Decoding an already translated astronomical tablet, that is, one of many, with an eye to understanding its most important symbols and significations. You don’t need to know Akkadian, the long dead ancient language of Mesopotamia, or have an advanced degree in Assyriology to appreciate what the astronomically related texts are telling us. The experts have translated the tablets for us. We will learn what the scholars say about their meanings. The topic is fascinating. It is littered with ancient astronomical, mathematical, and calendrical concepts and notation. So, as a stone skips across a pond, we’ll graze the surface of just one tablet – this about the mighty planet Jupiter – to capture its gist; just enough to discover what the tablet is trying to tell us after twenty-two centuries of silence.
The Tablet
We wrote earlier about how Jupiter, known as Peṣû (‘the white one’) in Akkadian [1], was the celestial manifestation of the great Babylonian god Marduk, a part of whose astronomical story was found written on a twenty-two century-old clay tablet from Uruk in Babylonia, translated from the Akkadian by antiquities’ scholar Otto Neugebauer [2]. It encompasses sixty years of the Seleucid Era, from S.E. 113 to 173 (calendar years -198 to -138) [3]. On its face, this tablet is an ancient almanac or train schedule of sorts showing when and where Jupiter would arrive each time at its ‘first station.’[4] Here we focus on the other languages found on the translated tablet: its astronomical, mathematical, and calendrical notations. These, coupled with techniques regarding their use, will allow us to decipher how tablet was organized and to discover its hidden contexts.
Tablet code
To us non-specialists, even the translated tablet appears to have been written in strange, muti-level codes. Deciphering it means we have to know – or discover – the rules of how to read it – in the way a cypher key in a code informs the recipient of the steps needed to read the encrypted message. These are the algorithms that tell how the code was put together – how the spatial and temporal arrangement of the contents of the tablet was determined – and so reveal the process necessary to unlock it. And to grasp the fuller implications of a code, we’d like to know the purpose for which it was written. In our case the purpose was to predict a specific type of synodic event for the planet Jupiter.
A synodic event is a repeat phenomenon where the planet-earth-sun alignment for each type of event is the same [5]. For an outer planet, the watched-for synodic events were first appearance, first station, opposition, second station, and last appearance. The ‘stations’ are the places where the planet appears to stop in the sky and change direction, in its retrograde loop. The Greek letter phi (Φ) to the right of the first three rows of the Uruk tablet (added by the translator) indicate that the tablet concerns the Jupiter’s first station [6]. Like finding a single page torn out of a compendious old almanac, we discover that the Uruk tablet (like many others) concerns predictions only for a single planet (Jupiter) and a single type of synodic event (first station) over a limited course of years (six decades). How do we know this? It is revealed in patterns on the tablet itself, as we shall see. We’ll start with a look at the astronomical, mathematical, and calendrical notations on the tablet: the ancient constellations, numbers and calendars.
Constellations
Constellations of the zodiac. The path of the sun, moon, and planets across the sky, along the ecliptic, then and now, is marked by the constellations of the zodiac, a Babylonian invention. Following eastward from the first sign of Aries, they are Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capricornus, Aquarius, and Pisces. Libra. Their near-universal symbols are:
Their names in Akkadian were hireling, bristle, twins, crab, lion, furrow, scales, scorpion, Pabilsag (i.e., archer), goat-fish, Gula (origin unclear), and tails [7]. From antiquity, the zodiac was deemed to begin at the vernal equinox located at the “first star of Aries.”[8]
Zodiacal longitudes. The Babylonian astronomers considered each of the 12 zodiacal constellations to be 30° in width, their sum constituting the entire 360° ecliptic. This chart shows the zodiacal constellations, their common symbols, and the longitudes of their westerly boundaries in the sky [9]:
Early astronomers described the longitudes of the moon and planets as being so many degrees within a particular constellation [10]. So for example, when Ptolemy tells us that the moon’s longitude on a particular date is
he means that it is 4º 20’ into Sagittarius, which begins at 240º. Hence the moon is then (240º + 4º 20’ =) 244° 20’ longitude on the ecliptic circle beginning with the zero point at Aries and moving counterclockwise until one reaches that longitude [11].Sometimes the order is reversed, as we see the Uruk tablet.[12].
Numbers
The Sexagesimal system. Babylonian mathematical astronomy is based on the sexagesimal system, where numbers are sequences of digits using base number 60 (‘sexagesimal’) [13]. The number before a colon is the whole number; those following are separated by a comma and are fractions of increasing powers of sixty in the denominator. Hence with angles, 48;43,18 means 48° + 43/60 + 18/3600 or (consistent with modern notation) 48º 43’ 18”, which in decimals is 48.722°. Without any colon, it means multiply by 60. Thus, the term 6, 0° corresponds to 360°. Here, we’ll mostly use simple degrees, minutes and seconds in a circle [14].
Synodic time, called “tithi.” Column II of the Uruk tablet is populated by a series of numbers representing a unique method of Babylonian timekeeping, where the numbers have a proportional correspondence to degrees of arc. Their function in the tablet is purely mathematical. The Greek letter tau (τ) used by scholars stands for the unfamiliar Sanskrit word tithi representing synodic time. It is the time interval between successive occurrences of the same synodic phenomenon [15].
You’ll notice that the numbers in Column II swing between about 48 and 42 as Jupiter goes round. They have a mean value of 45; 13,53, or 45.23,14. That value is actually equivalent to Jupiter's mean synodic arc – its average eastward shift (in degrees) from one synodic event to the next – plus a sort of magical (my word) constant called “c”. The constant is 12; 05, 10 or 12.086, and “plays a central role in computing the dates in System A theory of Jupiter [16]." It all sounds complicated, but the bottom line is this: subtracting c from any row of Column II yields the synodic arc for that row’s date, and vice versa. It’s pretty simple! What’s magical is that the Babylonians invented it. It is a clever way of making the tablet an easy calculating board with simple arithmetic, like the chalk slates of early schoolchildren. We’ll use c in the algorithms below. We could go deeper, but for now, think of tithis as units of time proportional to the synodic arc of the planet, and skip to the algorithms which use them [17].
Calendars
The Babylonian calendar. Babylonian dates on the tablet are expressed by year of the Seleucid Era, displayed in Column I of the tablet image below, and month and day in Column III. [18] The Babylonian month began with the first sighting of the crescent moon [19]. Months are commonly translated with Roman numerals [20]. You can wow your friends by reciting all the months of the Babylonian year: I Nisannu; II Ajjaru; III Simānu; IV Dûzu; V Abu; VI Ulūlu; VII Tašrītu; VIII Ara͜hsamna; IX Kislīmu; X Tebētu; XI Šabāṭu: XII Addaru [21]. Since the lunar month varies between 29 and 30 days, with a mean value of 29.53 days, a moon-based calendar will always short the solar year by about 11 days (29.5 x 12 = 354), or roughly a third of a month on average. The Babylonians discovered that adding a full month about every three years, a process called intercalation, helped true it up with the sun [22].
Babylonian calendar dates are not equivalent to modern Julian calendar dates. You may go to the Babylonian Converter Site and enter the date on the Babylonian calendar as shown on the tablet and it will tell you the Julian calendar equivalent and the Julian day number [23].
Intercalary months: Left to its own devices, any lunar calendar will become increasingly out of sync with the solar cycle. To fix it, the Babylonians periodically inserted an intercalary month in the autumn (in Ulūlu) or more commonly, spring (in Addaru), the choice probably dictated by agricultural considerations [24]. Years with intercalary months are denoted on the tablet (by the translator) by one asterisk in Column I of the tablet, signifying that an extra Addaru II [ XIIb] has been added, and two asterisks for intercalation of an extra Ulūlu II [VIb].[25]
Patterns and Algorithms
A remarkable feature of Babylonian mathematical astronomy is the use of algorithms to compute astronomical quantities. These mathematical instructions are the cypher keys to understanding the tablet. They are found in the so-called Procedure Texts, rule books found on separate Babylonian tablets that tell the scribe how to write it, and us how to unlock it [26].
My purpose is to show how, by noticing patterns in the tablet itself, anyone without any specialized mathematical training (much less knowledge of Assyriology) or access to scholarly journals, or knowing how or why the Babylonians did this or that, can uncover the tablet’s algorithms and appreciate the remarkable interplay of its rows and columns [27]. So, we’ll begin the way someone who wishes to decipher a code would: by inspecting it closely and seeing what reveals itself.
Jupiter’s orbital period
The sidereal period of a planet is the time it takes to complete an orbit around the sun. For Jupiter it is almost 12 years. Jupiter visits each of the 12 zodiacal constellations over the course of his 12-year orbit. As a first step in acquainting yourself with the tablet, scan down the list of zodiacal signs in each column IV of the Uruk tablet, and you can confirm Jupiter’s arrival at one sign a year, like a medieval monarch on a progress being hosted at each of his subject’s domains, one per year, as he lumbers ponderously along through his (zodiacal) realm. He started in S.E. 113 in Capricornus, and at the end of the twelfth tablet year, S.E. 125, he’s back at to where he began. And, if we didn’t already know, this is one way we’d confirm for certain that this tablet is about Jupiter.
If we look at Jupiter’s arrival at just one constellation along its orbital passage, say Gemini, we may notice something else:
Through these particular years, 12 apart, Jupiter has reached its first station in Gemini at a longitude increasing by 5º per visit. Why is this? Although these numbers are rounded, this positive longitude drift makes sense (and tallies mathematically) when we consider that the interval between each of the planetary longitudes is 12 years for a planet that completes its orbit in 11.859 years. (You can see the same Jovian drift using the NASA/JPL Horizons website, which attests, in a modest way, to the accuracy of the Babylonian predictions incised on clay.)
The Metonic cycle
Every 19 years the moon and sun fall into sync in an anciently discovered rhythm called the Metonic cycle. It is apparent that the Babylonians used it as the basis for intercalating their calendar, inserting 7 intercalary months every 19 years. How do we know this? Looked at in the right way, the Uruk tablet reveals the algorithm of the Metonic cycle. If you cut a copy of the Uruk table into three vertical strips and align them with their intercalated (asterisked) years, you’ll discover the pattern below. You have to account for the fact that that some years are missing on the tablet (shown in color below) because a Jovian synodic event didn’t happen that year. I added those to the following chart. Bold numbers are intercalated years:
What an elegant pattern emerges! Read across, the intercalary years are 19 apart; read down, there are 7 intercalary years in 19 year each set.
Synodic arcs
The synodic arc is the degree difference between one opposition (or any other synodic event, such as its ‘first station’) and the next [28]. This algorithm invites us to calculate the row-by-row differences in tablet longitudes on the last column of the Uruk tablet. The result is the synodic arc [29]. We illustrate this with a ten-year sample of Uruk tablet data, taking the differences in longitude in the 6th column, with results in the 7th.
The result echoes long-standing scholarship on how the Babylonians (in their ‘System A’ methodology) accounted for Jupiter’s variable speed by dividing its orbit into slow fast arcs of 30 and 36 degrees, respectively. The slow arc went from Gemini 25º to Saggitarius 0º and fast arc went Saggitarius 0º back to Gemini 25º. [30] This algorithm matches an instruction about updating the synodic arc found in a Jupiter procedure text: From 25 Gem until 30 Sco you add 30. From 30 Sco until [25] Gem you add 36 . [31]
You may wonder why the reference to 30 Sco[rpius] above would not be translated as 0 Sag[ittarius] as it is found in other sources. They are the same longitude (240º), so there should theoretically be no difference among translators. According to Dr. Teije de Jong of the Anton Pannekoek Astronomical Institute of the University of Amsterdam, however, the answer “touches on a very fundamental point in the sexagesimal number notation of the Babylonians.” Interestingly, “the Babylonians had trouble with the number 0 in their sexagesimal number notation. So, they wrote 30 Sco for what we would write as 0 Sag.” Different translators write it in different ways, but both are correct [32].
The Babylonians incorporated a few ‘transition arcs’ (in rows 6 and 11 of this sample) to roughly account for transitions between the extremes [33]. Here Jupiter’s predicted shift eastward one constellation per year along the zodiac is mathematically illustrated as it cycles through its fast and slow arcs [34].
And we can derive synodic arcs from τ. In this simple algorithm, we just subtract the ‘magic’ constant 12;5,10 (12.0861) from the applicable year’s τ in Column II of the tablet, and the result is the exact synodic arc for that year [35].
One can of course reverse the process and derive τ from the synodic arcs, by adding the same constant to them [36].
Calendar days
This intriguingly simple algorithm works by adding the Column II tithis from the tablet to the previous year’s predicted day in Column III for Jupiter’s first station, to get the next year’s predicted day for its first station. For those unfamiliar with the term “mod” in the chart, don’t be put off. “Mod 30” is short for “modulo” and means subtract 30 as many times as needed to bring the result below 30. We use it because a Babylonian month can never have more than 30 days. This algorithm can also be reversed to derive tithis in Column II from the date in Column III.
How fascinating it is to find from these examples that this ancient clay tablet has an overall internal algorithmic structure — crisscrossing it horizontally and vertically, the interaction of which elements tie the rows and columns together as a kind of mathematical work table, an ancient, single purpose laptop of sorts used — in this case — for prediction of a specific synodic occurrence for the planet Jupiter.
Image of the translated Uruk Tablet
The tablet was translated by Otto Neugebauer and appears in his book, A History of Ancient Mathematical Astronomy, p.393. The left-hand column marked “Obv.” stands for “Obverse, or front of the tablet; “Rev.” is the reverse side. Here’s an example showing line 5, on the front (obverse side) of the tablet, with explanatory notes:
Image of the transliterated Uruk Tablet
ACKNOWLEDGMENTS
Warm thanks to Dr. John Steele, Professor of Egyptology and Assyriology at Brown University, for directing me to important research sources for this article, and to Dr. Teije de Jong of the Anton Pannekoek Astronomical Institute of the University of Amsterdam for his help on some puzzling issues of text interpretation.
REFERENCES
de Jong, Teije. 2018. A Study of Babylonian Planetary Theory I. The Outer Planets. Archive for History of Exact Sciences (2019) 73:1–37. https://doi.org/10.1007/s00407-018-0216-0.
Neugebauer, O. 1975. A History of Ancient Mathematical Astronomy. 3 vols. Studies in the History of Mathematics and Physical Sciences 1. Berlin: Springer (cited as HAMA).
_________. (1957) 1969. The Exact Sciences in Antiquity. Reprint, New York: Dover Publications, Inc.
_________. 1955. Astronomical Cuneiform Texts. Sources in the History of Mathematics and Physical Sciences 5. Berlin: Springer (cited as ACT).
Ossendrijver, M. 2012. Babylonian Mathematical Astronomy: Procedure Texts. New York: Springer.
NOTES
[1] Ossendrijver, xxv. An occasional alternative name for Jupiter in Akkadian was Sagmegar. Ibid.
[2] In his three-volume book, Astronomical Cuneiform Texts, the late antiquities scholar Otto Neugebauer included a transliteration of this Uruk tablet. The text is identified as “ACT 600,” a copy of which is found in ACT III, Pl. 176. For text details, see ACT II, 339. For a brief discussion of theory, see ACT II, 307-308, and 339. A transliteration puts the Akkadian tablet language into our Roman alphabet. Its translation puts it English, or whatever other language is desired.
[3] HAMA, 392-393. The Uruk tablet here is only one of many different types of tablets that have a variety of purposes, styles, and content. An excellent introduction to the different collections of tablets and their history can be found in Ossendrijver, 1-12. All known texts with mathematical astronomy originate from Babylon and Uruk. The corpus of Babylonian mathematical astronomy comprises about 440 cuneiform tablets and fragments from those cities from the period 450–50 BC. The bulk of dateable texts are from SE 120–210 (192–102 BC); dating the so-called “procedure texts” is less clear. Ibid., 6.
[4] The first station for an outer planet is place at the beginning of its retrograde loop where the planet appears to stop in the sky before temporarily changing direction as the earth overtakes it in its faster orbit.
[5] The planetary texts are mostly about synodic events. Ossendrijver, 56.
[6] We know the tablet concerns Jupiter's first station because the first three lines of the transliterated tablet say uŝ igi-tu2, which means ‘first station’ in Akkadian. Ossendrijver, 58. Neugebauer has marked those rows on the tablet (Column IV) with his own first station symbol Φ. The use of Greek letters to identify synodic events is Otto Neugebauer’s invention. HAMA, 242, 386. He called them ‘characteristic phenomena.’ “In the modern literature synodic phenomena are also referred to as ‘phases’[;] in the context of ancient astronomy as ‘Greek letter phenomena’ … after a symbolic notation introduced by Neugebauer.” Ossendrijver, 215.
[7] Ossendrijver, xxv.
[8] As the seasons progress, each of these constellations appear first in the East before sunrise, rising out of the dawn sky. Over time they retreat from the sun, shifting slightly more westward each night, finally becoming lost in the glare of the evening sun about six months later. Aries will disappear in the west as Libra rises in the east. This movement is caused by the revolution of the Earth around the Sun, bringing each constellation into view as the earth’s night sky comes round to face it. Because of precession (unknown to the Babylonians), the entire celestial vault has shifted so that the vernal equinox is now in Pisces.
[9] “The zodiacal signs have nothing to do with the constellations (i.e. areas which contain certain configurations of stars) which carry the same name although, of course, the names of the signs originated from their relation to the constellations. In the usage of ancient and mediaeval mathematical astronomy, however, signs are nothing but names for sections of longitudes, counted from a properly defined vernal point.” HAMA, 1079.
[10] Ossendrijver, 33.
[11] Also, when Ptolemy writes that Mars was in the “20th degree of Libra” he means it was d 20°, or 200° longitude So too, Jupiter at 13; 6 g means it is at 283° 6’ λ. Note that Neugebauer in his translations from Akkadian cuneiform has often dispensed with the semicolon, writing 26,6 instead of 26; 6.
[12] The constellation notations on the translated tablet are the translator’s. Those on the transliterated tablet of course have the constellation names in Akkadian, on the right of column IV of it. For the curious, I include this table from Ossendrijver, xxv:
[13] In both Babylonian and Greek mathematics of the Hellenistic era, sexagesimal notation was a convenient way of expressing fractions, and is the first place-value system in history. It is considered one of the great Babylonian contributions, enabling them to describe celestial relations and positions with unparalleled accuracy.
[14] Neugebauer’s tablet translations will typically omit the first semicolon: hence 26,56,30 should read as 26;56,30 and means 26° 56’ 30”. The system is common in all the literature about ancient astronomy. Consider practicing yourself with an online calculator, https://planetcalc.com/9216/. The site includes cuneiform versions of the sexagesimal numbers.
[15] This came about because the Babylonians wanted a uniform system of timekeeping. So they defined the mean synodic month of the moon as the basic unit of time, dividing this into 30 artificial ‘days.’ Thus, the basic unit of time is 1/30th of the mean synodic month. Ossendrijver, 61. In both ACT and HAMA the synodic arc is Δλ, and the synodic time is Δt. Ossendrijver, 56, n.215. “Mean synodic time” for Jupiter is its mean synodic period of 398.88 days. Ossendrijver, 89.
[16] De Jong, 11. The constant itself is derived from a correction to the difference in days between the Babylonian sidereal year of 12.3689 months x 30 days and the Babylonian calendar year of 12 months x 30 days, which difference equals 11.067 days, to which another small tweaking correction is added. The total mean value (in sexagesimals) is 11; 04 + 33; 08,45 + 1; 01,08 = 45;13,53 tithis.
[17] We cannot do justice in this space to the nuances of tithi in Babylonian computational astronomy. For a concise explanation of it, see de Jong, 10 -11.
[18] Ossendrijver, 32. The Seleucid era began with the reconquest of Babylon by Seleucus I Nicator in 311 BCE, one of Alexander the Great’s Macedonian generals. He founded the Seleucid Empire. The Seleucid empire used the Babylonian Calendar in which the new year begins on 1 Nisanu (3 April in 311 BC).
[19] HAMA, 353.
[20] Ibid. That is, I–XII plus VIb and XIIb for the intercalary months (insertion of one extra month in 7 out of 19 years. Those notations are sometimes written VI2 and XII2. The Uruk tablet discussed contains an intercalary month (Addaru II [XII2]) in line 159. Texts written before the Seleucid era are dated by regnal years of the ruling king.
[21] Ossendrijver, xxv. The extra months (see above note) are called VI2 Ulūlu arkû (‘second Ulūlu’) and XII2 Addaru arkû (‘second Addaru’)
[22] For example, in S.E. 114 on the tablet, 7 of the months have 30 days and 5 have 29 days (which you can check on the Babylonian Calendar Converter site https://webspace.science.uu.nl/~gent0113/babylon/babycal_converter.htm) for a total year of 355 days, for an average month of 29.58 days. Yet in S.E. 116, those proportions are reversed, for a year of 353 days and an average month of 29.42 days. Together they average 29.5 days, closer to the mean synodic month than either is individually.
[23] See https://webspace.science.uu.nl/~gent0113/babylon/babycal_converter.htm. The Babylonian Calendar Converter site is a little tricky to use. Adjust dates only by the buttons at bottom. First change the years, plus or minus, concentrating on getting the Seleucid year right; then adjust months/days to land on the correct Babylonian month and day. Then you can pick off the corresponding Julian date information. The Babylonian Calander Converter site will not convert hours, but for most purposes it is sufficient to know year, month, and day.
[24] HAMA, 356.
[25] For example, in SE 115* Addaru [XII] is followed by the intercalary month, Addaru II [XIIb]. In SE 132** Ulūlu [VI] is followed by the intercalary month Ulūlu II [VIb]. The Uruk tablet, not itself being a calendar, doesn’t show an intercalary month where Jupiter was not at first station in that intercalary month. It shows Addaru II for year SE 159 because Jupiter was at first station in that intercalary month of that year.
[26] Ossendrijver, 58. By far the most complete and impressive resource on procedure texts is Ossendrijver, upon whom I have relied heavily.
[27] My purpose here therefore isn’t to attempt to restate scholarly views on Babylonian planetary theory – the references do that well – or how the Babylonian astronomers were able to accomplish so much with so little.
[28] “With each synodic cycle the position of the synodic phenomenon shifts along the zodiac by a distance known as the synodic arc.” Ossendrijver, 56. Ossendrijver uses the symbol σ for it. Otto Neugebauer in HAMA uses the symbol for mean synodic arc.
[29] In 427 sidereal years there are 391 synodic events for Jupiter. Dividing the former (converted to days) by the latter, the result for Jupiter’s synodic period is 398.89 days, only a hundredth of a day off the modern value of 398.88 days. During this time, Jupiter makes 36 passages through the 360-degree zodiac, and thus covers 12,960 degrees of arc through those 36 revolutions. This total arc divided by the number of synodic events yields the mean synodic arc of 33; 08,45 degrees (33.1458), a number attested in the Babylonian procedure text ACT 812. DeJong, 8. The variations from this number in the table reflect the Babylonian effort to empirically fit the actual movement of Jupiter through its orbit, by roughly dividing it into fast and slow parts (called System A by scholars), with some transitional data connecting them.
[30] HAMA, 394.
[31] Ossendrijver, 248-249, translating Procedure Text No. 18, British Museum number, BM 34081.
[32] Specifically, the tablet from Uruk here and Neugebauer’s commentary on it in HAMA, 392-393, tells of Jupiter’s fast and slow arcs bounded by 25 Gemini and 0 Sagittarius. See also de Jong, 16, on the procedure text for accounting for the evident changes in Jupiter’s orbital speed detected by the Babylonians. There, de Jong says: “Fragments of this computational procedure for Jupiter are preserved in tablet BM 34081 . . .where we read in section 2, lines 8 and 9: From one appearance to the next appearance. From 25 Gem to 0 Sag you add 30. From 0 Sag to 25 Gem you add 36.” [Italics are mine.] Yet Ossendrijver, 248-249 refers to the bound as 30 Scorpius. He mentions the translation of the BM 34081 tablet (section 2, lines 8 and 9) as fixing the bounds (in System A) as 25 Gemini and 30 Scorpius. It reads: Appearance (FA) to appearance. From 25 Gem until 30 Sco you add 30. From 30 Sco until [25] Gem you add 36. The “gir2[.tab]” reference after the ‘30’ in the transliteration is “Scorpius.” Ibid., xxv. Both are correct, according to Professor de Jong, but a translator desiring a purer translation of the original text will tend to use the notation without zeros.
[33] HAMA, 392. Later systems, A’, A’’ etc. added more transition arcs to further refine their predictions of Jupiter’s positions. In the sample charts, we see the transition arcs in rows 6 (S.E. 118) and 11 (S.E. 123). In the Uruk tablet, there are 10 transition arcs, in S.E. years 118, 123, 130, 135, 142, 148, 154, 159, 166, and 172. Each has been set off by the translator with top and bottom horizontal lines in Column II of the tablet.
[34] The mathematics of how the Babylonians likely computed the synodic arc transitions for System A of Jupiter on the Uruk tablet is illustrated by Neugebauer in ACT II, 307.
[35] For the derivation of this constant, see De Jong, 11 & ACT II, 308.
[36] “The time of the synodic phenomena is thus updated from one to the next event by adding the synodic time τ” to the synodic arc. Ossendrijver, 61. “The Babylonian day begins at sunset, a commonly used reference point for timing astronomical phenomena.” Ossendrijver, 32.
Comments