.47 days 22.5 days 5 for Ptolemy s parapegma. This range of days gives latitudes of best fit between 27.8 N and 31.5 N, which does indeed include the latitude where this parapegma was probably compiled. A point concerning the text of the Phaseis: Heiberg s text 30 is based substantially upon only two manuscripts, Vaticanus gr. 318 (fourteenth century, called A) and Vaticanus gr. 1595 (ninth century, called B), though of course in his apparatus he makes occasional references to a few other manuscripts and sometimes refers to the views of earlier editors. In quite a number of places, especially in the second half of the parapegma, there are uncertainties in the readings. Sometimes, for example,

the astronomical events of the parapegma 183 Fig. 5. Total sequence (e s solid curves) and month (e m dashed curves) error v. geographic latitude (φ), for k = 0.23, for the star events reported in Ptolemy s Phaseis for the clima of 14 hours, omitting the events of α Sagittarii and θ Eridani. The curves are plotted from 25 N to 33 N and represent the average of the e s and e m values calculated using the Nawar and the Koomen sky brightness values. one manuscript will mention a star event for the clima of 14 hours while the other will say it is for the clima of 14½ hours, and Heiberg will have to choose. To be as scrupulous as possible, we also analysed Ptolemy s parapegma for 14 hours, being careful to omit all entries for which there was any conflict between the two principal manuscripts. 31 This led to the elimination of nearly half the entries, mostly from the second half of the year. The inferred latitude range, 29.9 N 32.6 N range of days from the minimum of the curves: days compared to that obtained using Heiberg s entire text, is shifted a little to the north, still though reasonably consistent with the latitude of Alexandria. 4.5. Geminos s Parapegmata of Euctemon and Eudoxus The Geminos parapegma includes a substantial number of visible risings and settings atrributed to Euctemon, Eudoxus and Callippus. The analysis of 3, for the visible events, was also repeated for Euctemon s parapegma, where quite large sequence errors were found and for Eudoxus s parapegma, where false citations were detected (see Appendix E of the online edition). Callippus s statements are very scarce and, for many of them, it is difficult to be sure exactly to which stars they refer. For example, Callippus will say that Sagittarius begins to rise, or Sagittarius finishes rising. For these reasons, Callippus s parapegma was not investigated. A notable result is that the data from the Antikythera parapegma, while few in number, are of much higher quality than the preserved data of Eudoxus s parapegma. Indeed, they are of a quality comparable to the data in Ptolemy s parapegma. While the

184 Magdalini anastasiou and colleagues Antikythera data are too few to allow certainty, their general quality is not inconsistent with the notion that they were the results of actual observation at a particular latitude. 5. conclusions A small part of the parapegma, with six star events and three zodiac statements, is preserved on Fragment C of the Antikythera Mechanism. A new method of astronomical analysis of these events and statements reveals that the parapegma works best for geographic latitudes in the range 33.3 N 37.0 N. The method was also applied to other parapegmata and in the case of Ptolemy, the results corroborate very well the geographic region that Ptolemy describes. It should be pointed out, however, that (a) the limited number of preserved events and (b) the historical uncertainties of the way the parapegma was actually formed do not allow a strong statement about the geographic association of the Antikythera Mechanism to be made. Rather, the star phases can be taken as suggesting a likely band of latitudes 33.3 N 37.0 N; and they do make it appear unlikely that the parapegma could have been designed for the northern cities of Epirus that might have been acceptable based on the calendrical data. Acknowledgements The authors acknowledge the critical role of the National Archaeological Museum in Athens and its staff in obtaining the original data. They would like also to thank B. E. Schaefer for consultations on atmospheric extinction and T. Freeth and M. T. Wright for helpful comments on the original draft. Magdalini Anastasiou acknowledges a Ph.D. grant from the Aristotle University. REFERENCES 1. T. Freeth, Y. Bitsakis, X. Moussas, J. H. Seiradakis, A. Tselikas, H. Mangou, M. Zafeiropoulou, R. Hadland, D. Bate, A. Ramsey, M. Allen, A. Crawley, P. Hockley, T. Malzbender, D. Gelb, W. Ambrisco and M. G. Edmunds, Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism, Nature, cdxliv (2006), 587 91. For this paper there is substantial Supplementary information available at http://www.nature.com/nature/journal/v444/n7119/ suppinfo/nature05357.html. Also: M. T. Wright, The Antikythera Mechanism reconsidered, Interdisciplinary science reviews, xxxii/1 (2007), 27 43. 2. T. Freeth, A. Jones, J. M. Steele and Y. Bitsakis, Calendars with Olympiad display and eclipse prediction on the Antikythera Mechanism, Nature, cdliv (2008), 614 17. For this paper there is substantial Supplementary information available at http://www.nature.com/nature/journal/v454/n7204/ suppinfo/nature07130.html. Also: J. Evans, C. C. Carman and A. S. Thorndike, Solar anomaly and planetary displays in the Antikythera Mechanism, Journal for the history of astronomy, xli (2010), 1 39; and M. T. Wright private communication. 3. D. de Solla Price, Gears from the Greeks: The Antikythera Mechanism A calendar computer from ca. 80 B.C., Transactions of the American Philosophical Society, n.s., lxiv/7 (1974) (reprinted New York, 1975). 4. C1-b is part of the front plate which has been broken, detached and stuck inverted on side 1 (ref. 1) of Fragment C. 5. Price was the first to link the letters of the alphabet inscribed in order along the zodiac dial with the letters at the beginning of the lines of the parapegma text. See Price, Gears (ref. 3), 18.

the astronomical events of the parapegma 185 6. D. Lehoux, Astronomy, weather, and calendars in the ancient world: Parapegmata and related texts in classical and Near Eastern societies (New York, 2007). 7. Price, Gears (ref. 3), 49. 8. This observation was also publicly announced by T. Freeth and M. Anastasiou (et al.) in their oral presentations in the XXIII ICHST (International Congress of History of Science and Technology) Ideas and Instruments in Social Context, 28 July 2 August 2009, Budapest, Hungary. 9. J. Evans, The history and practice of ancient astronomy (New York, 1998), 191. 10. C. Schoch, The arcus visionis of the planets in the Babylonian observations, Monthly notices of the Royal Astronomical Society, lxxxiv (1924), 731 4; B. E. Schaefer, Predicting heliacal risings and settings, Sky and telescope, lxx (1985), 261 3; B. E. Schaefer, Heliacal rise phenomena, Archaeoastronomy, xi (1987), 19 33; B. E. Schaefer, Astronomy and the limits of vision, Vistas in astronomy, xxxvi (1993), 311 61; M. Robinson, Ardua et astra: On the calculation of the dates of the rising and setting of stars, Classical philology, civ (2009), 354 75; Alcyone software (Lange, R.), available at http://www.alcyone.de/ 11. S. Nawar, Sky twilight brightness and colour during high solar activity, The Moon and the planets, xxix (1983), 99 105. 12. M. J. Koomen, C. Lock, D. M. Packer, R. Scolnik, R. Tousey and E. O. Hulburt, Measurements of the brightness of the twilight sky, Journal of the Optical Society of America, xlii (1952), 353 6. 13. Freeth et al., Decoding (ref. 1), 587. 14. P. Duffett-Smith, Astronomy with your personal computer (New York, 1990). 15. Ptolemy, Almagest VII, 5; G. J. Toomer, Ptolemy s Almagest (London, 1984). 16. The total apparent magnitude of the Pleiades was calculated by finding the brightness of each of the ten stars from its apparent magnitude m and by adding these values of brightness. From the total brightness value, the total apparent magnitude of Pleiades was calculated. The mean equatorial coordinates of the Pleiades, thus the mean right ascension α and the mean declination δ, were calculated correspondingly as a mean value of the ten right ascension values of the ten stars, α, and as a mean value of the ten declination values of the ten stars, δ. 17. The autumn equinox taking place when Libra begins to rise was first publicly reported by T. Freeth in his oral presentation in Budapest The parapegma of the Antikythera Mechanism (ref. 8). The presentation was reflecting work by A. Jones, T. Freeth and Y. Bitsakis, work which is currently in preparation. In this work, the plate C1-b was adjusted to other fragments of the Mechanism and the lines of the zodiac statement and the autumn equinox were reconstructed revealing that the two events take place at the first degree of the zodiac sign of Libra. The presence of lines in the parapegma referring to equinoxes and solstices remained unknown until then. 18. J. Evans and J. L. Berggren, Geminos s Introduction to the Phenomena (Princeton, 2006), 231 40. 19. Meton is said by a scholiast to Aristophanes to have observed on the Pnyx, a small hill in Athens. See G. J. Toomer, Meton, Complete dictionary of scientific biography, ix (Detroit, 2008), 337 40. Theophrastus (De signis I, 4) says that matters ( signs to be understood) concerning the solstices were observed by Matriketas at Methymna from Mount Lepetymnos, by Cleostratus in Tenedos from Mount Ida, and by Phaeinos at Athens from Mount Lycabettos. See David Sider and Carl Wolfram Brunschön, Theophrastus of Eresus On Weather Signs (Leiden and Boston, 2007), 60 3. 20. Price, Gears (ref. 3), 13. and Freeth et al., Calendars (ref. 2), 614. 21. Freeth et al., Calendars (ref. 2), Supplementary Information, p. 17. 22. R. Tousey and M. J. Koomen, The visibility of stars and planets during twilight, Journal of the Optical Society of America, xliii (1953), 177 83 (Tables I and Fig. 1, p. 177). 23. H. R. Blackwell, Contrast thresholds of the human eye, Journal of the Optical Society of America, xxxvi (1946), 630. 24. For an explanatory and very thorough analysis of the true and visible events, how they are related, their annual cycle, as well as the understanding of this cycle already at the time of Autolycus of Pitane (320 b.c.), see Evans, op. cit. (ref. 9), 190 7. 25. For the Greek text of Ptolemy s Phaseis see J. L. Heiberg (ed.), Claudii Ptolemaei Opera quae exstant

186 Magdalini anastasiou and colleagues omnia, ii: Opera astronomica minora (Leipzig, 1907), 1 67. A. Jones, Provisional translation of Ptolemy s Phaseis is available at http://www.classicalastrologer.com/phaseis.pdf 26. The geographic latitude where the longest day has a length of 14 hours was calculated using the equation: cos H δ = tan φ tan δ, where H δ is the hour angle of the Sun at sunset and δ its declination on the longest day. For 14 hours day length, H δ = 7h and for the longest day δ is equal to the obliquity of the ecliptic ε. For a.d. 130, ε was calculated using the equation provided by the Astronomical Almanac for the Year 2012: ε = 84381.406 46.836769 Τ 0.0001831 Τ + 0.00200340 Τ 6 4 8 5 0.576 10 Τ 4.34 10 Τ = 85244.1177 = 23.67892 2 3 where T is the time in Julian centuries since the ephemeris epoch of 2000. The day length of 14 hours was found to correspond to φ = 30.55 N = 30 33 N. 27. The events missing are: the α Leonis (Regulus) mr, the α Tauri (Aldebaran) ms, the α Virginis (Spica) mr and er, the β Aurigae mr, the α Cygni (Deneb) mr, the β Geminorum (Pollux) mr, the α Librae mr and er and the α Scorpii (Antares) ms. 28. Koomen et al., Measurements (ref. 12). Nawar, Sky (ref. 11). 29. That θ Eridani should be problematical is perhaps not surprising. This star is characterized by anomalously large errors in longitude in both Ptolemy s star catalogue in the Almagest and in Hipparchus s Commentary on the Phenomena of Eudoxus and Aratus. See Gerd Grasshoff, The history of Ptolemy s star catalogue (New York, 1990), 97, 215 16. 30. Heiberg, Claudii (ref. 25). 31. The entries for which there is conflict between the two principal manuscripts are: the α Aurigae (Capella) mr, the α Lyrae (Vega) er, the α Boötis (Arcturus) mr, ms and er, the β Leonis (Denebola) ms, er and es, the α Tauri (Aldebaran) mr and es, the α Orionis (Betelgeuse) mr, ms, er and es, the β Orionis (Rigel) mr, er and es, the α Canis Majoris (Sirius) es, the α Piscis Austrini (Fomalhaut) mr, ms, er and es, the θ Eridani (Acamar) mr, the α Carinae (Canopus) mr, er and es, the α Centauri (Rigil Kent) mr and er, the α Persei (Mirfak) mr, er and es, the β Aurigae (Menkalinan) es, the α Cygni (Deneb) er, the α Coronae Borealis (Alphecca) mr, the α Geminorum (Castror) mr and ms, the β Geminorum (Pollux) ms and es, the α Andromedae (Alpheratz) mr and es, the γ Orionis (Bellatrix) mr, er and es, the α Hydrae (Alphard) mr, ms and es, the β Librae (Zubeneschamali) mr and ms, the ε Orionis (Alnilam) mr and es, the α Librae (Zubenelgenubi) ms, the α Scorpii (Antares) er and es and the α Sagittarii (Alrami) ms.

the astronomical events of the parapegma A 1 appendix a: method In order to calculate the visible risings and settings of the stars, a software program was written, from first principles, using the Mathematica software package. For any place with geographic latitude (φ) and for any object on the celestial sphere (e.g., a star or the Sun) with equatorial coordinates (α, δ) and altitude (υ) above the horizon, the azimuth (A) and the hour angle (H) of the object can be calculated, using the following basic equations of spherical trigonometry: sinδ sinϕ sinυ cos A = cosϕ cosυ (1) sinυ = sinϕ sinδ + cosϕ cosδ cos H (2) The sidereal time (ST) at which the object is located at position (A, υ) of the celestial sphere can be calculated from the equation: ST = α + H (3) The stars equatorial coordinates are practically stable during a year. However, due to the orbit of the Earth around the Sun and the obliquity of the ecliptic, the Sun s equatorial coordinates (α i-sun, δ i-sun, i = 1 to 365) need to be set on a daily basis. For the visible events, the Sun s depression and the star s altitude are not known. In order to avoid the ambiguity caused by these two free parameters, we used the above equations to calculate, for each of the 365 days of the year (i = 1 to 365), for all j values of the Sun s altitude, υ j-sun, between 15.0 and 1.0 (j in steps of 0.1, 141 values), the horizon coordinates of the Sun in the eastern (A i,j-suneast, υ j-suneast ) and the western sky (A i,j-sunwest, υ j-sunwest ) as well as the sidereal time (ST i,j-suneast ) and (ST i,j-sunwest ) for each position of the Sun. Having computed the position of the Sun, for the known sidereal time (ST i,j-suneast ) or (ST i,j-sunwest ), the hour angle of the star, H i,j-star-suneast or H i,j-star-sunwest, using Equation 3, can be calculated. The altitude of the star and its azimuth are computed using Equations 2 and 1 respectively. So for each set of horizon coordinates of the Sun, i.e. (A i,j-suneast, υ j-suneast ) in the eastern sky or (A i,j-sunwest, υ j-sunwest ) in the western sky, we have the respective horizon coordinates (A i,j-star-suneast, υ i,j-star-suneast ) and (A i,j-star-sunwest, υ i,j-star-sunwest ) of the star. Whether a star of apparent magnitude (m Star ) above the horizon is visible or not depends on the sky background brightness. Nawar 32 published numerical tables of the sky twilight brightness B as a function of three variables: the depression of the Sun below the horizon, the azimuthal distance of a sky patch away from the Sun (the bearing angle from the solar vertical) and the altitude of the sky patch above the horizon. Nawar actually gives two tables, one using a blue and one using a red filter. We interpolate between the two values to obtain an approximate value for the V (visible) band. Using the Mathematica software package, we interpolated these V band values, to any position on the sky for any altitude of the Sun between 15.0 and 1.0 (in steps of 0.1 ). Independent measurements of sky brightness B were also published by Koomen et al., 33 again in

A 2 Magdalini anastasiou and colleagues Table 3. Results using Nawar s sky brightness values (ref. 11) for the visible mr and es of Vega in Athens, in 150 b.c. (k = 0.23). Visible mr of Vega Sun s Date in the altitude Julian ( ) calendar Star s apparent altitude ( ) Difference between the Sun s and the star s altitude Visible es of Vega Date in the Julian calendar Sun s altitude ( ) Star s apparent altitude ( ) 15 16/11 1.51 16.51 15 15/1 1.70 16.70 14 15/11 1.54 15.54 14 16/1 1.76 15.76 13 15/11 2.13 15.13 13 17/1 1.81 14.81 12 14/11 2.16 14.16 12 18/1 1.87 13.87 11 13/11 2.18 13.18 11 19/1 1.92 12.92 10 12/11 2.21 12.21 10 20/1 1.98 11.98 9 11/11 2.23 11.23 9 21/1 2.04 11.04 8 11/11 2.86 10.86 8 21/1 2.67 10.67 7 11/11 3.50 10.50 7 22/1 2.73 9.73 6 11/11 4.16 10.16 6 21/1 4.01 10.01 5 13/11 6.18 11.18 5 20/1 5.35 10.35 4 18/11 10.48 14.48 4 15/1 9.49 13.49 3 28/11 19.05 22.05 3 5/1 17.63 20.63 2 21/12 39.10 41.10 2 8/12 39.68 41.68 1 30/1 69.56 70.56 1 17/10 71.30 72.30 Difference between the Sun s and the star s altitude the form of tables at several altitudes of a sky patch above the horizon and several azimuthal distances from the Sun, for several depressions of the Sun, for two sites: Sacramento Peak and Maryland. The measurements in Maryland (altitude 30m) were similarly interpolated. With these interpolating functions, the sky background brightness at the position (A i,j-star-suneast, υ i,j-star-suneast ) and (A i,j-star-sunwest, υ i,j-star-sunwest ) of any star was calculated. Having computed the sky brightness B at the position of the star, we use Tousey and Koomen s table of threshold stellar apparent magnitude 34 as a function of the brightness of the sky background. Tousey and Koomen published values of the maximum apparent magnitude, m, that a star can have and still be visible at a given sky brightness, B. Their values apply to an approximate 98% probability of seeing the star. Using these values, an interpolating function was formed in order for the visibility (yes or no) of a star at its position on the sky (A i,j-star-suneast, υ i,j-star-suneast ) or (A i,j-star-sunwest, υ i,j-star-sunwest ) to be determined. Looking through the 141 365 values of (A i,j-star-suneast, υ i,j-star-suneast ), we record for each depression of the Sun before sunrise the first day of the year that the star is visible, as well as the last day of the year that the star is visible. Looking through the 141 365 values of (A i,j-star-sunwest, υ i,j-star-sunwest ), we record again for each depression of the Sun after sunset the first day of the year that the star is visible, as well as the last day of the year that the star is visible. The dates of the visible mr, ms, er and es are then located by examining these recordings at all values of Sun s depression. Table 3 shows two examples of results, in steps of 1, referring to the visible mr and es of the star Vega in Athens in 150 b.c., with Nawar s sky brightness values 35 and atmospheric extinction coefficient k = 0.23. The day of the visible mr of Vega (11/11), the first day of the year that Vega is visible and is thus

the astronomical events of the parapegma A 3 at minimum altitude difference from the Sun, is easily located. The day of the visible es of Vega (22/1) is the last day of the year that Vega is visible and is again at minimum altitude difference from the Sun. The depression of the Sun and the altitude of the star when an event takes place, read in the corresponding columns, are not, therefore, free parameters. (Note that in Table 3, the star s apparent altitude (including the effects of atmospheric refraction) is displayed.) The dates of the ms (the first day the star is seen setting) and the er (the last day it is seen rising) are similarly located. The above procedures were repeated for the location of the dates of all star events on the parapegma of the Antikythera Mechanism for all geographic latitudes in the range 25 N 45 N (in steps of 0.5 ). appendix b: parameters and relations B.1. Atmospheric Refraction and Atmospheric Extinction Atmospheric refraction, R, can be significant when a star s visible risings and settings take place. Near the horizon, the altitude of a star or the Sun increases significantly (by up to about 0.5 ). This effect was taken into account, using the following semi-empirical formula, 36 1.02 R = (4) 10.3 tan h + h + 5.11 where h ( ) is the true altitude. The formula is reasonably accurate for all h 0. The refraction correction was applied to all star places. The Sun s position is always below the horizon so no correction for refraction was needed. (Or, to put it another way, both Koomen s and Nawar s sky brightness functions are tabulated in terms of true depressions of the Sun, so refraction does not enter in.) Atmospheric extinction reduces the brightness of a star as seen from Earth. The corrected apparent magnitude mʹ of a star was calculated using the formula: 37 11 cos Z 1 m = m m = k X = k (5) cos Z + 0.025 e where k is the extinction coefficient, X the air mass and Z ( ) the apparent zenith distance of the star, due to atmospheric refraction. The coefficient k varies with location ranging from 0.1 (perfect night on a very tall mountain) to 0.4 (poor night at a humid sea-level site). 38 In our analysis, three different cases with k = 0.17, 0.23, 0.30 covering reasonable values for ancient observations were examined. Schaefer s more accurate formulas for the extinction near the horizon (see Appendix B.2) were also tested, with insignificantly different results. Β.2. Accurate Formulas of the Atmospheric Extinction Near the Horizon The change in the apparent magnitude of a star near the horizon due to extinction was accurately calculated 39 as the sum of the contributing components of the three sources of atmospheric extinction (Rayleigh scattering by gas molecules, Mie

A 4 Magdalini anastasiou and colleagues scattering by aerosols and absorption by ozone in the stratosphere): m = m m = k (6) R X e(8.2 km) + koz X L (20 km) + ka X e(1.5 km) where k R, k oz, k a are respectively the Rayleigh, the ozone and the aerosol extinction coefficients, X e (8.2 km) the air mass for Rayleigh scattering (characterized by a scale height of 8.2 km), X e (1.5 km) the air mass for the aerosol component (scale height of 1.5 km) and X L the air mass for the ozone layer. The extinction coefficients were calculated from the equations: H /8.2 kr = 0.1066 e (7) k 0.031 ( 3.0 0.4 ( cos cos3 oz = + ϕ αs ϕ )) (8) 3.0 k = k ( k + α R koz ) (9) where H (km) is the altitude of the location of the observer, φ (radians) the geographic latitude and α S (h) the right ascension of the Sun. We used H = 0.3 km. For the aerosol extinction coefficient, k α, Schaefer 40 published a more complicated formula that depends on the relative humidity but has substantial uncertainties. In order to avoid these uncertainties, k α was calculated from Equation 9, for the three different cases of k = 0.17, 0.23, 0.30, that were mentioned above. The air masses were calculated from the equations: 1 30cos Z Χ (10) e ( He ) = cos Z + 0.01 He exp H e 2 0.5 sin Z Χ L ( H L ) = 1 (11) 1 + ( H L / R ) o where Z ( ) is the apparent (due to atmospheric refraction) zenith distance of the star, R o (km) is the equatorial radius of the Earth (R o = 6378 km), H e (km) is the exponential scale height of the air mass for the Rayleigh scattering and the aerosols (8.2 km and 1.5 km respectively) and H L (km) is the height of the ozone extinction layer above the observer, which for stratospheric ozone is roughly 20 km. appendix c: visible or true phenomena As the true risings and settings take place when the star and the Sun are on the horizon (υ = 0 ), one can easily calculate, using Equations (2) and (3), the hour angle of a star/sun, when it rises, H StarRise /H SunRise or when it sets, H StarSet /H SunSet as well as the sidereal time of these events ST StarRise /ST SunRise and ST StarSet /ST SunSet. The date of the true mr and the date of the true er of the star can be determined by comparing the ST StarRise with the 365 values of ST i-sunrise and the 365 values of ST i-sunset (i = 1 to 365) respectively. The date of the true ms and the date of the true es of a star can be similarly found by comparing the ST StarSet with the 365 values of ST i-sunrise and the 365 values of ST i-sunset (i = 1 to 365) respectively. The bold curves in Figure 6 show the total sequence error (e s ) and the total zodiac