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 These descriptions support our claim that CC is the oldest preserved historical document of the discovery of a variable star.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.
Citation: Jetsu L, Porceddu S (2015) Shifting Milestones of Natural Sciences: The Ancient Egyptian Discovery of Algol’s Period Confirmed. PLoS ONE10(12): e0144140. https://doi.org/10.1371/journal.pone.0144140
Editor: Mark Spigelman, Hebrew University, ISRAEL
Received: June 10, 2015; Accepted: November 3, 2015; Published: December 17, 2015
Copyright: © 2015 Jetsu, Porceddu. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Data Availability: Data are available from Dryad: http://dx.doi.org/10.5061/dryad.tj4qg.
Funding: The authors have no support or funding to report.
Competing interests: The authors have declared that no competing interests exist.
The Ancient Egyptians referred to celestial events indirectly [1–4] by relating them to mythological events. Many prognoses in the Calendars of Lucky and Unlucky Days have been connected to astronomical observations [1, 5–7]. Such connections between astronomical events and prognosis texts have been uncovered in most cases only for individual days [6, 8, 9]. The PM = 29.6 days period of the Moon has been discovered in CC . We have claimed that this document also contains the PA = 2.85 days period of the eclipsing binary Algol . However, it not a straightforward task to identify those indirect mythological references that are influenced by Algol in CC. Here we present a statistical analysis that reveals which CC prognosis texts describe Algol’s regular variability.
The Ancient Egyptian year contained 12 months (M) of 30 days (D) and five additional “epagomenal” days. CC gives three prognoses for each D of every M (G = “gut” = “good” and S = “schlecht” = “bad”) [11, 12]. CC also gives textual descriptions of the daily prognoses (S1 Fig). We study the dates of 28 selected words (hereafter SWs) in these mythological texts of CC. The dates are transformed into series of time points ti with Eq (2). The PA and PM signals were originally discovered  from six large samples of lucky prognoses (n = 6 × 564 = 3384). We use these six samples to determine the zero epochs tE of Eq (9) for the PA and PM signals. The time points leading to the discovery of these signals were close to phase, ϕ = 0, of Eq (5) using the ephemerides of Eqs (11) and (12) based on these zero epochs tE. The lucky prognoses of each SW are a subsample of the above mentioned large samples of lucky prognoses. We compute an impact parameter zx for the ti of each SW with Eq (10). The time points ti of the lucky prognoses of any particular SW may strengthen (if zx > 0) or weaken (if zx < 0) the PA and PM signals. The impact parameter zx is used for identifying the SWs having lucky prognoses close to phase, ϕ = 0, computed with the ephemerides of Eqs (11) and (12). We will show that Algol and the Moon were at their brightest close to phase ϕ = 0 with these two ephemerides. Hence, Algol’s eclipse and the New Moon occurred close to ϕ = 0.5.
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 ti. 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.
Identification of SW dates
CC is the best preserved Calendar of Lucky and unlucky Days. As in our two previous studies [10, 11], we use the best preserved continuous calendar found on pages recto III-XXX and verso I-IX of papyrus Cairo 86637. There are two CC translations, in English by Bakir  and in German by Leitz . Our SWs have been identified according to the hieroglyphic transcription in Leitz  and the two aforementioned translations. In case of discrepancy we have consulted the photocopies of the original hieratic text given by Leitz . For the sake of convenience, we quote sentences according to Bakir’s English translation despite its imperfections because there is neither space nor reason to discuss the linguistic details of the text in the present article. This approach should ascertain that our study of the CC sentences is objective. In other words, we do not ourself translate any CC sentences into English, but we do check which individual Ancient Egyptian SWs were also identified by Bakir  and Leitz . There is only one exception to our sentence quotation rule, i.e. the CC text connected to Horus where Bakir  did not identify Horus, but Leitz  and we did (Algol in lucky prognoses: the text at date gi(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 Table 1.
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 Table 2. It shows that all our 460 SW date identifications are the same as those made by Leitz  (Column 5: 460× “Yes”). However, 21 of our identifications were not made by Bakir  (Column 6: 21× “No”: 1× “Earth”, 2× “Enemy”, 4× “Fire”, 12× “Heart”, 1× “Horus” and 1× “Osiris”). Fortunately, most days have combinations “GGG” or “SSS” and we know that the lucky or unlucky SW prognosis is certainly correct. We ignore the heterogeneous combinations “HET” (like “SSG”at D = 6 and M = 1), because the correct SW prognosis is uncertain. The dates with an unknown prognosis combination, “- - -”, are naturally also ignored. Our notations for the number of lucky and unlucky dates for each SW are nG and nS. For example, “Abydos” has nG = 3 and nS = 2.
Transformation of SW dates into series of time points
The dating of CC does not influence the results of our currect analysis, because we transform the time points to unit vectors with Eq (6). The mutual directions between these unit vectors do not depend on the chosen zero epoch t0 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 (8) and (13) do not depend on the connection between Gregorian and Egyptian days. The only assumption made in our Eq (2) below is that the separation between two subsequent days is exactly one day during the particular year that CC happens to describe.
The transformation relations in Eqs (2) and (3) of Jetsu et al. were (1) where NE = 30(M − 1) + D and ai was a decimal part. This decimal part ai was different for each of the three parts of the day. The ai values depended on the chosen transformation between Egyptian and Gregorian year, and on the chosen day division. The PA and PM signals were discovered in samples of series of time points SSTP = 1, 3, 5, 7, 9 and 11 in Jetsu et al.. The size of each sample was n = 564. The period analysis results were the same for all these six samples, although their ai values were different for every NE. The time points ti of these six samples are given in Table 3.
The mean of the decimal parts ai of all these n = 6 × 564 = 3384 values of ti is mt = 0.33. In this study, the time point for an SW at the day D of the month M in CC is therefore computed from (2) This accuracy is sufficient, because we do not know to which part or parts of the day each SW refers to (σt ≈ ±0.d5) and some prognosis texts may refer to the previous or the next day (σt ≈ ±1.d5). The ti of Table 3 (n = 6 × 564 = 3384) are also later used to determine the zero epochs tE for the ephemerides connected to the PA and PM signals (Eqs (11) and (12)). Our “synchronization” of time points of Eqs (1) and (2) ensures that these ephemerides enable us to identify the SWs connected to the PA and PM signals. For a given t value, the inverse transformation is (3) (4) where INT removes the decimal part of (t + 1 − mt)/30. In other words, if the analysis our data gives any particular t value, the D and M values of this t can be solved from Eqs (3) and (4).
Let us assume that time is a straight line, where events are equidistant dots with a separation of 2π. If this line is wound on a d = 1 diameter wheel, the dots line up at the same point on the wheel. Removing some dots produces gaps in the time line, but the remaining dots will still line up on the wheel. However, they will not line up on a d ≠ 1 diameter wheel. This is an analogy for the Rayleigh test. It projects time points on a unit circle with the tested period P. These points line up in the same direction, if their time distribution is regular with the tested P.
If the Rayleigh method discovers the period P in a series of time points points t = [t1, t2, …, tn], it is possible to identify those subsamples t* of n* time points that strengthen this signal. In other words, the signal can be separated from noise. The phases of the n time points ti are (5) where t0 is an arbitrary zero epoch and FRAC removes the integer part of (ti − t0)/P. The unit vectors are (6) where Θi = 360°, ϕi are the phase angles. The test statistic of the Rayleigh test is (7) where vector points to ΘR = atan(Ry/Rx), and . The corresponding phase is ϕR = ΘR/(360°). Coinciding directions Θi give |R| = n, while random Θi give |R| ≈ 0. The critical level (i.e. significance) of the Rayleigh test is (8) We use the ephemeris zero epoch (9) The mutual directions of ri and the length |R| are invariant for any constant shift of mt, ti, t0 or tE. Using the above tE of Eq (9), vector R points to Θ = ΘR = 0°. All ri with −90° < Θi < 90° strengthen the P signal, while the remaining ri weaken it. The test statistic can be divided into . We fix t0 = tE in Eq (5) and compute the “impact” of any subsample t* on the P signal from (10) where Rx is computed only for the n = n* time points of t*. These t* may strengthen (zx > 0) or weaken (zx < 0) the P signal, or represent noise (zx ≈ 0).
These six large samples have tE = 0.53 ± 0.09 for PA and tE = 3.50 ± 0.09 for PM. Hence, we use the following two ephemerides (11) (12) for computing the phases ϕi of Eq (5). The lucky “GGG” prognoses of every SW are a subsample of the above six large samples of all “G” prognoses. We give the z and zx values of Eqs (7) and (8) for any particular SW, if the analysed ti of this SW reach Qz ≤ 0.2 with the ephemerides of Eqs (11) or (12). These periodicities are called weak if 0.05 < Qz ≤ 0.2.
In our Figs 1–13, we project the ti of each SW to ri = [cos Θi, sin Θi] on a unit circle, where time runs in the counter clock–wise direction. For the PA signal, we define four points Aa, Ab, Ac and Ad. The first one, Aa, is at ϕ = 0 ≡ 0° with the ephemeris of Eq (11). The next three points Ab, Ac and Ad are separated by Δϕ = 0.25 ≡ 90°. Vectors ri pointing between Ad ≡ −90° and Ab ≡ +90° give zx > 0 and strengthen PA signal, the other ones weaken it. Because PA equals 57d/20, the ϕi of ti separated by multiples of 57 days are equal. For clarity, we shift such overlapping ϕi values by Δϕ = 0.005 away from each other in our Figs 1–13. However, there are no such shifts in our computations. Our unambiguous terminology is:
“Connected to the PA signal” ≡ ti of an SW strengthen the PA signal ≡ zx ≥ 1.0 and Qz ≤ 0.2 with the ephemeris of Eq (11).
“Connected to Algol” ≡ ti of an SW show periodicity with PA, but their contribution to the PA signal is insignificant when 0 ≤ zx < 1.0 or they weaken this signal when zx < 0 ≡ zx < 1.0 and Qz ≤ 0.2 with the ephemeris of Eq (11).
We use similar terminology for the Moon (Eq (12)), and Ma–Md points similar to Aa–Ad.
Time runs in the counter clock–wise direction on these unit circles. We give the z, Qz and zx values only when Qz ≤ 0.2. The large black point indicates the ΘR direction. (a) gi with Eq (11). Point Aa is at ϕ = 0 ≡ 0°. The thick line centered on point Ac at ϕ = 0.5 ≡ 180° outlines the proposed phase for the 10 hr eclipse of Algol. (b) si with Eq (11). (c) gi with Eq (12). Point Ma at ϕ = 0 ≡ 0° is close to the proposed Full Moon phase. (d) si with Eq (12)
Our notations for the lucky and unlucky time points ti of each SW are gi and si. The notations for their unit vectors ri of Eq (6) are gi and si, respectively. The critical level Qz measures the probability for the concentration of all nG and nS directions of gi and si of each SW. These directions are embedded within the directions of all gi (Table 4: NG = 177) and si (Table 4: NS = 105). We first choose the direction ΘR of R for some SW. Then we identify the n1 directions of gi or si of this SW that are among the n2 of all NG or NS directions closest to ΘR. For each SW, this gives the binomial distribution probability (13) where N = NG or NS, and qB = nG/NG or nS/NS. This QB is the probability for that the directions of a particular SW occur n1 times, or more, among all n2 directions closest to ΘR. Many Qz estimates based on small samples (nG or nS) are unreliable, but the QB estimates based on large samples (Table 4: NG = 177 or NS = 105) are not.
All results of our analysis are given in S1 Table, where the results mentioned in text are marked with bold letters. The structure of S1 Table resembles the four panel structure of Figs 1–13. We give four separate tables for each SW. The results for the lucky and unlucky prognoses with PA are those shown in figure panels “a” and “b”. The corresponding results for PM are shown in figure panels “c” and “d”.
Algol in lucky prognoses
Of all 28 SWs, only the lucky prognoses of Horus, Re, Wedjat, Followers, Sakhmet and Ennead unambiguously strengthen the PA signal of Algol, because they have an impact of zx ≥ 1.0 and a significance of Qz ≤ 0.2 with the ephemeris of Eq (11). The lucky prognoses of Heliopolis and Enemy are connected to Algol (Qz ≤ 0.2), but they are not connected to the PA signal (zx < 1.0). In this section, we discuss these eight SWs in the order of their impact on the PA signal, i.e. in the order of decreasing zx with the ephemeris of Eq (11).
This SW has the largest impact zx = +3.5 on the PA signal and the highest significance of the above eight SWs (Qz = 0.03, nG = 14). The unit vectors gi and si of lucky and unlucky prognoses with the ephemeris of Eq (11) are shown in Fig 1ab. Point Aa is at ϕ = 0 ≡ 0°. Points Ab, Ac and Ad are separated by Δϕ = 0.25 ≡ 90°. Only the gi pointing between Ad ≡ −90° and Ab ≡ +90° strengthen the PA signal. Twelve out of all fourteen gi are within this interval (Fig 1a). The four Θi closest to ΘR = 11° reach a high significance of QB = 0.006 (n1 = 4, n2 = 10, NG = 177). The gi pointing closest to Aa and giving the strongest impact on the PA signal has the CC text 
gi(14, 2) ≡ +6°: “It is the day of receiving the white crown by the Majesty of Horus; his Ennead is in great festivity.”
gi(19, 12) ≡ +13°: “Horus has returned complete, nothing is missing in it.”
gi(27, 1) ≡ +19°: “Peace on the part of Horus with Seth.”
gi(24, 3) ≡ +19°: “He has given his throne to his son, Horus, in front of Re.”
gi(1, 7) ≡ +32°: “Feast of entering into heaven and the two banks. Horus is jubilating.”
gi(15, 11) ≡ +38°: “Horus hears your words in the presence of every god and goddess on this day.”
gi(27, 3) ≡ +38°: “Judging Horus and Seth; stopping the fighting.”
gi(18, 1) ≡ −38°: “It is the day of magnifying the majesty of Horus more than his brother, …”
gi(1, 9) ≡ +51°: “Feast of Horus son of Isis and … his followers … day”
gi(23, 7) ≡ −69°: “Feast of Horus … on this day of his years in his very beautiful images.”
gi(29, 3) ≡ −69°: “White crown to Horus, and the red one to Seth.”
gi(7, 9) ≡ +88°: “The crew follow Horus in the foreign land, examining its list … therein when he smote him who rebelled against his master.”
gi(1, 10) ≡ −120°: “Horus … Osiris … Chentechtai … land”
gi(28, 3) ≡ +164°: “The gods are in jubilation and in joy when the will is written (lit. made) for Horus, …”
These passages of lucky prognoses are suggestive of Algol at its brightest. The “white crown”, Horus having “returned complete” and “entering into heaven” (i.e. into the sky) are not easy to explain as symbols for the eclipse. Among the gi of all 28 SWs, the gi of Horus are the “best hit” on Aa (zx = +3.5). If these gi represent Algol at its brightest, then Aa is in the middle of this brightest phase and the thick line centered at Ac in Fig 1a outlines Algol’s eclipse. In this case, the gi(7, 9) ≡ +88° text may refer to an imminent eclipse and “the will is written” in gi(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 Algol.
si(26, 1) ≡ −107°: “… It is the day of Horus fighting with Seth. …”
si(11, 11) ≡ −107°: “Introducing the great ones by Re to the booth to see what he had observed through the eye of Horus the elder. They were with heads bent down when they saw the eye of Horus being angry in front of Re.”
si(20, 9) ≡ −69°: “Mat judges in front of these gods who became angry in the island of the sanctuary of Letopolis. The Majesty of Horus revised it.”
si(5, 8) ≡ 6°: “The Majesty of Horus is well when the red one sees his form. As for anybody who approaches it, anger will start on it.”
If the gi that described feasts were connected to the brightest phase of Algol, these si describing anger would have occurred after Algol’s eclipse. “Horus is well” for the last si(5, 8) would seem natural for a lucky prognosis of Horus (as it should be close to Aa) but it is deemed unlucky for some other reasons. This type of “conflict of interest” prognoses may explain, why there are significant concentrations of directions accompanied by a few irregular directions (e.g. Fig 7c).
The gi and si of Horus have Qz > 0.2 with the ephemeris of Eq (12), and are therefore not connected to the Moon, except for some gi texts mentioning both Horus and Seth. We argue that, as Leitz  also did, Mc ≡ 180° in Fig 1c coincides with the New Moon (see paragraph Seth). All the aforementioned lucky prognoses mentioning both Horus and Seth are close to Md ≡ −90° in Fig 1c, i.e. gi(27, 1) ≡ −82°, gi(27, 3) ≡ −73° and gi(29, 3) ≡ −48° with the ephemeris of Eq (12). The texts of these three days may describe the “luminosity competitions” between Horus and Seth which come to an end when more than half of the lunar disk becomes illuminated immediately after Md. The legend of the Contendings of Horus and Seth (hereafter LE1) has inspired these descriptions. The text “White crown to Horus, and the red one to Seth” in gi(29, 3) would describe the brightening of Horus with Algol (Fig 1a: Θ = −69°) and the brightening of Seth (Fig 1c: Θ = −48°) with the approaching Full Moon at Ma. The most simple explanation for the context of these texts is that the lucky prognoses of Horus are connected to Algol at its brightest.
The lucky prognoses reach Qz = 0.07 (nG = 32) with the ephemeris of Eq (11) and give the second largest impact zx = +2.5 on the PA signal (Fig 2a). Absence of small QB values, i.e. gi concentrations, may indicate that Re (the Sun) was casually following the undertakings of Horus. The si of Re reach Qz = 0.2 (nS = 26) with the ephemeris of Eq (12), and explicitly avoid Ma, the proposed Full Moon phase (Fig 2d).
The lucky prognoses show weak periodicity (Qz = 0.1, nG = 4) with the ephemeris of Eq (11). They give the third largest impact zx = +2.0 on the PA signal (Fig 3a). However, their impact on the PM signal is even larger, zx = +2.9 (Fig 3c). Wedjat may represent Algol observed at its brightest close to the Full Moon. The gi and si distributions of Horus and Wedjat are similar (Figs 1ab and 3ab) with the ephemeris of Eq (11). Wedjat is the Eye of Horus in Ancient Egyptian mythology.
The lucky prognoses have an impact of zx = +1.4 on the PA signal (Fig 4a). This periodicity is weak (Qz = 0.2, nG = 15). Six si reach Qz = 0.01 (Fig 4b). The five si closest to ΘR reach a high significance of QB = 0.003 (n1 = 5, n2 = 18, NS = 105) and may refer to an approaching eclipse of Algol. These si also show a weak connection to the Moon (Fig 4d). It is tempting to suggest that Followers would be Pleiades following very close behind Algol in the revolving sky, e.g. in gi(7, 9) ≡ 88° “The crew follow Horus in the foreign land” (Figs 1a and 4a).
The gi and si reach Qz = 0.06 (nG = 4) and 0.05 (nS = 3) with the ephemeris of Eq (11). The impact of gi on the PA signal is zx = +1.3 (Fig 5a). The three si at Ad, after the proposed eclipse at Ac, are strongly connected to Algol, because they reach the most extreme significance in this study, QB = 0.0004 (n1 = 3, n2 = 6, NS = 105). The texts  are
si(27, 8) ≡ −95°: “Re sets because the Majesty of the goddess Sakhmet is angry in the land of Temhu.”
si(13, 6) ≡ −82°: “It is the day of the proceeding of Sakhmet to Letopolis. Her great executioners passed by the offerings of Letopolis on this day.”
si(7, 10) ≡ −82°: “It is the day of the executioners of Sakhmet.”
These three unlucky prognoses (Fig 5b) are immediately followed by lucky ones (Fig 5a). The gi and si distributions of Sakhmet (Fig 5ab) resemble those of Horus (Fig 1ab) with the ephemeris of Eq (11). The Eye of Horus (Wedjat) was transformed into the vengeful goddess Sakhmet in the legend  of the Destruction of Mankind (hereafter LE2). The si vectors of Horus, Wedjat and Sakhmet point close to Ad which is after Algol’s proposed eclipse at Ac (Figs 1b, 3b and 5b), and may refer to the abrupt pacification of enraged Sakhmet in LE2.
The lucky prognoses show weak periodicity (Fig 6a: Qz = 0.1, nG = 18) and an impact of zx = +1.1 on the PA signal with the ephemeris of Eq (11), as well as some concentration (QB = 0.02, n1 = 12, n2 = 63, NG = 177). Ennead was a group of nine deities in Ancient Egyptian mythology. We discussed earlier, why Followers may have represented Pleiades. Ennead may have been another name for Pleiades, having the modern name “Seven sisters”. However, the number of Pleiades members visible with naked eye depends on the observing conditions and the observer, the maximum number of such members being fourteen [15, 16]. The unlucky prognoses of Followers could be connected to Pleiades following the disappearing Algol before eclipse (Fig 4b), while the unlucky prognoses of Ennead could be connected to Algol reappearing in front Pleiades after eclipse (Fig 6b). Furthermore, the lucky prognosis distributions of Followers and Ennead are very similar (Figs 4a and 6a).
The lucky prognoses show weak periodicity with PA, but their impact on this signal is insignificant, zx = +0.2, with the ephemeris of Eq (11).
These lucky prognoses weaken the PA signal, because their impact is zx = −1.0 with the ephemeris of Eq (11).
The Moon in lucky prognoses
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 PA signal.
The lucky prognoses of Earth, Heaven, Busiris, Rebel, Thoth and Onnophris are connected to the PM signal, because they have zx ≥ 1.0 and Qz ≤ 0.2 with the ephemeris of Eq (12). The lucky prognoses of Nut are weakly connected to the Moon.
These lucky prognoses reach the highest impact parameter value of this study, zx = +5.3, on the PM signal. This periodicity also reaches the highest Rayleigh test significance of all, Qz = 0.001 (nG = 19). The good moments on Earth occurred before and during Ma, the proposed Full Moon phase (Fig 7c). The unlucky prognoses also show a weak connection to Algol (Fig 7b: Qz = 0.06, nS = 5) and an even weaker connection to the Moon (Fig 7d: Qz = 0.2, nS = 5).
The second largest impact zx = +3.4 on the PM signal comes from these lucky prognoses. Again, the good moments coincide with Ma, the proposed Full Moon phase (Fig 8c). This is significant periodicity (Qz = 0.03, nG = 19) combined with a very significant concentration (QB = 0.002, n1 = 12, n2 = 45, NG = 177). The unlucky prognoses also show a weak connection to the Moon (Fig 8d: Qz = 0.06, nS = 4).
The third largest impact on the PM signal, zx = +3.0, comes from the lucky prognoses of Busiris. This periodicity reaches Qz = 0.05 (nG = 4) with the ephemeris of Eq (12). And again, the lucky prognoses are close to Ma, the proposed Full Moon phase (Fig 9c)
The lucky prognoses show weak periodicity (Qz = 0.2, nG = 3) with the ephemeris of Eq (12) and have an impact of zx = 1.6 on the PM signal.
Thoth and Onnophris.
The lucky prognoses of these SW have a weaker impact on the PM signal, i.e. 1.0 ≤ zx ≤ 1.3 with the ephemeris Eq (12).
The lucky prognoses show a weak connection to the Moon. They have no impact on PM, because zx = −0.1 with the ephemeris of Eq (12).
Algol in unlucky prognoses
The PA and PM signals were detected from the lucky prognoses gi[10, 11]. It is therefore self–evident that the unlucky prognoses si had no impact on these two signals. However, this does not rule out the possibility that the si of some SW may be connected to Algol or the Moon. Most of these si vectors point away from Aa or Ma, i.e. zx < 0 with the ephemerides of Eqs (11) or (12). Man and Flame are the only exceptions to this general rule (zx ≥ 0).
The unlucky prognoses have zx = −3.1 with the ephemeris of Eq (11). They point towards Ac, the proposed eclipse phase of Algol (Fig 10b). This periodicity reaches a significance of Qz = 0.04 (nS = 5) and QB = 0.04 (n1 = 5, n2 = 39, NS = 105).
The three unlucky prognoses of this SW reach Qz = 0.06 and a high significance of QB = 0.003 (n1 = 3, n2 = 11, NS = 105) with the ephemeris of Eq (11). They also show a weaker connection to the Moon.
The Moon in unlucky prognoses
We will first discuss the unlucky prognoses of SWs having negative zx values with the ephemeris of Eq (12), and then the two exceptions of Man and Flame.
“See you on the dark side of the Moon” sums up the unlucky prognoses of Seth (Fig 11d). The significance is Qz = 0.05 (nS = 9) with the ephemeris of Eq (12). Leitz  has argued that the following texts  at two consecutive days
si(16, 7) ≡ 173°: “Do not look, darkness being on this day (or, do not see darkness on this day).”
si(17, 7) ≡ 185°: “Do not pronounce the name of Seth on this day.”
take place during the New Moon. The si vectors of these two particular texts point at the opposite sides of Mc ≡ 180°, which supports both our “prediction” formula of Eq (12) and Leitz’ attribution  of the texts to the New Moon. We conclude that Seth is connected to the Moon and strongly suggest that Mc computed with Eq (12) is close to the New Moon. Hence, the Full Moon is close to Ma.
The four unlucky prognoses of this SW also point to the dark side of the Moon, assuming that Mc is close to the New Moon (Fig 12d). The significance estimates are Qz = 0.05 (nS = 4) and QB = 0.02 (n1 = 3, n2 = 15, NS = 105) with the ephemeris of Eq (12).
The significance estimates for the unlucky prognoses are Qz = 0.02 (nS = 6) and QB = 0.009 (n1 = 5, n2 = 23, NS = 105) with the ephemeris Eq (12). These unlucky moments of Man concentrate on a few days after Ma, the proposed Full Moon phase (Fig 13d).
The significance estimates for these unlucky prognoses are Qz = 0.03 (nS = 4) and QB = 0.003 (n1 = 4, n2 = 17, NS = 105) with the ephemeris of Eq (12).
No Algol or the Moon in lucky or unlucky prognoses
Some general remarks
This concludes our analysis of 28 SWs. Numerous other  SWs in CC need to be analysed in the future. Combining the inverse relations of Eqs (3) and (4) to the ephemerides of Eqs (11) and (12) will have countless applications. For example, the first eclipse of Algol would have occurred on t(2.6, 1) = 1.96 at D = 2.1 in M = 1 or the last New Moon on t(14.6, 12) = 343.9 at D = 14.6 in M = 12. Any question about CC can now be studied within this precise framework, e.g. was some meaning given to the nights when an eclipse of Algol (Eq (11): ϕ = 0.5) coincided with the New Moon (Eq (12): ϕ = 0.5)?
Previously, we  applied four tests to the astrophysical hypothesis
This is a summary of those tests:
- test i: The mass transfer in this binary system should have increased the period in the past three millennia. The period value in CC is the first evidence for such an increase since Goodricke  discovered this periodicity over two centuries ago.
- test ii: The period change of 0.017 days from 2.850 to 2.867 days gives a reasonable estimate for the rate of this mass transfer.
- test iii: If eclipses were observed in Ancient Egypt, the orbital plane of the Algol A–B system must be nearly perpendicular to that of the Algol AB–C system [18, 19].
- test iv: Algol and the Moon are the most probable objects, where naked eye observers could have discovered periodicity that we could then rediscover in CC.
tests i and iv supported H1, while tests ii and iii indicated that it could be true.
Algol’s observable night time mid eclipse epochs occur in groups of three separated with a period of 19 days and we also discovered this period in CC . This phenomenon is displayed in Fig 14. First, a mid eclipse epoch occurs in the end of the night. After three days, the next one occurs close to midnight. After another three days, a mid eclipse epoch occurs in the beginning of the night. Then, the next observable night-time mid eclipse epoch occurs after 13 days. Naked eye observations could easily lead to the discovery of this 3 + 3 + 13 days regularity. One could speculate that this is one of the reasons, why the prime number 13 is still considered unlucky. This would be consistent with our result that, if the brightest phases of Algol were considered lucky then the eclipses (i.e. the dimmer phases) were considered unlucky. The 2.85 days period is exactly equal to 57/20 days. This means that after 57 = 3 × 19 days the eclipses returned exactly to the same moment of the night (see Fig 14). All D = 1 days in CC have a prognosis combination “GGG”, while all D = 20 days have “SSS”. Perhaps this regular separation of 19 days was also inspired by Algol.
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 TA1 = 10 hour time intervals of eclipses are denoted with thick continuous or thin dashed lines. The tilted open and closed triangles show the TA2 = 7 and TA3 = 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. Algol’s eclipses last TA1 = 10 hours. Naked eye can detect brightness differences of 0.m1 in ideal observing conditions. Hence, an eclipse detection is theoretically possible for TA2 = 7 hours when Algol is more than 0.m1 dimmer than its brightest suitable comparison star γ And (Fig 14: tilted open triangle limits). This detection could become certain for TA3 = 3 hours when Algol is also at least 0.m1 dimmer than all its other suitable comparison stars ζ Per, ϵ Per, γ Per, δ Per and β Tri (Fig 14: tilted closed triangle limits). During the 57 days eclipse repetition cycle, only two mid eclipse epochs outside the 10 hour night time limits would qualify as certain observable eclipses (Fig 14: open circles at 19th and 48th days). However, a certain detection of these two events would have been very difficult so close to dawn and dusk. The same argument is true for three additional possible eclipse detections (Fig 14: open circles at 11th, 31st and 54th days).
Here, our statistical analysis of SWs giving the largest impact on the PA signal reveals that Algol was represented as Horus. The lucky prognoses were most likely connected to Algol’s brightest phase. Sakhmet may have represented Algol after eclipses, and Wedjat during periods close to the Full Moon. To the Ancient Egyptians, Algol’s cycle may have symbolised the familiar events of LE1 and LE2. At Aa, Re sends the Eye of Horus (Wedjat) to destroy the rebels, as in LE2. At Ab, Horus enters the “foreign land” in gi(7, 9), where he “smote him who rebelled”, as in LE1 or LE2. The “will is written” for him in gi(28, 3) at the beginning of an eclipse—the only gi vector of Horus overlapping the thick line centered at Ac in Fig 1a. After an eclipse, Wedjat returns as Sakhmet who is pacified immediately after Ad, as in LE2. And a new cycle begins.
Followers and Ennead may have represented Pleiades. Thus, these two, together with Horus, Re, Wedjat and Sakhmet, give the largest impact on the PA signal.
The two periods, PA and PM, regulate the assignment of mythological texts to specific days of the year. The Moon strongly regulates the times described as lucky for Heaven and for Earth (Figs 7c and 8c). The unlucky prognoses of Seth are clearly associated with the phases of the Moon (Fig 11d). Other SWs follow PA or PM, like Busiris, Heart, Osiris and Man (Figs 9, 10, 12 and 13). We show no figures for Heliopolis, Enemy, Rebel, Thoth, Onnophris, Nut, Nun, Abydos, Lion and Flame which also reach Qz ≤ 0.2 with PA or PM. All these regularities can not simply be dismissed as a coincidence, let alone with the possible errors of σt ≈ ±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 Algol. In 1596, Fabricius discovered the first variable star, Mira. Holwarda determined its eleven month period 44 years later. In 1669, Montanari discovered the second variable star, Algol. Goodricke  determined the 2.867 days period of Algol in 1783. All these astronomical discoveries were made with naked eye. Since then, they have become milestones of natural sciences. Our statistical analysis of CC confirms that all these milestones should be shifted about three millennia backwards in time.
S1 Fig. Text of Cairo Calendar page rto VIII.
Inside our superimposed rectangle is the hieratic writing for the word Horus. Reprinted from Leitz  under a CC BY license, with permission from Harrassowitz Verlag, original copyright .
S1 Table. Analysis results for all SWs.
Day (D), month (M) of lucky (gi) and unlucky (si) time points, their phase (ϕi), phase angle (Θi), direction of their R vector (ΘR) and differences ΔΘi = Δi − ΘR with Eq (11) for PA = 2.85 days and Eq (12) for PM = 29.6 days. The binomial distribution parameters are n1, n2, qB for QB. Note that the parameters are given in the order of increasing ΔΘi, n1 and n2. All values mentioned in text are marked in bold. We also make available the code of a Python 3.0 program tableS1.py which can be downloaded on Dryad (http://dx.doi.org/10.5061/dryad.tj4qg). This program can be used to reproduce and replicate all analysis results given in S1 Table.
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.
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.
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