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The authors have declared that no competing interests exist.

Conceived and designed the experiments: LJ SP. Performed the experiments: LJ SP. Analyzed the data: LJ SP. Contributed reagents/materials/analysis tools: LJ SP. Wrote the paper: LJ SP.

The Ancient Egyptians wrote Calendars of Lucky and Unlucky Days that assigned astronomically influenced prognoses for each day of the year. The best preserved of these calendars is the Cairo Calendar (hereafter CC) dated to 1244–1163 B.C. We have presented evidence that the 2.85 days period in the lucky prognoses of CC is equal to that of the eclipsing binary Algol during this historical era. We wanted to find out the vocabulary that represents Algol in the mythological texts of CC. Here we show that Algol was represented as Horus and thus signified both divinity and kingship. The texts describing the actions of Horus are consistent with the course of events witnessed by any naked eye observer of Algol. These descriptions support our claim that CC is the oldest preserved historical document of the discovery of a variable star. The period of the Moon, 29.6 days, has also been discovered in CC. We show that the actions of Seth were connected to this period, which also strongly regulated the times described as lucky for Heaven and for Earth. Now, for the first time, periodicity is discovered in the descriptions of the days in CC. Unlike many previous attempts to uncover the reasoning behind the myths of individual days, we discover the actual rules in the appearance and behaviour of deities during the whole year.

The Ancient Egyptians referred to celestial events indirectly [_{M} = 29.6 days period of the _{A} = 2.85 days period of the eclipsing binary

The Ancient Egyptian year contained 12 months (_{i} with _{A} and _{M} signals were originally discovered [_{E} of _{A} and _{M} signals. The time points leading to the discovery of these signals were close to phase, _{E}. The lucky prognoses of each SW are a subsample of the above mentioned large samples of lucky prognoses. We compute an impact parameter _{x} for the _{i} of each SW with _{i} of the lucky prognoses of any particular SW may strengthen (if _{x} > 0) or weaken (if _{x} < 0) the _{A} and _{M} signals. The impact parameter _{x} is used for identifying the SWs having lucky prognoses close to phase,

Our statistical analysis also confirms two general things regarding the origin of the mythological texts of CC. First, the appearances and feasts of various deities are not independent of the prognoses, or randomly assigned, but regulated by the same periodic patterns. Second, the deities are used to represent the same astronomical phenomena that were also used to choose the prognoses for the days of the year.

In this section, we transform the dates of 28 SWs in the mythological texts of CC into series of time points _{i}. Our main aim is that all stages of the production of these data can be replicated. With these instructions, similar series of time points can be produced for any particular SW in CC or other similar calendars, where the SW dates are available. We create the data in two stages: Identification of SW dates and Transformation of SW dates into series of time points.

CC is the best preserved Calendar of Lucky and unlucky Days. As in our two previous studies [_{i}(1, 10)).

Naturally, we can not analyse all words in CC. Our main selection criterion is to include deities, nouns or locations that could have been used to indirectly describe periodic phenomena, due to their significant mythological properties and multiple occurrences in the text. Our list of SWs is not absolute and we give all the necessary information for other researchers to repeat our experiment on other SWs we may have ignored. Our 28 SWs in Ancient Egyptian language are given in

We do not use the occurrences of our SWs in compound words and composite deities (e.g. House of Horus or Ra-Horakhti), because it is uncertain to which word, if not both, the prognosis is connected to. Our identifications of 28 SWs in CC are given in _{G} and _{S}. For example, “Abydos” has _{G} = 3 and _{S} = 2.

SW | Prog | Ltz | Bkr | ||
---|---|---|---|---|---|

Abydos | 13 | 3 | SSS | Yes | Yes |

Abydos | 17 | 3 | - - - | Yes | Yes |

Abydos | 11 | 4 | GGG | Yes | Yes |

Abydos | 18 | 5 | GGG | Yes | Yes |

Abydos | 27 | 6 | - - - | Yes | Yes |

Abydos | 28 | 7 | GGG | Yes | Yes |

Abydos | 13 | 8 | SSS | Yes | Yes |

Abydos | 23 | 8 | - - - | Yes | Yes |

Busiris | 26 | 2 | SSS | Yes | Yes |

Busiris | 14 | 5 | SSS | Yes | Yes |

Busiris | 26 | 5 | SSS | Yes | Yes |

The selected word (SW) identified on day (

The dating of CC does not influence the results of our currect analysis, because we transform the time points to unit vectors with _{0} in time. Adding any positive or negative constant value to these time points rotates all unit vectors with the same constant angle. Hence, our significance estimates of Eqs (

The transformation relations in Eqs (_{E} = 30(_{i} was a decimal part. This decimal part _{i} was different for each of the three parts of the day. The _{i} values depended on the chosen transformation between Egyptian and Gregorian year, and on the chosen day division. The _{A} and _{M} signals were discovered in samples of series of time points SSTP = 1, 3, 5, 7, 9 and 11 in Jetsu et al.[_{i} values were different for every _{E}. The time points _{i} of these six samples are given in

SSTP = 1 | SSTP = 3 | SSTP = 5 | SSTP = 7 | SSTP = 9 | SSTP = 11 |
---|---|---|---|---|---|

0.080 | 0.095 | 0.076 | 0.120 | 0.142 | 0.114 |

0.239 | 0.284 | 0.227 | 0.359 | 0.426 | 0.341 |

0.399 | 0.473 | 0.379 | 0.739 | 0.784 | 0.727 |

1.080 | 1.095 | 1.076 | 1.120 | 1.142 | 1.113 |

1.240 | 1.284 | 1.227 | 1.360 | 1.425 | 1.340 |

The _{i} values of SSTP = 1, 3, 5, 7, 9 and 11 from Table 3 in Jetsu et al. [

The mean of the decimal parts _{i} of all these _{i} is _{t} = 0.33. In this study, the time point for an SW at the day _{t} ≈ ±0.^{d}5) and some prognosis texts may refer to the previous or the next day (_{t} ≈ ±1.^{d}5). The _{i} of _{E} for the ephemerides connected to the _{A} and _{M} signals (Eqs (_{A} and _{M} signals. For a given _{t})/30. In other words, if the analysis our data gives any particular

The time points _{i} for all dates with a “GGG” or “SSS” prognosis combination in CC are given in _{i} are needed in computing the binomial distribution probabilities _{B} of

_{i} |
Prog | ||
---|---|---|---|

1 | 1 | 0.33 | GGG |

2 | 1 | 1.33 | GGG |

5 | 1 | 4.33 | GGG |

7 | 1 | 6.33 | GGG |

9 | 1 | 8.33 | GGG |

10 | 1 | 9.33 | GGG |

11 | 1 | 10.33 | SSS |

12 | 1 | 11.33 | SSS |

16 | 1 | 15.33 | SSS |

17 | 1 | 16.33 | SSS |

The day (_{i}) for the days with the prognosis (Prog) combinations “GGG” or “SSS”. There are _{G} = 177 and _{S} = 105 days with a “GGG” and “SSS” combination, respectively. These data are from Table 1 in Jetsu et al. 2013 [

Let us assume that time is a straight line, where events are equidistant dots with a separation of 2

If the Rayleigh method discovers the period _{1}, _{2}, …, _{n}], it is possible to identify those subsamples _{i} are
_{0} is an arbitrary zero epoch and _{i} − _{0})/_{i} = 360°, _{i} are the phase angles. The test statistic of the Rayleigh test is
_{R} = atan(_{y}/_{x}), _{R} = Θ_{R}/(360°). Coinciding directions Θ_{i} give |_{i} give |_{i} and the length |_{t}, _{i}, _{0} or _{E}. Using the above _{E} of _{R} = 0°. All _{i} with −90° < Θ_{i} < 90° strengthen the _{i} weaken it. The test statistic can be divided into _{0} = _{E} in _{x} is computed only for the _{x} > 0) or weaken (_{x} < 0) the _{x} ≈ 0).

Using the zero epoch _{0} = 0 for the _{i} of the G prognoses in _{E} values of _{A} and _{M} signals with

SSTP = 1 | SSTP = 3 | SSTP = 5 | SSTP = 7 | SSTP = 9 | SSTP = 11 | |
---|---|---|---|---|---|---|

2.85 | 0.45 | 0.45 | 0.44 | 0.61 | 0.61 | 0.60 |

29.6 | 3.42 | 3.42 | 3.42 | 3.58 | 3.58 | 3.58 |

These six large samples have _{E} = 0.53 ± 0.09 for _{A} and _{E} = 3.50 ± 0.09 for _{M}. Hence, we use the following two ephemerides
_{i} of _{x} values of Eqs (_{i} of this SW reach _{z} ≤ 0.2 with the ephemerides of Eqs (_{z} ≤ 0.2.

In our Figs _{i} of each SW to _{i} = [cos Θ_{i}, sin Θ_{i}] on a unit circle, where time runs in the counter clock–wise direction. For the _{A} signal, we define four points Aa, Ab, Ac and Ad. The first one, Aa, is at _{i} pointing between Ad ≡ −90° and Ab ≡ +90° give _{x} > 0 and strengthen _{A} signal, the other ones weaken it. Because _{A} equals 57^{d}/20, the _{i} of _{i} separated by multiples of 57 days are equal. For clarity, we shift such overlapping _{i} values by Δ

_{A} signal”_{i} of an SW strengthen the _{A} signal ≡ _{x} ≥ 1.0 and _{z} ≤ 0.2 with the ephemeris of

_{i} of an SW show periodicity with _{A}, but their contribution to the _{A} signal is insignificant when 0 ≤ _{x} < 1.0 or they weaken this signal when _{x} < 0 ≡ _{x} < 1.0 and _{z} ≤ 0.2 with the ephemeris of

We use similar terminology for the

Time runs in the counter clock–wise direction on these unit circles. We give the _{z} and _{x} values only when _{z} ≤ 0.2. The large black point indicates the Θ_{R} direction. (a) _{i} with _{i} with _{i} with _{i} with

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

otherwise as in

Our notations for the lucky and unlucky time points _{i} of each SW are _{i} and _{i}. The notations for their unit vectors _{i} of _{i} and _{i}, respectively. The critical level _{z} measures the probability for the concentration of _{G} and _{S} directions of _{i} and _{i} of each SW. These directions are embedded within the directions of _{i} (_{G} = 177) and _{i} (_{S} = 105). We first choose the direction Θ_{R} of _{1} directions of _{i} or _{i} of this SW that are among the _{2} of all _{G} or _{S} directions closest to Θ_{R}. For each SW, this gives the binomial distribution probability
_{G} or _{S}, and _{B} = _{G}/_{G} or _{S}/_{S}. This _{B} is the probability for that the directions of a particular SW occur _{1} times, or more, among all _{2} directions closest to Θ_{R}. Many _{z} estimates based on small samples (_{G} or _{S}) are unreliable, but the _{B} estimates based on large samples (_{G} = 177 or _{S} = 105) are not.

All results of our analysis are given in _{A} are those shown in figure panels “a” and “b”. The corresponding results for _{M} are shown in figure panels “c” and “d”.

Of all 28 SWs, only the lucky prognoses of _{A} signal of _{x}_{z} ≤ 0.2 with the ephemeris of _{z} ≤ 0.2), but they are not connected to the _{A} signal (_{x} < 1.0). In this section, we discuss these eight SWs in the order of their impact on the _{A} signal, i.e. in the order of decreasing _{x} with the ephemeris of

This SW has the largest impact _{x} = +3.5 on the _{A} signal and the highest significance of the above eight SWs (_{z} = 0.03, _{G} = 14). The unit vectors _{i} and _{i} of lucky and unlucky prognoses with the ephemeris of _{i} pointing between Ad ≡ −90° and Ab ≡ +90° strengthen the _{A} signal. Twelve out of all fourteen _{i} are within this interval (_{i} closest to Θ_{R} = 11° reach a high significance of _{B} = 0.006 (_{1} = 4, _{2} = 10, _{G} = 177). The _{i} pointing closest to Aa and giving the strongest impact on the _{A} signal has the CC text [

_{i}(14, 2) ≡ +6°:

The texts [_{i} closest to Aa are

_{i}(19, 12) ≡ +13°:

_{i}(27, 1) ≡ +19°:

_{i}(24, 3) ≡ +19°:

_{i}(1, 7) ≡ +32°:

_{i}(15, 11) ≡ +38°:

_{i}(27, 3) ≡ +38°:

_{i}(18, 1) ≡ −38°:

_{i}(1, 9) ≡ +51°:

_{i}(23, 7) ≡ −69°:

_{i}(29, 3) ≡ −69°:

_{i}(7, 9) ≡ +88°:

_{i}(1, 10) ≡ −120°:

_{i}(28, 3) ≡ +164°:

These passages of lucky prognoses are suggestive of _{i} of all 28 SWs, the _{i} of _{x} = +3.5). If these _{i} represent _{i}(7, 9) ≡ +88° text may refer to an imminent eclipse and _{i}(28, 3) ≡ +164° to the moment when the beginning of the eclipse is just becoming observable with naked eye. These passages could certainly describe naked eye observations of the regular changes of

Three _{i} of _{B} = 0.07 (_{1} = 3, _{2} = 25, _{S} = 105). The fourth vector _{i} points close to Aa. Their CC texts [

_{i}(26, 1) ≡ −107°:

_{i}(11, 11) ≡ −107°:

_{i}(20, 9) ≡ −69°:

_{i}(5, 8) ≡ 6°:

If the _{i} that described feasts were connected to the brightest phase of _{i} describing anger would have occurred after _{i}(5, 8) would seem natural for a lucky prognosis of

The _{i} and _{i} of _{z} > 0.2 with the ephemeris of _{i} texts mentioning both _{i}(27, 1) ≡ −82°, _{i}(27, 3) ≡ −73° and _{i}(29, 3) ≡ −48° with the ephemeris of _{i}(29, 3) would describe the brightening of

The lucky prognoses reach _{z} = 0.07 (_{G} = 32) with the ephemeris of _{x} = +2.5 on the _{A} signal (_{B} values, i.e. _{i} concentrations, may indicate that _{i} of _{z} = 0.2 (_{S} = 26) with the ephemeris of

The lucky prognoses show weak periodicity (_{z} = 0.1, _{G} = 4) with the ephemeris of _{x} = +2.0 on the _{A} signal (_{M} signal is even larger, _{x} = +2.9 (_{i} and _{i} distributions of

The lucky prognoses have an impact of _{x} = +1.4 on the _{A} signal (_{z} = 0.2, _{G} = 15). Six _{i} reach _{z} = 0.01 (_{i} closest to Θ_{R} reach a high significance of _{B} = 0.003 (_{1} = 5, _{2} = 18, _{S} = 105) and may refer to an approaching eclipse of _{i} also show a weak connection to the _{i}(7, 9) ≡ 88°

The _{i} and _{i} reach _{z} = 0.06 (_{G} = 4) and 0.05 (_{S} = 3) with the ephemeris of _{i} on the _{A} signal is _{x} = +1.3 (_{i} at Ad, after the proposed eclipse at Ac, are strongly connected to _{B} = 0.0004 (_{1} = 3, _{2} = 6, _{S} = 105). The texts [

_{i}(27, 8) ≡ −95°:

_{i}(13, 6) ≡ −82°:

_{i}(7, 10) ≡ −82°:

These three unlucky prognoses (_{i} and _{i} distributions of _{i} vectors of

The lucky prognoses show weak periodicity (_{z} = 0.1, _{G} = 18) and an impact of _{x} = +1.1 on the _{A} signal with the ephemeris of _{B} = 0.02, _{1} = 12, _{2} = 63, _{G} = 177). Ennead was a group of nine deities in Ancient Egyptian mythology. We discussed earlier, why

The lucky prognoses show weak periodicity with _{A}, but their impact on this signal is insignificant, _{x} = +0.2, with the ephemeris of

These lucky prognoses weaken the _{A} signal, because their impact is _{x} = −1.0 with the ephemeris of

We discuss the remaining other 20 SWs in this section and in sections

Algol in unlucky prognoses

The Moon in unlucky prognoses

No Algol or the Moon in lucky or unlucky prognoses

These SWs are discussed only briefly, because they are not connected to the _{A} signal.

The lucky prognoses of _{M} signal, because they have _{x} ≥ 1.0 and _{z} ≤ 0.2 with the ephemeris of

These lucky prognoses reach the highest impact parameter value of this study, _{x} = +5.3, on the _{M} signal. This periodicity also reaches the highest Rayleigh test significance of all, _{z} = 0.001 (_{G} = 19). The good moments on _{z} = 0.06, _{S} = 5) and an even weaker connection to the _{z} = 0.2, _{S} = 5).

The second largest impact _{x} = +3.4 on the _{M} signal comes from these lucky prognoses. Again, the good moments coincide with Ma, the proposed Full _{z} = 0.03, _{G} = 19) combined with a very significant concentration (_{B} = 0.002, _{1} = 12, _{2} = 45, _{G} = 177). The unlucky prognoses also show a weak connection to the _{z} = 0.06, _{S} = 4).

The third largest impact on the _{M} signal, _{x} = +3.0, comes from the lucky prognoses of _{z} = 0.05 (_{G} = 4) with the ephemeris of

The lucky prognoses show weak periodicity (_{z} = 0.2, _{G} = 3) with the ephemeris of _{x} = 1.6 on the _{M} signal.

The lucky prognoses of these SW have a weaker impact on the _{M} signal, i.e. 1.0 ≤ _{x} ≤ 1.3 with the ephemeris

The lucky prognoses show a weak connection to the _{M}, because _{x} = −0.1 with the ephemeris of

The _{A} and _{M} signals were detected from the lucky prognoses _{i}[_{i} had no impact on these two signals. However, this does not rule out the possibility that the _{i} of some SW may be connected to _{i} vectors point away from Aa or Ma, i.e. _{x} < 0 with the ephemerides of Eqs (_{x} ≥ 0).

The unlucky prognoses have _{x} = −3.1 with the ephemeris of _{z} = 0.04 (_{S} = 5) and _{B} = 0.04 (_{1} = 5, _{2} = 39, _{S} = 105).

The three unlucky prognoses of this SW reach _{z} = 0.06 and a high significance of _{B} = 0.003 (_{1} = 3, _{2} = 11, _{S} = 105) with the ephemeris of

We will first discuss the unlucky prognoses of SWs having negative _{x} values with the ephemeris of

“See you on the dark side of the Moon” sums up the unlucky prognoses of _{z} = 0.05 (_{S} = 9) with the ephemeris of

_{i}(16, 7) ≡ 173°:

_{i}(17, 7) ≡ 185°:

take place during the New _{i} vectors of these two particular texts point at the opposite sides of Mc ≡ 180°, which supports both our “prediction” formula of

The four unlucky prognoses of this SW also point to the dark side of the _{z} = 0.05 (_{S} = 4) and _{B} = 0.02 (_{1} = 3, _{2} = 15, _{S} = 105) with the ephemeris of

These unlucky prognoses show a weak connection to the

The significance estimates for the unlucky prognoses are _{z} = 0.02 (_{S} = 6) and _{B} = 0.009 (_{1} = 5, _{2} = 23, _{S} = 105) with the ephemeris

The significance estimates for these unlucky prognoses are _{z} = 0.03 (_{S} = 4) and _{B} = 0.003 (_{1} = 4, _{2} = 17, _{S} = 105) with the ephemeris of

These SWs are not connected to _{i} and _{i} have _{z} > 0.2 with the ephemerides of Eqs (

This concludes our analysis of 28 SWs. Numerous other [

Previously, we [

_{1}: _{A} = 2.^{d}850 in CC was P_{orb} of Algol.”

This is a summary of those tests:

_{1}, while

The horizontal continuous lines show the beginnings and ends of 10 hours long nights. The filled and open circles denote mid eclipse epochs occurring inside and outside such nights. The _{A1} = 10 hour time intervals of eclipses are denoted with thick continuous or thin dashed lines. The tilted open and closed triangles show the _{A2} = 7 and _{A3} = 3 hour limits.

Only a skilled naked eye observer would have been able to discover the minor exceptions from the 3 + 3 + 13 days regularity. _{A1} = 10 hours. Naked eye can detect brightness differences of 0.^{m}1 in ideal observing conditions. Hence, an eclipse detection is theoretically _{A2} = 7 hours when ^{m}1 dimmer than its brightest suitable comparison star _{A3} = 3 hours when ^{m}1 dimmer than all its other suitable comparison stars

Here, our statistical analysis of SWs giving the largest impact on the _{A} signal reveals that _{i}(7, 9), where he _{i}(28, 3) at the beginning of an eclipse—the only _{i} vector of

_{A} signal.

The two periods, _{A} and _{M}, regulate the assignment of mythological texts to specific days of the year. The _{A} or _{M}, like _{z} ≤ 0.2 with _{A} or _{M}. All these regularities can not simply be dismissed as a coincidence, let alone with the possible errors of _{t} ≈ ±0.5 or ±1.5 days.

What was the origin of the phenomenon that occurred every third day, but always 3 hours and 36 minutes earlier than before, and caught the attention of Ancient Egyptians? Our statistical analysis leads us to argue that the mythological texts of CC contain astrophysical information about

Inside our superimposed rectangle is the hieratic writing for the word

(EPS)

Day (_{i}_{i}_{i}_{i}), direction of their _{R}) and differences ΔΘ_{i} = Δ_{i} − Θ_{R} with _{A}_{1}, _{2}, _{B}_{B}. Note that the parameters are given in the order of increasing ΔΘ_{i}, _{1} and _{2}. All values mentioned in text are marked in bold. We also make available the code of a Python 3.0 program

(PDF)

We thank L. Alha, T. Hackman, T. Lindén, K. Muinonen and H. Oja for their comments on the manuscript. This work has made use of NASA’s Astrophysics Data System (ADS) services.