- Lunation length of the previous month: 29 (hollow) or 30
(full) [days]

In Babylonian text the lunation length of the previous month can be derived as they stated if the day of the first crescent being on the 30^{th}day (so previous month was 29 days: hallow month) or being on the 1^{st}(previous month was 30 days: full month). - na (
*na in the middle of the month*: first mentioned around 567 BCE) [*uš*]

The time between sunrise and moonset on the day the moon sets for the first time after sunrise (in evening)

- KUR (first mentioned around 418 BCE) [
*uš*]

The time between sunrise and moonrise on the day the moon is seen for the last time (last crescent) (in morning)

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- na [
*uš*]

The time between sunrise and moonset on the day the moon sets for the first time after sunrise (in morning)

- ŠÚ [
*uš*]

The time between moonset and sunrise on the day the moon sets for the last time before sunrise (in morning)

- ME [
*uš*]

The time between moonrise and sunset on the day the moon rises for the last time before sunset (in evening)

- GE
_{6 }[*uš*]

The time between sunset and moonrise on the day the moon rises for the first time after sunset (in evening)

- na
_{N}[*uš*] (*na of the 1*) [Lis Brack-Bernsen. "Goal-Year tablets: Lunar data and predictions." In Ancient Astronomy and Celestial Divination, edited by N.M. Swerdlow: MIT Press, 1999, page 168]^{st}day

The time between sunset and moonset on the day the moon is seen for the first time (first crescent) (in evening)

- na [
*uš*]

The time between sunrise and moonset on the day the moon sets for the first time after sunrise (in morning)

- ŠÚ [
*uš*]

The time between moonset and sunrise on the day the moon sets for the last time before sunrise (in morning)

- ME [
*uš*]

The time between moonrise and sunset on the day the moon rises for the last time before sunset (in evening)

- GE
_{6 }[*uš*]

The time between sunset and moonrise on the day the moon rises for the first time after sunset (in evening) - KUR [
*uš*]

The time between sunrise and moonrise on the day the moon is seen for the last time (last crescent) (in morning)

In interesting aspect of the Lunar Four is that if one make sums
like: ŠÚ+na; ME+GE_{6}; and ŠÚ+na+ME+GE_{6},
these sums become almost independent of longitude, latitude and
horizon altitude. This is ideal for determining a parameters
independent of location. Below picture is calculated using ARCHAEOCOSMO:

**The spiky na curve (green) and smoother ŠÚ+na curve
(blue)**

<**Remark: **above na curve is slightly different
then Brack-Bernsen's Figure 4 as that had a small ephemeris
error [Pers. Comm. Brack-Bernsen, 2012]>

Brack-Bernsen showed e.g. that ŠÚ+na+ME+GE_{6} is a
very good way to determine the periodicy of Lunation-length P_{Φ}
[Lis Brack-Bernsen, and Matthias Brack. "Analysing shell structure
from Babylonian and modern times." In 10th Nuclear physics
workshop, 1-13. Kazimierz Dolny Poland, 2003, page 10.]: ~411.8
days (being the HarmonicDifference(Lunar
apse cycle, Tropical year)). (Meeus [2002], page 21-23)

From ŠÚ+na and ME+GE_{6} (above picture) one can
relate the Tropical year with 12.368 Synodic months.

(na_{N})_{i}=(na_{N})_{i-223}
+ 1/3*(ŠÚ+na)_{i-229}_{} |
mod(ŠÚ+na)_{i-229} |

ŠÚ_{i}=ŠÚ_{i-223}
+ 1/3*(ŠÚ+na)_{i-223}na _{i}=na_{i-223} - 1/3*(ŠÚ+na)_{i-223} |
mod(ŠÚ+na)_{i-223} |

ME_{i}=ME_{i-223} +
1/3*(ME+GE_{6})_{i-223}(GE _{6})_{i}=(GE_{6})_{i-223}
- 1/3*(ME+GE_{6})_{i-223}_{} |
mod(ME+GE_{6})_{i-223} |

KUR_{i}=KUR_{i-223} +
1/3*(ME+GE_{6})_{i-229} |
mod(ME+GE_{6})_{i-229} |

My proposed modulus function maps nicely on the additional correction rules mentioned in BM 42282+42294 (Lis Brack-Bernsen, and Hermann Hunger. "BM 42282+42294 and the Goal-Year method." SCIAMVS 9 (2008): 3-23).

The 223 (Synodic months) is the Saros cycle, and was used by the Babylonians in these formula and recorded this method around 600 BCE. The below picture is calculated using ARCHAEOCOSMO:

Green curve is na_{i}while Blue curve is na_{i-223}- 1/3*(ŠÚ+na)_{i-223}

As can be seen in the above graph, there can be
still large discrepancies (e.g. at lunation 236), but if one
realises this modulus function, the difference between predicted
and actual value is not that large.

- First a matching of the Babylonian lunation length with
the calculated lunation length (using my implementation of Schaefer's criterion).

- When best match found; the ŠÚ+na and ME+GE
_{6}are calculated.

- With that info the below pictures are graphed

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[LBAT 1251+1252, Hunger, 1999, page 91-94]

[LBAT 1285, Hunger, 2006, page 269-275]

So this larger than expected error is unlikely to be explainable due to the impreciseness of the Babylonian water clocks around 200 BCE. Would there be a systematic measurement procedure difference?

Major content related changes: Aug. 8, 2012