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BM 76829: A small astronomical fragment with important implications for the Late Babylonian Astronomy and the Astronomical Book of Enoch Abbreviations


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BM 76829, a fragment from the mid-section of a small tablet from Sippar in Late Babylonian script, preserves what remains of two new unparalleled pieces from the cuneiform astronomical repertoire relating to the zodiac. The text on the obverse assigns numerical values to sectors assigned to zodiacal signs, while the text on the reverse seems to relate zodiacal signs with specific days or intervals of days. The system used on the obverse also presents a new way of representing the concept of numerical 'zero' in cuneiform, and for the first time in cuneiform, a system for dividing the horizon into six arcs in the east and six arcs in the west akin to that used in the Astronomical Book of Enoch. Both the obverse and the reverse may describe the periodical courses of the sun and moon, in a similar way to what is found in astronomical texts from Qumran, thus adding to our knowledge of the scientific relationship between the two cultures.
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Archive for History of Exact Sciences
BM 76829: A small astronomical fragment with important
implications for the Late Babylonian Astronomy
and the Astronomical Book of Enoch
Jeanette C. Fincke1·Wayne Horowitz2·Eshbal Ratzon3
Received: 3 October 2019
© Springer-Verlag GmbH Germany, part of Springer Nature 2021
BM 76829, a fragment from the mid-section of a small tablet from Sippar in Late
Babylonian script, preserves what remains of two new unparalleled pieces from the
cuneiform astronomical repertoire relating to the zodiac. The text on the obverse
assigns numerical values to sectors assigned to zodiacal signs, while the text on the
reverse seems to relate zodiacal signs with specific days or intervals of days. The
system used on the obverse also presents a new way of representing the concept
of numerical ‘zero’ in cuneiform, and for the first time in cuneiform, a system for
dividing the horizon into six arcs in the east and six arcs in the west akin to that
used in the Astronomical Book of Enoch. Both the obverse and the reverse may
describe the periodical courses of the sun and moon, in a similar way to what is found
in astronomical texts from Qumran, thus adding to our knowledge of the scientific
relationship between the two cultures.
Communicated by Alexander Jones.
BJeanette C. Fincke
Wayne Horowitz
Eshbal Ratzon
1Nederlands Instituut voor het Nabije Oosten, Leiden University, Leiden, The Netherlands
2Hebrew University, Mount Scopus, Jerusalem, Israel
3Ariel University, Kiryat Hamada, Ariel, Israel
J. C. Fincke et al.
ACT See Neugebauer (1955)
A.H. Tell Abû H
.abbah, modern name of ancient Sippar; abbreviation for the
tablets found at Sippar in the British Museum registration books
BM Museum number of the British Museum, London
BRM IV A. T. Clay, Epics, Hymns, Omens and Other Texts, Babylonian Records
in the Library of J. Piermont Morgan, Part 4. Yale University Press, New
Haven 1923
LBAT Th. G. Pinches and J. N. Strassmaier, Late Babylonian Astronomical and
Related Texts. Brown University Studies 18, Providence, Rhode Island:
Brown University Press
TU F. Thureau-Dangin, Tablettes d’Uruk à l’usage des prêtres du Temple d’Anu
au temps des Séleucides. Musée du Louvre: Textes cuneiforms tome VI
(TCL VI), Paris: Paul Geuthner
1 Introduction
BM 76829 (A.H. 83-1-18, 2201)1is a fragment from the middle part of a small, on
both sides very curved tablet from Sippar in Late Babylonian cursive script, that now
preserves just part of its right edge. Our study of BM 76829 began as part of the
search by J. C. Fincke (JCF) and W. Horowitz (WH) for new exemplars of “The Great
Star List.”2We quickly realized that the fragment did not belong to this compendium,
but were puzzled by its content, which found no clear parallels in the Babylonian
corpus of astronomical text that were known to us. After completing a preliminary
edition of the tablet, JCF and WH decided to present the tablet to our informal ancient
mathematical-astronomy group which meets on a monthly basis at various sites in
Israel. Our ponderings by the group finally led to a proposal by Eshbal Ratzon (ER)
that the astronomical information surviving on the fragment might best be understood
by comparison with astronomical materials embedded in the Book of Enoch. Below,
JCF and WH first present a standard Assyriological edition of the tablet and examine
the place of the tablet in the Babylonian cuneiform astronomical tradition. In particular,
they will discuss an interesting means of expressing the notion of a mathematical “zero”
in cuneiform texts. ER leads the study in light of the Astronomical Book of Enoch
(AB) and the Qumran Scroll 4Q318. The astronomical interpretation of BM 76829
is a joint effort of all three authors. Our article concludes with some more general
observations of the significance of BM 76829 by all three authors together.
2 Edition
BM 76829 (A.H. 83-1-18, 2201) is a fragment from the mid-section of a small tablet
from Sippar in Late Babylonian cursive script. A small part of the right edge survives,
1We publish this fragment by the courtesy of the Trustees of the British Museum.
2For the project “The Great Star List and Related Texts: Astronomy, Mysticism, and Learned Knowledge
in the Ancient Near East” funded by the Israel Science Foundation. See also Horowitz (in press).
BM 76829: A small astronomical fragment with important…
and the left edge is almost reached in one line of the obverse. The tablet itself is
convex on both sides, but we take the rounder of the two sides to be the reverse. The
measurements are: 40 ×38+×23 mm (width ×height×thickness). For the autograph,
see Figs. 1and 2.
Fig. 1 BM 76829, obverse
Fig. 2 BM 76829, reverse
J. C. Fincke et al.
2.1 Transliteration and translation
obv. 1[ki mu]l.ab.sín a-na 30
2[ki mu] a-na 20
3[k]imul.gí a-na 10
4[ki mu]<sag.>a-na
5[ki mul.suhur.]ša-na 10
6[ki]a-na 20
7[ki mul.kun.meš a-n]a30
(rest is missing)
rev. 1[ud.x.kam mul]. [maš]
2[ud.x.kam mul.a]l.lul
3[ud.x.kam mu]l.[u]
4[ud.2]9.kam mul.ab.sín
5[sag i]tu (erasure)
8[ud.x.kam mul].suhu[r?.]š?x[
(rest is missing)
obv. 1[The stellar segment of Th]e Furrow to/for 30. Virgo
2[The stellar segment of Th]e Scales to/for 20. Libra
3[The stellar seg]ment of The Scorpion to/for 10. Scorpio
4[The stellar segment of] Pabilsag to/for (vacat). Saggitarius
5[The stellar segment of The Goat-fi]sh to/for 10. Capricorn
6[The stellar segment of The Great One] to/for 20. Aquarius
7[The stellar segment of The Tails t]o/for 30. Pisces
(rest is missing)
rev. 1[Day x (equals) The] Twins. Gemini
2[Day x (equals) The C]rab. Cancer
3[Day x (equals) Th]e [L]ion. Leo
4[Day 2]9 (equals) The Furrow. Virgo
5[The beginning of a m]onth (equals) The Scales. Libra
6[Day x] (equals) the Tail of The Scorpion. Scorpio
7[Day x] (equals) Pabilsag. Saggitarius
8[Day x (equals) The] Goa[t-fis]h..[…]. Capricorn
(rest is missing)
2.2 The date of the fragment
The scribe must have written this tablet after 400 BCE, since the use of the zodiac
provides this as a terminus post quem (Britton 2010, pp. 638–649). The contents of
the reverse, which correlates zodiacal signs with calendar dates, allow us to suggest a
BM 76829: A small astronomical fragment with important…
more specific dating. Such correlations are to be found on BRM IV 19 and BRM IV
20, both edited by Ungnad (1940–41, pp. 255–284), and BM 36644 (1880-06-17, 376;
unpublished). Of these, only BRM IV 20 has a colophon with the name of its scribe:
Iq¯ıšâ son of Ištar -šuma-¯eriš descendant of Ekur -zakir who was active under the reign
of Philip Arrhidaeus in the late fourth-century BCE (323–316 BCE) (Hunger 1976,
p. 13a). We believe that our text is the oldest of this group, since it makes use of the
traditional names for Leo ( and Virgo (mul.ab.sín) that go back into
the 2nd millennium BCE. In contrast, the others use names that were first introduced
during the Late Babylonian period, such as mul.ur.aand mul.ab.sín (e.g., BRM
IV 20), ur.aand ki..(e.g., BRM IV 19), aand ki (absin0) (e.g., ACT 200),
and ur.aand ab (BM 36644). Thus, we posit that BM 76829 was most likely written
during the first half of the fourth-century BCE, but it also could be that a later scribe
made use of older star-names or that our fragment is from a later copy of an earlier
2.3 Commentary
What survives on BM 76829 at first appearance seems to be two separate and dis-
tinct sections of astronomical discussion: The pattern of the text on the obverse
can be established on the basis of the most complete lines, obv. 3–4,as“ki—star-
name—ana—numeral,” with the exception of line 4where the space where we expect
a numeral is left vacant. The format of the reverse is best preserved in rev. 4–5: “day
of the month—star-name.” In both obverse and reverse, the sequence of star-names
follows that of the zodiac (see Table 1).
Table 1 Star-names on the tablet BM 76829
Star-name Meaning Zodiac Obverse Reverse
[mul.lú.hun.gá]* The Hireling Aries
[]* The Bull of Anu Taurus
mul.maš The Great Twins Gemini 1 The Crab Cancer 2 The Lion Leo 3
mul.ab.sín The Furrow Virgo 14 The Scales Libra 25
mul.gí The Scorpion Scorpio 3
kun mul.gí The Tail of The Scorpion 6
mul.dpa.bíl.<sag.>gá Pabilsag Saggitarus 47
mul.suhur.máš The Goat-fish Capricorn 58
[]* The Great One Aquarius 6
[mul.kun.meš]* The Tails Pisces 7
*Star-names restored
J. C. Fincke et al.
Both obverse and reverse show an interest in the zodiac. Our discussion will attempt
to find common ground between the two portions of the tablet. We begin below with
a discussion of the obverse.
2.3.1 The obverse
The lines on the obverse begin by naming a zodiacal sign or rather the qaqqaru of
a zodiacal constellation and then gives the preposition ana (to/for) and a numerical
value. Akkadian qaqqaru, here written with the logogram ki, has the basic meaning
“ground, terrain, territory,” but the more specific meaning in astronomical contexts
of “location, area, region, blank space” (CAD Q 121a), or better “stellar segment”
(see e.g., Horowitz 1998, p. 166 and 256 with note 15). In the context of the zodiac,
this must refer to each of the twelve 30° sections assigned to each of the twelve
zodiacal signs, since the surviving sequence of constellations in our text follows that
of the zodiac, from Virgo to Capricorn. The values given for the constellations form
a zigzag function: 30–20–10–vacant space–10–20–30. Given that the values increase
or decrease by 10 in each line, it is clear that the vacant space represents “zero” (see
below 3.) Thus, the maximum 30 is assigned to the signs Virgo and Pisces, i.e., the
6th and 12th signs. At Sagittarius, i.e., the 9th sign, there is a vacant space, in lieu of
a dedicated cuneiform numeral for zero. Given that the portion of the original tablet
both above and below the surviving fragment is lacking, we cannot be sure as to where
in the zodiac our list originally began. If we extrapolate our sequence above and below
into the missing portions of the tablet, the most likely place to begin the list is with
zero at the sign Gemini, i.e., the 3rd sign.3
If we translate these signs into their time of year, we find that our list begins around
the time of the Summer Solstice (SS). It is important, however, to note that the date
of the SS on the obverse is not expressed here in terms of the lunar calendar, but
only in terms of the zodiac. This may very well be to escape the problem that the SS
would sometimes fall in Month III and sometimes in Month IV in Late Babylonian
astronomical texts of the middle of the fourth-century BCE (Britton 2002, p. 44;
Horowitz 2014, pp. 132–133). This problem finds expression in the context of the
zodiac when one compares our text, with the SS in Gemini as opposed to, e.g., LBAT
1494 and 1495, where the summer solstice falls in Cancer (Brack-Bernsen and Hunger
1999). If one translates back from zodiacal signs to lunar months, we find the SS here
too alternately in Month III (Gemini) and Month IV (Cancer). We suggest that this
anomaly results from problems in converting what were originally Sirius dates for the
SS into zodiacal dates, because Sirius is not a zodiacal constellation. More generally,
this may reflect confusion in conceptualizing the difference between constellations
and zodiacal signs, the former of which refers to actual observed stars in the sky, and
the latter to time and/or arcs on the celestial sphere.4Thus, the zodiacal nomenclature
3Beginning with the first sign Aries would demand that the list ended with the 12th sign Pisces at our obv.
7, the last surviving line on the obverse of the fragment, leaving space for a number of lines at the bottom
of the original tablet unaccounted for.
4This would have been a problem for the early history of the zodiac rather than for the later and thus seems
to confirm our hypothesis of the early fourth-century BCE date for the fragment as given in Sect. 2.2.
BM 76829: A small astronomical fragment with important…
Table 2 Restored full cycle of
values assigned to the zodiacal
signs on the obverse of BM
Zodiacal sign Numeric value Approximate modern equivalent
[Gemini] [0] May/June
[Cancer] [10] June/July
[Leo] [20] July/August
Virgo 30 August/September
Libra 20 September/October
Scorpio 10 October/November
Sagittarius 0 November/December
Capricorn 10 December/January
[Aquarius] 20 January/February
[Pisces] 30 February/March
[Aries] [20] March/April
[Taurus] [10] April/June
Fig. 3 Relation between zodiacal
signs and numerical values
for the position of the sun at the SS alternatively in Cancer or Gemini is ultimately a
question of where one places the border between the two zodiacal signs.
In any case, it seems certain that the tablet when complete would have given at
least one full annual cycle which we reproduce on the basis of an extrapolation of the
surviving data in Table 2. The zigzag function that derives from the extrapolation can
be seen in Fig. 3.
The correspondence of the zodiacal signs to the numerical values suggests that
the obverse refers to the sun’s monthly rising points on the horizon. Unlike popular
belief, the sun does not always rise from the true east and set in the true west. During
summer months, the sun rises from northeast and sets in northwest. During winter
months, the sun rises from southeast and sets in southwest. In Babylonia, the sun’s
maximal inclination on the horizon is approximately 30° (Neugebauer 1964,1979,
p. 156). Thus, the numerical values may refer to horizontal degrees. Assuming that
30° is at the true east where the sun rises at the equinoxes, zero will refer to both
summer and winter solstices at the two extremes of the horizontal arc. Brack-Bernsen
and Hunger (1999) found in LBAT 1494 and 1495 evidence for horizontal aspects
of the zodiac. However, while they concluded that this system in these two texts is
J. C. Fincke et al.
based on observation, the symmetrical data of our text, on the other hand, seem to
reflect a mathematical scheme. However, our tablet does not allow us to answer what
may ultimately be the ‘chicken and egg’ question: Did a recognition of arcs along
the horizon prompt the invention of the zodiac, or were the arcs themselves actually
at first simply projections of the rising and setting of zodiacal signs (Ben-Dov 2008,
pp. 185–189; Ratzon 2014,2015a; Jacobus 2014a)?5A similar horizontal system that
describes the sun’s yearly motion through dividing the horizon into six arcs in the east
and six arcs in the west is also known from the Astronomical Book of Enoch. The AB
refers to these arcs as gates through which the sun exits and enters when it rises and
sets (see below 4, Fig. 4).
In essence, numerical values in our scheme measure the distance of the sun month-
by-month, zodiacal-by-zodiacal sign, from the two extremes, rather than from the
middle. The result of this is that we apparently have a system with two separate zero
points at the two opposite ends of the arc. Although it is tempting to describe this
system as a horizontal coordinate system, the fact that we have two points of origin of
the axis (zero) renders this impossible.
We would have been much more comfortable if the zero point would have been
at the equinoxes and the maximum and minimum would have been 30 (i.e., + 30
and 30 depending upon how one defined his/her axis) at the solstices. While the
Babylonians did not know negative numbers in their modern sense, they did in fact
develop a solution to this type of problem in their mathematical-astronomy by defining
numbers as tab and sìg, additive and subtractive (Ossendrijver 2012, pp. 29–31).
Another option would have been to define one solstice as a minimum point (zero)
and the other as a maximum (60). Below, we shall see that this is exactly the strategy
adopted by the Book of Enoch (see below 4.). We would suggest that BM 76829
obverse makes use of minimum and maximum values 0–30 under the influence of the
division of each sign of the zodiac into 30°. This created a division of the year into four
seasons each consisting of three zodiacal signs and three horizontal sections of 10°.
In effect, the system creates two mathematical axes—one being the ecliptic divided
into 30° zodiacal signs, and the other being the horizon divided into two segments of
3×10 30° each. By combining the traditional division of the annual motion of the
Sun into four seasons (MUL.APIN II i 9-21; Hunger–Pingree 1989, pp. 72–76) with
the later system of the zodiac, our text has produced something completely new in the
known Mesopotamian astronomical corpus.
2.3.2 The reverse
The reverse is even less complete than the obverse, giving a set of star-names from
Gemini to Capricorn, but with little context other than surviving traces in rev. 4–5
where we may find reference to the last day(s) of 1 month and first day of the next
month (the 29th and sag itu, Akkad. eš arh
˘i, “the head of the month”). If we assume
a month of 30 days, we have here an interval of 2 days. The trace of the date formula
5For the stellar Paths of Anu, Enlil, and Ea as bands crossing the sky from east to west rather than arcs
along the horizon as argued earlier by Pingree and Reiner see Horowitz (2014, pp. 11–15) with further
BM 76829: A small astronomical fragment with important…
Table 3 Reconstruction of the reverse according to three above patterns
Zodiacal sign Numbers according to the 3–2–3–2
4Q318 Numbers according to the 2–2–2
pattern of the AAB
[Aries] [16] [17.5] [19.6]
[Taurus] [19] [19.5] [21.6]
Gemini [21] [22.5] [23.6]
Cancer [24] [24.5] [25.6]
Leo [26] [26.5] [27.6]
Virgo 29 29[.5] 29[.6]
Libra 1 1[.6] 1[.7]
Scorpio [4] [3.6] [3.7]
Saggitarius [6] [5.6] [5.7]
Capricorn [9] [8.6] [7.7]
[Aquarius] [11] [10.6] [9.7]
[Pisces] [14] [12.6] [11.7]
[Aries] [16] [15.6] [13.7]
in rev. 7confirms that this pattern continues. Thus, it seems that the reverse relates
days or intervals of days with zodiacal signs.
If our reconstruction of the obverse and our hypothesis that the obverse has only
this one cycle written on it is correct, we can restore two more lines at the beginning
of the reverse and a few lines at the end. If so, the first entry of the reconstructed text
would be for Aries, which corresponds to the first month of the Babylonian calendar.
On the other hand, the more fragmentary state of the reverse makes a much larger
number of reconstructions possible. In fact, any reconstruction which calculates the
position of the moon on the 29th of the old month in Virgo and then the first of the
new month in Libra cannot be excluded. Nonetheless, we below present three options
that can be justified on the basis of parallels in contemporary sources.
A. The first requires an assumption of an underlying ideal lunar month of 30 days
divided into the 12 sectors of the micro-zodiac of 2½ degrees for each day
(Rochberg 2004). The mean value of 2½ degrees is achieved by alternating
between 3 and 2 days between each line, resulting in a 3–2–3–2 pattern. Based on
this assumption, the dates assigned to each of the zodiacal signs are as in Table 3
col. ii. An interesting mathematical phenomenon emerges. From Aries to Virgo,
we have 15 days inclusive (days 16–30 of the first month), and then from Libra to
Pisces, there are again 15 days (see Table 3). Thus, we have a system of 30 days
which could parallel the system using 30 units on the obverse.
B. The second option follows 4Q318, an astronomical scroll from Qumran, written
in Aramaic, which, according to most scholars, correlates the zodiacal signs with
dates in the 360-day ideal year expressed with Babylonian month names. There,
the moon is placed on the 29th of each month in one sign, and on the 1st of the next
month in the preceding sign, just as in the reverse of our tablet. This phenomenon
is related to the sidereal month, which is the period in which the moon returns to
J. C. Fincke et al.
the same position in relation to the stars. This second option assumes that while
the obverse describes the path of the sun along the horizon, the reverse discusses
the same phenomenon in relation to the moon over an ideal sidereal month of
28 days, rather than an ideal synodic month of 30 days.6The period of 28 days
for the sidereal month is also used in a Babylonian intercalation scheme (Hunger
and Reiner 1975; Ratzon 2016). 4Q318 presents the period of a 28-day sidereal
month as four sets of 2–2–3 days, see Table 3col. iii. Then 2 additional days are
added at the end of each month to achieve the 30-day synodic month (Greenfield
and Sokoloff 2000).7This 28 + 2 day pattern allows the moon to enter a different
zodiacal sign at the beginning of each synodic month over the course of the year.8
Since BM 76829 reverse names specific zodiac signs, the surviving text cannot
refer to a generic month in accordance with a theoretical monthly scheme, but
rather must refer to a specific month, in this case the second half of month six and
the first half of month seven (Libra as day 1). If BM 76829 reverse indeed reflects
a pattern similar to 4Q318, it seems logical to assume that similar Babylonian
tables or tablets existed for the other months of the year.9
C. The Astronomical Book of Enoch likewise refers to a similar lunar phenomenon.
It describes the motion of the moon along the horizon. The moon passes through
the same six gates that are used by the sun. The period of the motion of the moon
along the horizon is called the Draconian Month, which is slightly longer than
27 days. However, the Aramaic version of the Astronomical Book (AAB) gives
a very schematic approximation of this motion, assuming that the moon stays
in each gate for exactly 2 days, which sums up to a too short 24-day Draconian
Month. Although the AAB uses a lunar calendar, in its earlier versions this system
was used in a 360-day year. The moon’s movement of one gate for every 2 days
can be expressed as a pattern of 2–2–2 (Ratzon 2017,2019). Applying this pattern
to the 360-day calendar predicts that in the first day of the seventh month the moon
will be in gate 3, corresponding in this case to Libra (see below 4.).
3 Writings of zero
Upon initial reading of the text, it was obvious that the obverse of BM 76829 gave
the numerical sequence 30–20–10–0–10–20–30, which presented the scribe with a
problem since cuneiform has no dedicated cuneiform numeral or sign for zero. In late
cuneiform astronomical tablets, zero is most often expressed by the Glossenkeil (sep-
aration marker) (Neugebauer 1941). In each case, Neugebauer (1941) gives examples
of what he later (Neugebauer 1955) calls medial zero where the Glossenkeil represents
an empty sexagesimal column. Two such sub-uses may be identified: (A) Glossenkeil
6The true synodic month is approximately 29.53 days, and the sidereal month is approximately 27 1/3 days.
7For Wise (1994), who assumes a 364-day year, there are four 31-day long months. At the end of these
months, 3 days will be added to the general pattern.
8Since the sidereal month is shorter than the synodic month, the dates in which the moon correlates to a
specific zodiacal sign shift along the year.
9The present fragment is not thick enough to be from a large enough tablet to allow for sections for all
12 months, but could give an excerpt from such a full system for a month or season.
BM 76829: A small astronomical fragment with important…
marking zero units between two numbers larger and smaller by a power of 60 (Friberg
1987–90, p. 536); for example, 1, 0, 1 would represent 1×602+0×60 + 1, which is
3601. (B) Glossenkeil representing zero tens or zero units within sexagesimal columns;
for example 10 + 0 for ten, or 0 + 5 for five.10
Neugebauer (1955) also makes reference to two other usages of the Glossenkeil
in this way, what he calls “initial zero” and “final zero.” Initial zero is when the
Glossenkeil occurs at the start of a numeral, as for example in a rendering of 89/100
as 089. For “initial zero,” Neugebauer can point to ACT 122 rev. v 8 where two
Glossenkeile and a numeral provide the value for three sexagesimal columns.11 This
use of the Glossenkeil to fill an otherwise valueless sexagesimal column of zero
10s and zero digits may also be found in ACT 122 rev. v 9 (Glossenkeil—empty
space—54—nim)12 and ACT 123 obv. iii 4 (Glossenkeil—7—empty space—zodia-
cal sign). Orthographic variants of the Glossenkeil have the appearance of the gam sign
(two small Winkelhaken forming one diagonal line from top-left to bottom-right),13
or sometimes even what can also be read as numeral 9 (three small Winkelhaken also
forming one diagonal line from top-left to bottom-right).14 Final zero is when zero
occurs at the end of a numeral as in our 10, 100, or 1000. The index to Neugebauer
(1955, p. 511) suggests some occurrences of “final zero” but inspection of the exam-
ples shows that they are either in broken contexts, or placed between two numerals
(ACT 222 rev. iii 16).15
Another option used to mark the numerical value zero is to omit any sign at all,
i.e., by leaving empty space. For example in our case of 1, 0, 1, the open space for
zero insures that a reader does not mistake the two vertical strokes for the numeral 2
(1 + 1), or for 3660 (1 ×602+ 60) or 61 (1 ×60 + 1). Examples for initial, medial,
and final zero marked this way are all available, e.g., on ACT 122.16 For examples of
empty space final zero outside the ACT-tradition see LBAT 1499 obv. 4, 10, 13 passim
(Astrolabes). Friberg (1987–90, p. 536b) makes mention of earlier examples from Old
Babylonian mathematical tablets where medial zero is represented by empty space in
lieu of writing a Glossenkeil.
What is novel in BM 76829 is the graphic use of empty space for zero without ref-
erence to any other numeral or number. Thus, the empty space here does not represent
an empty column in a larger number but rather the actual absence of any numerical
value at all, i.e., mathematical zero. To our knowledge, this may be the earliest attested
evidence for a concept of mathematical zero. This scribal practice is anticipated almost
2000 years earlier in the Ur III period where blank space before a class of animals in
10 For this usage in the second millennium BCE see Horowitz (2014, p. 227).
11 Hørup (2002, pp. 293–294), previously said that initial and final zeros are never marked in cuneiform at
12 Or Glossenkeil—53—nim.
13 See e.g., Ossendrijver (2012, pp. 18–19), who describes all Glossenkeile as gam sign.
14 See Sachs (1952, p. 148) and Neugebauer (1955, p. 418) reference to ACT 813 rev. III 14, 14 (section 28).
15 The case of 3, 0, 6 is to be interpreted 3,0 (180 degrees) 6danna (geometric leagues of 30° each).
16 See e.g., the hand copy of ACT 122, LBAT 66 obv. ix 13 (initial + final),×10–11 (final), rev. v 14 (initial
+ final), 19 (medial).
J. C. Fincke et al.
administrative documents can in fact be understood as zero animals of this type (HUJI
07001; Dimenstein 2004; Conlan 201317).
The closest example of a sense of zero to our text is to be found in the auxiliary
tablet at the end of ACT 135 (TU 24 + 26), rev. col. i 24, where we find nu tuk-ši
“there is none”18 in the following context:
(20) h
˘ab-rat.meš 18 ana 18 ana igi-ka
(21) 30 zib 42 ana ´
-šú lal
(22) 30 [l]u28 lal
(23) 30 múl 14 lal
(24) 30 maš nu tuk-ši
(25) 30 kúšu 14 lal
(26) 30 a28 lal
(27) 30 ábsin 42 lal
(20) In order for you to find the eclipse magnitudes (from) 18 to 18 (years)19
(21) The Moon Pisces 42 behind it—near the ascending
(22) The Moon Aries 28 near the ascending node
(23) The Moon Taurus 14 near the ascending node
(24) The Moon Gemini it does not have (a value)
(25) The Moon Cancer 14 near the ascending node
(26) The Moon Leo 28 near the ascending node
(27) The Moon Virgo 42 near the ascending node
Here, though, we have a statement about a numerical value which the scribe is
not able to represent in graphic form as a numeral. Another solution to this prob-
lem that also uses words rather than numerical notation can be found in en¯uma anu
enlil tablet 14 (Al-Rawi and George 1991–92; Ossendrijver 2014), table A, for the
duration of lunar visibility in the equinoctial month. The tablet gives a numerical
value for the length of lunar visibility for days 1–29 of the months, but for the
last day, when the moon is not visible in the sky at night at all, we find dingir
ina ud gub, “the god stands in the day.”21 Similar fomulations to describe zero
17 With reference to additional examples noted by M. Sigrist.
18 Neugebauer (1955: 477a) takes this for Akkadian ul iš¯ı, which could be translated “it does not have (a
value),” while Ossendrijver (2012, p. 18) footnote 100 gives it as ul ibašši. Neither CAD B 144b nor AHw
112b give tuk as a standard sumerogram for bašû. Therefore, nutuk-ši should rather be rendered as ul
irašši “it has not acquired (a value),” as is commonly used in omen texts and elsewhere.
19 Neugebauer (1955, p. 197).
20 For lal interpreted as increasing latitude and referring to “near the ascending node” in the context of
eclipse magnitudes see Steele (2000, p. 128).
21 Al-Rawi and George (1991–92,p.55).
BM 76829: A small astronomical fragment with important…
include anu22 “there is not,” eq23 “empty, without” and mimma ula “there is noth-
4 Comparison with the Astronomical Book of Enoch and 4Q318
If our interpretation of the astronomical meaning of BM 76829 is correct, a simi-
lar system can be found in the Astronomical Book of Enoch. Enoch was a biblical
antediluvian figure. He is mentioned only briefly in the Hebrew Bible, where it is
implied that Enoch acquired astronomical knowledge from divine beings and was
taken to stay with God after 365 years of life (Genesis 5:19–24) instead of dying. The
earliest explicit descriptions of Enochic visions date to the Second Temple period
on fragments that have been preserved in Aramaic among the Dead Sea Scrolls.
From Aramaic, they were translated into Greek, and from Greek into Geez (Clas-
sical Ethiopic). The Book of Enoch (1 Enoch), a collection of sources composed
during the second half of the first millennium BCE, is now part of the Ethiopic bible
and is divided into five sections and two appendices (Nickelsburg 2001, pp. 15–17;
Nickelsburg and VanderKam 2012, pp. 350–356; Ratzon 2012, pp. 45–48). Its third
section is today called “The Astronomical Book of Enoch” (AB). In Ethiopic, it
was known as The Book of the Revolution of the Lights of Heaven (Knibb 1978,
p. 167).
It is generally accepted that the first versions of the AB and the astronomical
knowledge contained within originated in the Land of Israel between the fourth- and
third-century BCE, i.e., the Persian and Hellenistic periods, i.e., the early Second Tem-
ple period.25 Several studies have demonstrated the significant influence of Babylonian
astronomy on the AB.26
The main focus of the astronomy of the Aramaic version of the AB is the rising and
setting times of the moon in relation to sunrise and sunset. However, both the Aramaic
22 E.g. Ossendrijver (2012, p. 18) footnote 101 with reference to text 52 Ri9; Pearce–Wunsch (2014,
pp. 118–119) text no. 15 line 19 (zero date dates assessed).
23 See Aaboe–Sachs (1966, p. 6b) for Text A (BM 36300+) rev. iv 22 and vi 14, where riq (i.e., the sign
.U) is written where initial zero is expected.
24 In a mathematical text, see Friberg (2007, p. 269) with reference to MS 3052 § 1d (transliteration
pp. 267–268) line 16.
25 The terminus ante quem is determined according to the Dead Sea Scroll 4Q208, the earliest copy of
the AB that was transcribed at the end of the third-century or beginning of the second-century BCE. For a
summary of the debate on the paleographical and C14 dating of this scroll, see Ratzon (2015a). The terminus
post quem relies on the AAB’s knowledge of the zodiac. See Brack-Bernsen and Hunger (1999), Ben-Dov
(2008), Drawnel (2011), Ratzon (2014) and Jacobus (2014a,b). Three other copies of the Aramaic AB have
survived among the Dead Sea Scrolls: 4Q209 dated to the end of the first-century BCE or beginning of
the first-century CE. It contains an astronomical treatise similar to the one in 4Q208 and parts of 1 Enoch
76–79, 82, known from the Ethiopic version of the book; 4Q210 dated to the middle of the first-century
BCE. It contains only parts of 1 Enoch 76–78 without the astronomical treatise found in 4Q208-9; 4Q211
dated to the first half of the first-century BCE. It does not contain any parallel to the Ethiopic version of 1
Enoch nor to the astronomical treatise, but it contains a section that was probably lost from the end of 1
Enoch 82, and a very fragmented section dealing with the stars.
26 The relationship between Enochic and earlier Mesopotamian astronomy has been studied intensively.
For a summary, see VanderKam 2008, pp. 965–978; Ratzon 2012, pp. 299–301; and note the bibliography
given in the previous note.
J. C. Fincke et al.
Fig. 4 Gates for the Sun and Moon in the Astronomical Book (72–75)
and the Ethiopic versions occasionally address spatial aspects, like the positions of the
rising and setting of the moon in relation to the horizon.27 The sun and moon rise and
set through gates, openings in the firmament, that stand on the edges of the earth and are
used for the passage of astronomical and meteorological phenomena from heaven to
earth and back (Ratzon 2012,2014,2015a,2017). Heavenly gates are also known from
Babylonian literature (Heimpel 1986; Horowitz 2011, 266–267).28 However, only in
the AB these gates are used as a horizontal coordinate system (Ratzon 2014).29
A full description of the gates appears only in the Ethiopic version of the AB, but
this description seems to correspond to the gates system used by the Aramaic scrolls
(Ratzon 2015a, p. 97). The gates system for the sun includes 12 gates: six in the
east for rising and six in the west for setting (Fig. 4). Neugebauer suggests that the
concept of these gates was created in order to explain the fact that the sun rises from
the northeast in summer and from the southeast in winter, rather than rising in the
true east and setting in the true west (Neugebauer 1964,1979,1985). The authors of
the AB noticed the correlation between the seasonal variations in the lengths of day
and night and the sun’s movement on the horizon, but described this movement with
the gates system rather than in terms of the ecliptic. According to 1 Enoch 72, the
sun rises in summer from the northeast, because it passes through a northeastern gate,
and from the southeast in winter, because it rises through a southeastern gate. Hence,
the sun’s yearly cycle begins the day after the vernal equinox—on the first day of the
first month—when it passes through the fourth gate. From there, it continues to rise
through the northeastern gates, taking one full month to pass through each gate until
27 Ben-Dov (2008) demonstrates that while the Aramaic scroll discusses both spatial and temporal aspects
of lunar theory together, the Ethiopic translation distinguishes between them and addresses each aspect in
a separate chapter.
28 For a possible Mesopotamian antecedent to this diagram, compare the Late Babylonian compass card
from Uruk most recently discussed in Horowitz (2015), and the diagram on the reverse of the ziqpu-star
fragment (BM 61677) in Horowitz and Al-Rawi (2001, p. 180).
29 While only the Ethiopic version of the AB explicitly describes the gate system, the Aramaic version
assumes and uses the same system, since every one of the gates is mentioned in Aramaic. See Ratzon
(2015a, fn. 11).
BM 76829: A small astronomical fragment with important…
the sun finishes passing through the sixth gate for the first time from south to north at
the summer solstice. The sun then begins to return southward to the first gate, to whose
southernmost boundary it reaches at the winter solstice; it then returns northward so
that at the equinox, at the end of the twelfth month, the sun has just finished passing
through the third gate again; then it repeats its cycle.
Ethiopic tradition, influenced by Arabic astronomy, identifies the gates with the
signs of the zodiac (Neugebauer 1979, pp. 156–161). Traces of this traditional inter-
pretation can be found in early modern scholarship (for a summary see Jacobus 2014b).
However, Neugebauer was strongly opposed to the identification of the gates with the
signs of the zodiac, and until recently his authoritative opinion was quoted by every
subsequent commentary (Neugebauer 1964). This consensus is changing gradually.
Brack-Bernsen and Hunger (1999) compare the AB gates to LBAT 1494 and 1495.
Ben-Dov (2008) first suggests that the gates are in fact a projection of the signs of the
zodiac on the horizon. Further evidence and development of this argument is found in
Lourié (2010), Ratzon (2014), (2015a,b), Jacobus (2014b). Our interpretation of BM
76829 strengthens the connection between the zodiacal signs and arcs on the horizon.
It is uncertain whether all gates were of equal length or the inner gates were bigger
than the others (Ratzon 2014, pp. 501–502).30 Both possibilities have Babylonian
analogues. In BM 76829, the horizontal arcs are all equal in length, while in LBAT
1494 and 1495 the arcs are determined by observations, and therefore the middle arcs
must be longer than those at the edges (Brack-Bernsen and Hunger 1999).
Two different reconstructions have been proposed in previous studies for the AAB’s
scheme for lunar movement through the gates. Neugebauer (1979, pp. 159–161) sug-
gests that the pattern of the moon’s passage through the gates has been preserved
in late Ethiopic texts that list the number of days the moon stays in each gate. The
exact numbers in this text may fit the data in BM 76829 rev. 4–5, but the Ethiopic text
assumes lunar months of alternating 29 and 30 days, rather than the ideal 360-day year
used by Babylonian astronomical texts of the type BM 76829. Applying Neugebauer’s
scheme to the 360-day year will not allow for the moon to be in Virgo on the 29th and
then in Libra on the 1st of the seventh month as preserved in BM 76829 rev. 4–5.In
contrast, Ratzon (2017) mathematically reconstructs the lunar scheme in the Aramaic
fragments of the AAB. According to her reconstruction, the moon stays in each gate
30 The Ethiopic version of the AB refers to one of the middle gates, the fourth gate, as “the big gate” (1
Enoch 72:6). The first commentators of 1 Enoch, Dillman (1853) and Charles (1912) explain that the gate
is considered big compared to the windows surrounding it. In response Neugebauer (1985) asks why only
the fourth gate is called big, if all other gates are also surrounded by windows. However, in his figures
and computations, he refers to all the gates as equal in size (Neugebauer 1964), and so do other scholars
(Albani 1994; Glessmer 1996). Barker (1989) and Brack-Bernsen and Hunger (1999) have proposed that
the middle gates were larger. However, the term ‘big gate’ might not reflect astronomical considerations at
all, but instead perhaps be understood from a more architectural perspective with the “great gate” being like
the main gate of a city (Sumerian-Akkadian ká.gal abullu “gate”) rather than ordinary gates (abu
“door, gate,” Sumerian gal rabû “big, large.” For an astronomical example of ká.gal used in this way
for thunder at ’The Gate of the Moon,’ with no apparent observational inference see Hunger (1992: 74–75,
no. 119: 5). For architectural themes in cosmography see e.g., Van Leeuwen (2010), and in relation to the
Enochic gates see Barker (1989), Thiering (2004), and Nickelsburg and VanderKam (2012, p. 423). In any
case, it is important to note that nowhere in 1 Enoch is the third gate, which is also a middle gate, referred
to as big. Another issue is the fact that the adjective “big” is only mentioned in the Ethiopic version, and
might be a later insertion.
J. C. Fincke et al.
for 2 days, and the entire period is 24 days. Ratzon’s reconstruction applied to the
360-day calendar predicts that in the first day of the seventh month the moon will be
in gate 3, corresponding in this case to Libra.
As mentioned above, BM 76829 has two zero points in both summer and winter
solstices, while the AB has only one zero point at the winter solstice, and the numbering
of the gates continues to increase until the sixth gate at summer solstice. This difference
allows us to refer to the Enochic gates system as a unique horizontal coordinate system
(Ratzon 2014).
The existence of BM 76829 may change our evaluation of the originality of the
astronomers of the AB. Lourié (2010) claims that there is nothing original in the AB,
and all of its astronomy is derived from Babylonia. He partly based his opinion on
the hypothesis of Brack-Bernsen and Hunger (1999) that the horizontal observations
predated the development of the zodiac. Ratzon (2014), on the other hand, claimed
that as long as the AB remains the sole example for a horizontal coordinate system,
it should be regarded as an original development of the authors of the AB. However,
BM 76829 is obviously another piece of evidence that this concept may predate the
composition of the first drafts of the AB. Our analysis above demonstrates that the
concepts in these two compositions are similar. However, until another tablet appears
with even more similarities to the AB, we must still adhere to the position that the
authors of AB adopted Babylonian ideas to their own unique cosmology and world
Another scroll from Qumran that refers to the moon’s position in relation to the
zodiacal signs is 4Q318. This Selenodromion (the position of the moon in the zodiacal
signs throughout the year) is followed by a Brontologion (meteorological omens) and
dated paleographically to the late first-century BCE. Most scholars understand that
unlike most of the Qumran calendars, this text assumes a 360-day year.31 Nothing
in this scroll points to sectarian context. The names of the zodiacal signs in this text
are derived from the Greek tradition. Thus, it uses (virgin) for Virgo, and not
the Babylonian ab.sín (Furrow).32 Rather than in Aries, the moon begins the year in
Taurus. Wise (1994, pp. 43–48) explains the beginning of the year in Taurus within
the background of western astronomy. He suggests that the authors of 4Q318 were
familiar with the concept of precession. According to Wise, 4Q318 describes the first
year of the world after creation, which he thinks that the Qumran community dated
to approximately 4000 years before their time, when the vernal equinox coincided
with Taurus. However, Albani (1993, pp. 23–35) correctly argues that the astronomy
of 4Q318 is not sophisticated enough to show awareness of precession, which was
known to only few professional astronomers of the time. He connects the beginning
of the year in Taurus to Babylonian tradition, as in MUL.APIN II A 8–9.33 From
31 An exceptional is Wise (1994) who reads this text within the Qumran context and sees an underlying
364-day year.
32 However, note the drawing of the zodiacal sign Virgo with a female figure, presumably the Virgin, in
VAT 7847 + AO 6448 (see Weidner 1967, Tafel 9–10 and now Beaulieu–Frahm–Horowitz–Steele (2018,
p. 99).
33 The date of the beginning of annual astronomical cycle is not fixed in cuneiform astronomical texts. For
example, some begin with the month of the Spring Equinox (Months III or IV) while others begin with
the month of the Summer Solstice (Months IX or X). For the parallel Babylonian systems of placing the
BM 76829: A small astronomical fragment with important…
there, the moon continues traveling through the zodiacal signs according to a pattern
of 2–2–3 days in each sign, except for the end of the month when an additional 2 days
are added. Although it is not explicitly stated, the 7-day pattern may be related to the
length of a week, but the last 2 days at the end of each month do not allow the pattern
to begin in a constant day of the week. Another connection to western astronomy is
manuscript CCAG 7.193-7 (Paris S. GR. 1191), which is very similar in content. It was
written in Greek in Byzantine times, but probably preserves Hellenistic knowledge
(Albani 1993,1994,1999;Wise1994; Yardeni 2000; Greenfield and Sokollof 2000;
Pingree 2000; Jacobus 2010,2014a). As mentioned before, together with its western
orientation, 4Q318 is also related to Babylonian astronomy. The additional 2 days
allow a correlation between the position of the moon in the zodiacal signs and the
beginning of each month. This pattern fits the data in BM 76829 rev. 4–5, since every
month in 4Q318 begins in its corresponding zodiacal sign. However, unlike 4Q318,
BM 76829 reverse probably begins its cycle in Aries rather than Taurus.
5 Conclusions
Above we have provided a first edition of the previously unpublished Babylonian
astronomical fragment BM 76829. Although but a small piece of the original tablet,
even in its current state the fragment allows for a number of new insights into Baby-
lonian astronomy and mathematics that find parallels in Jewish sources including the
Astronomical Book in Aramaic from Qumran and the Dead Sea Scroll 4Q318. Thus,
our fragment may be seen as a missing link between the two traditions. Of particular
importance are parallel systems to describe the position of the Sun at Sunrise in terms
of the six arcs along the horizon on the obverse of the tablet. The fact that the reverse of
BM 76829 concerns the Moon and the Zodiac also places our tablet firmly within the
realm of contemporary Babylonian astronomical concerns, and perhaps those of the
Book of Enoch and/or Qumran scroll 4Q 318 depending on one’s reconstruction of the
Babylonian text. Thus, our fragment, when complete, might have been a missing link
between Judean and Babylonian astronomy, perhaps even offering some type of unified
explanation of some solar, lunar, and stellar phenomena within the general time frame
that the Babylonian Mathematical-Astronomy of the ACT-tradition emerges on clay
tablets. It is within this context that the new way of expressing zero that we find on BM
76829 may be particularly important. Unfortunately, given that much of our ‘missing
link’ is still missing, we are unable to reach full conclusions on a number of impor-
tant issues on the basis of the surviving text on our fragment. Within the cuneiform
world, is the new way of writing ‘zero’ may be part of the intellectual background
which lies behind the emergence of the ACT-tradition, or is this new way of writing
‘zero’ simply a matter of scribal convenience? Likewise, our fragment in its current
state cannot be of assistance in solving questions such as whether the existence of the
zodiac prompted the idea of defining the six arcs along the horizon, or if these arcs
were already recognized when the concept of the zodiac was emerging and becoming
Footnote 33 continued
solstices and equinoxes in Month XII, III, VI, IX and/or Months I, IV, VII, X see e.g., Horowitz (2014,
pp. 16–18).
J. C. Fincke et al.
popular. Either way, the answers to this question are relevant to better understanding
how the Enochic gate system might have emerged out of an earlier Babylonian model
based on degrees of arc. For reasons such as these we see our edition of BM 76829 as
a starting point for further research into interrelationships between the astronomical
traditions of Babylonia in the Late Babylonian period and Second Temple Judaea,
rather than a final word on any of the topics that we have addressed above. We trust
that BM 76829 and its contents will now become part of the interdisciplinary discus-
sion of these matters relating to the separate, but clearly connected worlds of what we
might call Babylonia and Judean intellectual thought.
Compliance with ethical standards
Conflict of interest All authors declare that they have no conflict of interest.
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... In a recent paper in this journal, Fincke et al. (2021;hereafter FHR) publish BM 76829, a small but highly interesting astronomical fragment from Babylonia. The obverse of the tablet contains a list of zodiacal signs accompanied by numerical values, which FHR suggest represents the varying rising place of the sun over the horizon at different times of the year. ...
Full-text available
In this paper I present an alternative reading and interpretation of the cuneiform tablet BM 76829. I suggest that the obverse of the tablet contains a simple astrological scheme linking the sign of the zodiac in which a child is born to the maximum length of life, and that the reverse contains a copy of a scheme relating parts of the body to the signs of the zodiac.
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This article analyzes changes in the reference system of the Jewish Calendar. Initially, the Jewish Calendar reference was based on the movement of the Sun. However, the calendar system used by the Jewish community worldwide now refers to two celestial objects, the Sun and the Moon. This study is literature research with a qualitative approach and uses the Book of Enoch and the Sefer Yetzirah as primary sources. The authors found that the first reference system of the Jewish Calendar, which was based on the movement of the Sun (Solar), follows the description in the Book of Enoch and the Sefer Yetzirah. These two books explain the calendar system used by Prophet Idris to Prophet Moses. The Solar Calendar reference system was later replaced with the Lunar Calendar in the second century BCE. Six centuries later, the Jewish Calendar returned to using a Solar Calendar combined with the Lunar Calendar, known as the Lunisolar Calendar. The latest reference system is a mathematical calendar with the Metonic cycle that refers to the movement of the Sun and Moon. The authors argue that the influence of the Greek civilization and the adjustment of the difference between the Lunar and Solar Year caused the changes in the reference system of the Jewish Calendar.
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Enochic cosmology is often regarded as bewildering and unrelated to reality. The article explores the main features of the ancient Hebrew universe showing that the prophet's revelation was accurate; it is concluded that the heliocentric model is false.
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This paper proposes a new reconstruction of 4Q208, a copy of the Aramaic Astronomical Book of Enoch (AAB) that was preserved in 36 small fragments. The new reconstructed text demonstrates that alongside the textual continuity between 4Q208 and its later and fuller copy 4Q209, these two manuscripts also reveal textual plurality. The difference in their schemes is not merely an outcome of scribal error but seems to derive from an ongoing scientific creativity. Later Jewish astronomers adapted the spatial astronomical theory present in 4Q208 to the more accurate, and probably already sanctified, 364-day year. They also attempted to suit 4Q208's lunar visibility scheme to reality by reversing the order of full and hollow months.
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Argument The Astronomical Book of Enoch describes the passage of the moon through the gates of heaven, which stand at the edges of the earth. In doing so, the book describes the position of the rising and setting of the moon on the horizon. Otto Neugebauer, the historian of ancient science, suggested using the detailed tables found in later Ethiopic texts in order to reconstruct the path of the moon through the gates. This paper offers a new examination of earlier versions of the Astronomical Book, using a mathematical analysis of the figures and astronomical theories presented throughout the Aramaic Astronomical Book; the results fit both the data preserved in the scrolls and the mathematical approach and religious ideology of the scroll's authors better than the details found in the late Ethiopic texts. Among other new insights, this alternate theory also teaches about the process of the composition of the Astronomical Book in the first centuries of its composition.
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This paper examines the two intercalation schemes found in the Mesopotamian astronomical compendium MUL.APIN from the beginning of the first millenium BCE, and the lunar theory that they imply. It demonstrates that the two schemes do not agree with each other. Two intercalation rules in the second scheme use the conjunction of the moon and the Pleiades. This paper concludes that the intercalation rules are based on the assumption of a 28-day ideal sidereal month. These rules work with a triennial cycle of intercalating one additional month every third lunar year. Two similar intercalation schemes from other compositions, likewise dating from the beginning of the first millenium BCE, are known: a seventh-century intercalation scheme from Babylonia that also assumes a 28-day ideal sidereal month and an intercalation scheme from an unpublished astronomical commentary that, like the scheme in MUL.APIN, uses a triennial cycle. Previous scholars believed that discrepancies exist between the dates of the conjunctions of the moon and the Pleiades across all three schemes. However, this paper proposes that the astronomical assumptions of the three schemes are identical.