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On dating Hero of Alexandria

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Abstract

The dating of Hero of Alexandria has been linked with the lunar eclipse of March 13, ad 62, since Otto Neugebauer discovered that this eclipse is the only one that can fit the one described in Hero’s Dioptra 35. Although only a number of scholars claim that Hero himself observed the eclipse, almost all of them take Neugebauer’s identification for granted. We use statistical and linguistic methods to criticize this assumption: all indices we have found point to the fact that the eclipse was merely invented as an example and, for that reason, that it cannot be used to determine Hero’s life span.

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Notes

  1. The direct observation of the eclipse by Hero has only been asserted explicitly by Drachmann (1972, n. 4).

  2. Neugebauer (1938, p. 23, n. 39) used the Oppolzer and Ginzel Tables, whereas Souffrin (2000, p. 14) and Sidoli (2005, p. 251) consulted tables of ancient eclipses and performed additional calculations of their own.

  3. In fact, Souffrin does not argue his probabilistic results: «un calcul simple indique alors que la probabilité a priori qu’une éclipse «convenable» ait eu lieu au moins une fois précisément un dixième jour avant le printemps dans un intervalle de 500 ans est de 0.75» (Souffrin 2000, p. 15). Souffrin does not show us how to do this «calcul simple», but perhaps it is \(\frac{365}{500}= 0.73 \). This result (and the calculation likely producing it) is not correct; the probability that at least one eclipse occurs on a particular day, if we consider a uniform distribution of eclipses, is \(1-\frac{CR_{364,500}}{CR_{365,500}}\), approximately equal to 0.58 (\(CR_{a,b}\) means combinations with repetition of \(a\) elements grouped in sets of \(b\) elements. \(b\) can be greater than \(a\) because the operation allows repetition of elements). This calculation is not difficult, but cannot be described as «simple». Souffrin also performed a calculation taking into account the hour and not only the day of the eclipse. This too appears to be wrong: The correct calculation is \(1-\frac{CR_{4{,}379,500}}{CR_{4{,}380,500}}= 0.10248\) (where \(4{,}380=365\times 12\), and \(4{,}379=365\times 12-1\)). We do not know the exact operations that Souffrin used as he only states that the result «est un peu supérieur[e] à 0.1»; since his previous result of 0.75 is incorrect, it is reasonable to think that this is incorrect too, and that it is close to the result of the exact calculation only by chance. In fact, Souffrin’s second result could be based on a simple operation, close to his former one: \(\frac{500}{365 \times 12}\), that is approximately equal to 0.11. The resemblance between this result and the exact result is due to the fact that the formulae are asymptotic.

  4. We have used the complete tables of ancient eclipses provided by NASA Eclipse Web Site (2013).

  5. This is not a minor problem, since the error (that Sidoli himself acknowledged: Sidoli 2011, p. 7) in Sidoli (2005) is precisely due to a misinterpretation of the dating method used by the table of eclipses he himself had used.

  6. See Meeus (1998) and (Mosshammer (2008), chapter 2) for part of the information contained in this section.

  7. In recent years, some scholars have advocated the use of the religiously neutral abbreviations bce (for «Before Common Era») to replace bc, and ce (for «Common Era») to replace ad. See (Mosshammer (2008), p. 34)

  8. The famous eclipse of Arbela, for example, is placed by some texts in year 331 bc (e.g., Berggren and Jones 2000, p. 29), in other texts in \(-330\) (e.g., Neugebauer 1975, p. 668, n. 30 and the NASA website). Steele (2000) even uses both notations: the Year Zero notation in the tables, the Anno Domini notation in the text. For a concise but complete discussion of dating conventions see http://eclipse.gsfc.nasa.gov/SEhelp/dates.html).

  9. See (Mosshammer (2008), p. 34). There are other minor issues on dating, but they do not affect our analysis. One of them, however, must be briefly mentioned: Souffrin (2000, p. 14) assigns the date of March 14 and not March 13 (namely that provided by all other sources) to the eclipse of year 62. This is probably due to the fact that in 1979 the Terrestrial Dynamical time (TD) or, simply, Terrestrial Time (TT), was introduced. This parameter, which takes into account relativistic effects in the measurement of time, is used to calculate the TD of Greatest Eclipse, namely «the instant when the distance between the center of the moon and the axis or earth umbral shadow cone reaches a minimum». Therefore, the TD of Greatest Eclipse is expressed without reference to a place, which is the usual way in modern tables. The TD of Greatest Eclipse of that observed in Alexandria on March 13, 62, was March 14 at 00:53:05. As a consequence, the date of the eclipse without reference to a place is March 14 and not March 13.

  10. Note that the meaning of a specific time expressions in Greek culture can be unclear because of our inaccurate understanding of its exact meaning, not because it is intrinsically obscure. We shall return to this issue presently.

  11. In the introduction to his translation of the Almagest, Toomer (1984, p. 23), defines and discusses these terms.

  12. Recall that a lunar eclipse can only happen during night.

  13. For example, Sidoli (2005, p. 28) says that the starting time of the eclipse of March 13, 62, in Alexandria was 22:50. Assuming that the UTC time is intended (though it is not specified), the eclipse must occur at 21:50 in Rome, the time in its time zone, 1 h earlier than in Alexandria, and not at 21:40 as Sidoli claims, because in the UTC system the exact longitude of a place does not matter, only its time zone does. Nocturnal hours could be used to take into account differences of time due to longitude, but the two systems must not be mixed.

  14. The duration of an eclipse starting at exactly 1:00 a.m. and ending at 3:30 a.m. is the difference of two numbers, \(3{:}30-1{:}00 = 2{:}30\); so the eclipse lasted 2 h and a half.

  15. If an eclipse occurs in the third hour, i.e., in the interval \([2, 3]\), and ends in the sixth hour, i.e., in the interval \([5,6]\), it has not lasted \(6-3 = 3\) h, but \([5, 6]-[2, 3]=[2, 4]\), i.e., it has lasted «between 2 and 4 h»: the result of an operation on intervals is also an interval (see Cloud et al. 2009).

  16. Called by Ptolemy , night and day. Toomer’s translation does not use a specific term for it, and translates it as «day» (Toomer 1984, p. 23). We shall adopt the lexical calque nychtemeron.

  17. «ipsum diem alii aliter observauere: babylonii inter duos solis exortus, athenienses inter duos occasus, umbri a meridie ad meridiem, uulgus omne a luce ad tenebras, sacerdotes romani et qui diem finiere civilem, item aegyptii et hipparchus a media nocte in mediam.» Historia Naturalis, 2.79, ed. E. H. Warmington, transl. H. Rackham, HUP, 1938.

  18. This issue assumes quite bizarre connotations when considering paradigmatic relations between adjectives linked to time intervals. For example, the Great Panathenaea or the Olympica were called penteteric festivals, [lit. period of 5 years], but were repeated, according to our way of counting, every 4 years (the same as our modern Olympic games). In the LSJ, [lit. period of 5 years] is defined as «happening every five years, quinquennial, cf. »; if we look up this last term, it is identified with [lit. period of 5 years], and the definition reads «festival celebrated every fifth year (inclusively)», whereas the entry [lit. period of 5 years] reads «falling every four (=five inclusive) years, quinquennial». One cannot but get confused by these definitions. The entry [lit. period of 4 years] directs back to [lit. period of 4 years], and this entry reads «a quadrennial festival», or «period of four years», without indication of inclusivity. Still, another paradigmatically related adjective, [lit. period of 3 years] reads «triennial festival, i.e., celebrated every third year (inclusively), \(=\) in alternate years», and shortly afterward, «cycle or period of three (two) years», which is obviously self-contradictory. There are no adjectives * and * [lit. period of 2 years] in the LSJ. Another group of adjectives of time is even more telling in this regard and perhaps worse treated by the LSJ: those ending with , with the same meaning as the former group, «period of \(n\) years ». In the LSJ, there are adjectives of this kind for \(n=2, 3, 4, 5, 6, 9, 10, 100\), but the definitions are not always consistent with each other. For \(n=3, 5, 9\), the definition is «\(n\) years old ». For \(n=5\), there is another adjective with the same ending: \(\upvartheta \), «in the fifth year (inclusively)». For \(n=10\), , «ten-yearly: , a space of 10 years ... more freq. as Subst., period of 10 years, a space of 10 years». For \(n=100\), , «of a hundred years». Probably, \(n=2\) shows most clearly the issue; in this case, the prefix is not a numeral, but a preposition indicating duality, , and the adjective is , «celebrated in yearly festivals». In other words, in this case, we read ‘two’ literally, but we must interpret ‘one’ (inclusive interpretation), whereas for \( n> 2 \), we must conclude the contrary (except in one of the two words for \( n = 5 \)). Therefore, either the system of these adjectives is not entirely consistent or it is not well understood—or both.

  19. When preceded by the Greek text, the references to Toomer’s translation of the Almagest include some numbers preceded by ‘H’, which is a reference to Heiberg’s edition as is also partly (i.e., without the indication of lines) given in Toomer’s book. Thus, H302.16-17 means ‘Heiberg’s edition, p. 302 lines 16–17’ (all passages quoted happen to be contained in the first volume of this edition). This double reference makes also easier to find these passages in Toomer’s translation.

  20. This is important since it is the way in which the eclipse of Dioptra 35 is referred to.

  21. All of this can be explained if we consider the meaning of «eclipse» and its relation with the verb from which it derives: , «leave out», «abandon». An «eclipse» can be understood as the entire period in which the phenomenon occurs (range meaning) and also as its most remarkable feature (punctual meaning), namely the moment in which the moon is covered by the earth’s shadow at a maximum degree—that is, when the moon is in the strictest sense of the verb «left out»; this very moment can be defined with the same word as the entire period: «eclipse». It is telling that sometimes, in the description of a lunar eclipse, when the earth’s shadow covers the moon to the maximum extent (i.e., at the middle of the eclipse), the verb is used without any complement, and in this case indicates the central phase of the eclipse (Toomer 1984, p. 253, H419.16): , «the moon was eclipsed half its diameter from the north»; whereas when the beginning of the eclipse is referred to, the main verb used is «to begin» and not «to be eclipsed» (Toomer 1984, p. 284, H478.2): , «the moon began to be eclipsed». To sum up, the verb «to be eclipsed» and the derived noun «eclipse» contain in their semantic field the complete duration of the event as well as the midpoint of it. It should be noted, however, that in astronomical texts the noun is usually accompanied by complements that specify the exact phase even if the phase is the central phase.

  22. Together with Hero’s passage in Dioptra 35, they are the only recorded eclipses in antiquity observed in two places simultaneously.

  23. According to Berggren and Jones (2000, pp. 29–30), «Ptolemy’s report is [...] in serious error for Arbela whether it refers to the middle or the beginning of the eclipse». This conclusion seems to be the communis opinio. But the fifth hour in Arbela is, approximately, the interval [22:00, 23:00], and the central phase of the eclipse took place in Arbela, as we have seen, in the interval [20:52, 21:56]. Therefore, the end of this phase of the eclipse almost coincides with the beginning of the fifth hour, which suggests that the error is not so serious if we compare it with the mean value of the error on eclipse timings made by the Greek astronomers listed in Ptolemy’s Almagest: \(-0.38\) (Steele 2000, p. 101) (i.e., \(-23\) min).

  24. A comment by Sidoli (2005, p. 258, n. 37) seems to contradict this conclusion: «Steele (2000, pp. 100–102) shows that early Greek astronomers generally recorded the times of eclipse observations by the beginning of the eclipse». We have not found anything that could justify this statement in the whole book of Steele. It appears to be a mistake, probably due to a misinterpretation of these words (Steele 2000, p. 101): «The eclipse timings made by the early Greek astronomers are listed in Table 3.4. Unlike the Babylonian observations, there appears to be a systematic error in the times of all of these observations. The mean value of this error is \(-\)0.38 h. The time of the start of the eclipse in 141 bc is almost an hour early. This is noticeably less accurate than the other records in this group which suggests that there may be some problem with the record. Britton has suggested that the time may relate to the middle of the eclipse rather than the beginning. This would reduce the error to \(-\)0.09; however, there is no real justification for making this correction and so it seems better simply to say that this record is somehow corrupt» (emphasis added). But it is the expression in Ptolemy’s Almagest of the eclipse of 141 bc that leaves no doubt as to the exact time of the eclipse, not the usage of ancient astronomers: «the moon began to be eclipsed», we read in Toomer (1984, p. 284). It is worth noting that (Neugebauer 1975, p. 846, n. 12) suggests, from the eclipse reported in Dioptra 35, that «for the discussion of ancient eclipse records it is of interest to notice that there the “time” of the eclipse refers to the beginning» without adducing evidence other than the actual eclipse of 62. This is a circular argument.

  25. Note that, in both cases, the word that immediately precedes the feminine article ends with the Greek letter (êta). It may be that a copyist transcribed only one of the letters (this kind of quite common scribal error is called «haplography»). It is not easy to assess this possibility, but in the whole Dioptra, we have found 55 pairs of consecutive words such that the first of them ends with an êta and the second begins with the same letter; the feminine article features in 40 such pairs. On the other hand, only one restitution presents a situation as the one in our passage: it is at Schöne (1903, 306.14). Therefore, it seems unlikely that the absence of both articles in our passage is the result of a mistake of copy.

  26. In Greek, when the meaning of is «the same », as in this case, the term must be preceded by an article.

  27. The Teubner edition of Dioptra has only 20 article restitutions. As we have said, the first restitution is required but the second is not. A demonstrative \(+\) noun may not require an article when the noun with which agrees stands as its Predicate or when the Predicate is not so distinctly separated from the Subject; the Greek , «this eclipse», is correct. In any rate, even in the presence of the article, the meaning of this noun phrase can still be indefinite (see note 29).

  28. In fact, there is also another article restitution a few words before (Schöne 1903, 302.16), the dative singular masculine, .

  29. The article is required by grammar prescriptions (so that the syntagma is determinate), but its meaning can still be definite or indefinite, because the semantic opposition definite/indefinite is there neutralized. The same applies to English language, because one must write «the same eclipse», with a determinate article, and not «*a same eclipse» which is an incorrect grammatical construction. Other languages, such as Catalan, allow using use both determinate and indeterminate articles in this context (indeterminate: «un mateix eclipse», determinate: «el mateix eclipse»). Therefore, in Greek (and in English), the semantic opposition definite/indefinite is neutralized, because the (determinate) article is mandatory, and the definite/indefinite meaning of the expression is dictated by the context. In the present instance, the facts that the passage is formulated in a mathematical style (see also infra) and that the next reference to «eclipse» does not need the article restitution, as we have seen in note 27 (i.e., it is an indefinite reference, because Greek has no indefinite articles), suggest that the syntagma «the same eclipse» must be read as indefinite. In mathematical texts, neutralization is very common. For example, we read in \(El.1.30\)      «straight lines parallel to the same straight line are parallel to each other» (in Catalan the article must be indefinite: «rectes paral\(\cdot \)leles a una mateixa recta són paral\(\cdot \)leles entre elles»). There is no doubt that the noun phrase «the same line» is indefinite even though a definite article is mandatory in Greek and English (but not in Catalan). The first article , «the» (feminine and plural), is also mandatory in Greek (for the noun is qualified by a complement in form of a complex syntagma), but not in English « straight lines parallel to the same straight line», because the meaning of the expression is indefinite. Here, we have in Greek, again, the neutralization of the opposition definite/indefinite.

  30. The Greek text does not use the noun «observations» but a participle qualifying the subject of the substantive clause, «\(<\)it will be possible for us,\(>\) when observing».

  31. The sentence «to find an eclipse in the stated regions» sounds a bit strange in Greek, and it is much more reasonable to repeat the complement of the same verb where the verb last appeared, as Greek language frequently avoids repeating verb complements when the verb is repeated. Therefore, we must translate «to find \(<\)in the records\(>\) an eclipse in the stated region»). The expression without the repeated complements sounds so strange, even in translation, that Souffrin changed the verb («to find») to «to observe» (Souffrin 2000, p. 13): «Admettons que cette éclipse est observée dans les régions mentionnes». A. Rome makes the same slip: «supposons que la même éclipse ait été observée dans les deux régions [...]» (Rome 1923, p. 237).

  32. In the translation of Sidoli (2011) the demonstrative could be confused with the adjective , «the same ».

  33. This was already pointed out in Acerbi (2007).

  34. Acerbi and Vitrac (2014, p. 106) proposes , «and in the same night in the 3rd hour», deleting the subsequent sintagma , «obviously the same night ».

  35. For example, sunset in Alexandria on March 13, 2013 was at 18:07 and in Rome at 18:14 (see http://www.timeanddate.com). We do not know if these times change throughout history—probably they do, but the difference will not change, or the change is very small, because the difference in longitude does not change, and this is the most important factor in this difference.

  36. As said above, we have taken the 2013 times, because we do not know the exact hour of sunset in the year 62.

  37. On March 13, the nocturnal hour is almost 1 h long: in Alexandria, 1:00:30, and in Rome, 1:00:55.

  38. It is worth noting that this range of 87 min (10 if 1 takes into account only the difference in longitude between Alexandria and Rome) is 13 % of a whole hour. In other words, 13 % of the fifth nocturnal hour in Alexandria matches 13 % of the third nocturnal hour in Rome.

  39. Nor do the hours set up by other researchers. Neugebauer and Sidoli give 22:50 for the beginning of the eclipse in Alexandria, Souffrin calculates it at 22:34, and the NASA tables we use set it at 22:39. There are no great differences, but none of these could fit the description of the eclipse in Dioptra 35. However, the NASA website offers an estimate of the error (http://eclipse.gsfc.nasa.gov/SEcat5/uncertainty.html): «Morrison and Stephenson (2004) propose a simple parabolic relation to estimate the standard error (\(\sigma \)) which is valid over the period 1000 bce to 1200 ce: \(\sigma = 0.8 * t^2\) seconds, where: \(t = \frac{\hbox {year}-1820}{100}\)». If we substitute \(\hbox {year}=62\), the error is 247.25 s, just over 4 min. Therefore, a range for the beginning of the eclipse with an almost absolute confidence is [22:35, 22:43]. The lower limit is greater by 18 min than the third nocturnal hour in Rome. In any case, our ignorance of the exact time of sunset in Alexandria and Rome on that day introduces uncertainties that cannot be estimated.

  40. In fact, the optimal eclipse to calculate the distance between two cities is one that lasts only one nocturnal hour in both cities, because the observers of the eclipse need not describe any phase of it (beginning, end, etc.) to accurately delimit the exact moment of the eclipse. We might think that the number of lunar eclipses that lasts only one nocturnal hour is very low, but this is not true: Whatever period of years is chosen, the proportion of eclipses that lasts a single nocturnal hour is always close to 18 %.

  41. For the interpretation of the eclipse, we have not used at all the analemmatic construction and its explanation, set forth in Dioptra 35. Sidoli (2005, 2011) and Acerbi and Vitrac (2014, pp. 110–115) discussed such diagrams and text. The diagram is based essentially on a semicircle representing hours, divided into 12 equal parts. Each such part represents a nocturnal hour. We did not use it in our interpretation because, besides being a very simplified geometric representation of the situation, it contains a notable inconsistency: The arc of circumference that represents night, called , is divided into 12 parts; represents five of these 12 parts, because «the eclipse was observed in Alexandria in (or from) the fifth hour; therefore, \(<\)point\(>\) M is in the same position as where the sun was when the eclipse occurred», . But it is clear that point M marks the beginning of the sixth hour (or the end of the fifth hour) and not the beginning of the fifth hour! To put it in modern terms, rather than choosing a corresponding range of the fifth hour, the author has preferred to simplify matters by choosing the first exact point in time after the fifth hour, corresponding to the very beginning of the sixth hour. In short, neither the analemmatic diagram nor the comments on it could be used in the interpretation of the text of the eclipse, because they amount to nothing more than a graphic simplification.

  42. There is only one eclipse after \(-\)2000 and before 350 in the NASA database fitting quite well our text of Dioptra 35: the eclipse of March 13, \(-\)459. It was almost complete (magnitude 0.931), its middle point occurred at 22:16 in Alexandria (i.e., in the spot where Alexandria would be built) and at 21:16 in Rome, i.e., at the fifth hour in Alexandria and at the third in Rome.

  43. As far as we know and we have said in Sect. 3, there is no reason to undermine these facts, but most scholars seem to pass by them. In the next section, we will assess statistically the consequences of this interpretation, even if we have not found any indication to support it.

  44. Removing these two restrictions of our interpretation (eclipse in Rome, reference to the central hour of the eclipse), there are other eclipses in the entire NASA database before 350 that fit as well as the one of 62 of even better our text of Dioptra 35 (March 11, \(-\)1631 and March 13 \(-\)1520, \(-\)999, \(-\)980).

  45. There are exceptions to this unanimity: A. Rome, the first scholar who studied the passage exhaustively, concluded that «le chapitre 35 de la Dioptra ne se rapporte pas à une éclipse réellement observée,» because the data associated with the eclipse in Rome and in Alexandria «sont tellement contradictoires qu’il est impossible qu’ils soient obtenus par l’observation» (Rome 1923, p. 248 and p. 247, respectively).

  46. The lower limit of the interval of Hero’s life span cannot go beyond \(-\)286 (birth of Archimedes; Hero mentions him by name), the upper limit beyond, approximately, 320 (the date at which we can locate the acme Pappus, the first author quoting Hero in Coll. 8). Raïos (2000), Drachman (1972) reduced this interval from different considerations.

  47. We consider a year of 365 days and 1/4, i.e., \(365.25\times 12 = 4{,}383\) nocturnal hours.

  48. Although Souffrin and Neugebauer use probabilities, they do not make their hypotheses explicit. Neugebauer only observes that in a period of five centuries, only one eclipse fits closely enough the one reported by Hero; Souffrin also introduces the probability that an eclipse had occurred on a given day of the year in the same period of five centuries, but does not explain, as we have seen in note 3 above, what this means or what exactly we can deduce from this probability.

  49. The complete table of eclipses can be found in this file: http://www.webcitation.org/6WbQxULEf.

  50. After almost 2000 years, some of these data may have slightly changed but the error is small. Moreover, this error is probably much less than the error in the calculation of time in antiquity and, therefore, can be considered negligible. The data are drawn from http://www.timeanddate.com/worldclock/astronomy.html?n=426&month=12&year=2012&obj=sun&afl=-11&day=1.

  51. For example, the triad [v, 13, March], i.e., the nocturnal time interval \([4,5] \) of March 13 (starting at 22.139 and ending at 23.144 in decimal format and UTC at Alexandria) is associated to the eclipse of March 13, 62 (whose beginning is at 22.65 and the end at 1.67 the next day, in decimal format) because the interval of the eclipse and the triad have a non-zero intersection.

  52. We read in (Sidoli 2011, p. 57): «If the time of the equinox was observed using an instrument like the equinox rings that Hipparchus and Ptolemy, quoting and discussing him, say were set up in Alexandria then it will not have been observed until the sunrise of 23 March. Another possibility is that the date of the equinox was taken from a calendrical type device, like a parapegma. In either case, the lunar eclipse could have been recorded as 10 days prior to the equinox and we are left with only some uncertainties concerning the stated times of the observations.» However, Sidoli does not seem to note that 10 days could well be interpreted as 9 days. His intention is to make the date and time of the actual equinox (18:33 in year 62 as stated in http://www.timeanddate.com/calendar/seasons.html?year=50&n=426) to agree with the data of the text: 10 days before the vernal equinox must be 13 March. What is plainly true, however, is that the exact time that the author of the text (or his source) considered to be the vernal equinox is totally unknown. In fact, the vernal equinox occurs in a movable date over long periods of time, phenomenon due mainly to the precession of equinoxes and already known to Hipparchus, about 150 bc (see Neugebauer 1975, pp. 292ss.).

  53. As a matter of fact, this suggestion seems to be accepted by all scholars, though not explicitly. So, Sidoli finds eclipses that «were 9–12 days before the equinox for this period»; an eclipse in this range would immediately become an acceptable candidate precisely because the exact day of the eclipse cannot be accurately determined.

  54. In the first case, we extend the lower limit to the foundation of Alexandria. In scenario 2, we extended the lower limit by 150  years to make it coincide, roughly, with the birth of Hipparchus. In any case, it is very difficult to give precise margins without knowing what and how ancient almanacs were actually used.

  55. We assume that the error margin is not too big, but we cannot assure what is the precise meaning of not too big.

  56. Hypothesis launched in Acerbi and Vitrac (2014, p. 111(xv)) to justify some geometric operations performed in Dioptra 35: «Ici, on mesure tout l’intérêt d’avoir placer l’éclipse dans la troisième heure à Rome : si on la plaçait dans la quatriéme heure, il faudrait faire la trisection de l’arc (ou de l’angle au centre correspondant) ; en toute généralité, l’opération n’est pas si simple».

  57. See Sect. 3.1. It is in fact reasonable to extend the lower limit of the scenario to consider eclipses recorded up to \(-\)200. Moreover, there is no plausible argument that justifies assuming uniqueness (apart from the circular argument that the eclipse of year 62 is unique); on the contrary, there are many triads with a unique eclipse and, in fact, it is merely a prejudice that it is more likely that the eclipse is real if the eclipse fits a unique eclipse in tables. Finally, the error margin of 1 day is needed because the text does not specify the actual day of the eclipse—in fact it specifies no day at all—and contains textual and factual ambiguities that make an accurate calculation of the day impossible.

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Acknowledgments

I am grateful to Bernard Vitrac for his insightful comments and remarks at every stage of elaboration, which improved the argument and avoided a serious mistake. Special thanks are given to Fabio Acerbi for critical scrutiny of the typescript and detailed remarks.

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  1. Centre Alexandre Koyré, 27, rue Damesme, 4e étage, 75013, Paris, France

    Ramon Masià

  2. Universitat Oberta de Catalunya, Rambla del Poblenou, 156, 08018, Barcelona, Spain

    Ramon Masià

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  1. Ramon Masià
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Correspondence to Ramon Masià.

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Communicated by: Bernard Vitrac.

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Masià, R. On dating Hero of Alexandria. Arch. Hist. Exact Sci. 69, 231–255 (2015). https://doi.org/10.1007/s00407-014-0148-2

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