Introduction
The Ancient Egyptians referred to celestial events indirectly [1 , 2 , 3 , 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 , 6 , 7 ] . Such connections between astronomical events and prognosis texts have been uncovered in most cases only for individual days [6 , 8 , 9 ] . The P M = 29.6 P_{M} = 29.6 P M = 29.6 Moon has been discovered in CC [10 ] . We have claimed that this document also contains the P A = 2.85 P_{A} = 2.85 P A = 2.85 Algol [11 ] . 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 M M D D D D D D M M M [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 t i t_i t i (2) . The P A P_A P A P M P_M P M [11 ] from six large samples of lucky prognoses (n = 6 × 564 = 3384 n = 6 \times564 = 3384 n = 6 × 564 = 3384 t E t_E t E P A P_A P A P M P_M P M ϕ = 0 \phi= 0 ϕ = 0 t E t_E t E z x z_x z x t i t_i t i t i t_i t i z x > 0 z_x > 0 z x > 0 z x < 0 z_x < 0 z x < 0 P A P_A P A P M P_M P M z x z_x z x ϕ = 0 \phi= 0 ϕ = 0 Algol and the Moon were at their brightest close to phase ϕ = 0 \phi= 0 ϕ = 0 Algol ’s eclipse and the New Moon occurred close to ϕ = 0.5 \phi= 0.5 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.
Materials
In this section, we transform the dates of 28 SWs in the mythological texts of CC into series of time points t i t_i t i
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 [13 ] and in German by Leitz [12 ] . Our SWs have been identified according to the hieroglyphic transcription in Leitz [12 ] and the two aforementioned translations. In case of discrepancy we have consulted the photocopies of the original hieratic text given by Leitz [12 ] . 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 [13 ] and Leitz [12 ] . There is only one exception to our sentence quotation rule, i.e. the CC text connected to Horus where Bakir [13 ] did not identify Horus , but Leitz [12 ] and we did (Algol in lucky prognoses: the text at date g i ( 1 , 10 ) g_i(1, 10) g 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 Table 51 .
Table 51. List of SWs in Ancient Egyptian language.
Table 51.
Table 51.
Table 51.
Table 51.
Table 51.
Table 51.
Table 51.
Table 51.
Table 51.
Table 51.
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 52 . It shows that all our 460 SW date identifications are the same as those made by Leitz [12 ] (Column 5: 460דYes”). However, 21 of our identifications were not made by Bakir [13 ] (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 D = 6 D = 6 M = 1 M = 1 M = 1 n G n_{\mathrm{G}} n G n S n_{\mathrm{S}} n S n G = 3 n_{\mathrm{G}} = 3 n G = 3 n S = 2 n_{\mathrm{S}} = 2 n S = 2
Table 52. SWs identified in CC.
Table 52.
Table 52.
Table 52.
Table 52.
Table 52.
Table 52.
Table 52.
Table 52.
Table 52.
Table 52.
The selected word (SW) identified on day (D D D M M M [12 ] (Ltz=”Yes” or “No”) and by Bakir [13 ] (Bkr=”Yes” or “No”). The twelve first lines of all 460 lines are shown here for guidance regarding the contents of this ASCII file which can be downloaded on Dryad (http://dx.doi.org/10.5061/dryad.tj4qg ).
The dating of CC does not influence the results of our currect analysis, because we transform the time points to unit vectors with equation (6) . The mutual directions between these unit vectors do not depend on the chosen zero epoch t 0 t_0 t 0 (8) and (13) do not depend on the connection between Gregorian and Egyptian days. The only assumption made in our equation (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 equations (2) and (3) of Jetsu et al. [11 ] were
t i = N E − 1 + a i , (1) t_i = N_{\mathrm{E}} - 1 + a_i, \tag*{(1)} t i = N E − 1 + a i , (1) where N E = 30 ( M − 1 ) + D N_{\mathrm{E}} = 30(M - 1) + D N E = 30 ( M − 1 ) + D a i a_i a i a i a_i a i a i a_i a i P A P_{\mathrm{A}} P A P M P_{\mathrm{M}} P M [11 ] . The size of each sample was n = 564 n = 564 n = 564 a i a_i a i N E N_{\mathrm{E}} N E t i t_i t i Table 53 .
Table 53. The time points t i t_i t i [11 ] .
Table 53.
Table 53.
Table 53.
Table 53.
Table 53.
Table 53.
Table 53.
Table 53.
Table 53.
Table 53.
The t i t_i t i Table 53 in Jetsu et al. [11 ] . The five first lines of all 534 lines are shown here for guidance regarding the contents of this ASCII file which can be downloaded on Dryad (http://dx.doi.org/10.5061/dryad.tj4qg).
The mean of the decimal parts a i a_i a i n = 6 × 564 = 3384 n = 6 \times564 = 3384 n = 6 × 564 = 3384 t i t_i t i m t = 0.33 m_t = 0.33 m t = 0.33 D D D M M M
t i = t i ( D , M ) = N E − 1 + m t . (2) t_i = t_i(D,M) = N_{\mathrm{E}} - 1 + m_t. \tag*{(2)} t i = t i ( D , M ) = N E − 1 + m t . (2) This accuracy is sufficient, because we do not know to which part or parts of the day each SW refers to (σ t ≈ ± 0.5 \sigma_t \approx\pm0.5 σ t ≈ ± 0.5 σ t ≈ ± 1.5 \sigma_t \approx\pm1.5 σ t ≈ ± 1.5 t i t_i t i Table 53 (n = 6 × 564 = 3384 n = 6 \times564 = 3384 n = 6 × 564 = 3384 t E t_{\mathrm{E}} t E P A P_{\mathrm{A}} P A P M P_{\mathrm{M}} P M P A P_{\mathrm{A}} P A P M P_{\mathrm{M}} P M t t t
M = INT [ ( t + 1 − m t ) / 30 ] + 1 (3) \begin{aligned}
M &= \operatorname{INT}[(t + 1 - m_t)/30] + 1 \tag*{(3)}
\end{aligned} M = INT [( t + 1 − m t ) /30 ] + 1 (3) D = t − m t + 1 − 30 ( M − 1 ) , (4) \begin{aligned}
D &= t - m_t + 1 - 30(M - 1), \tag*{(4)}
\end{aligned} D = t − m t + 1 − 30 ( M − 1 ) , (4) where INT removes the decimal part of ( t + 1 − m t ) / 30 (t + 1 - m_t)/30 ( t + 1 − m t ) /30 t t t D D D M M M t t t
The time points t i t_i t i Table 54 . These t i t_i t i Q B Q_{\mathrm{B}} Q B
Table 54. The time points t i t_i t i
Table 54.
Table 54.
Table 54.
Table 54.
Table 54.
Table 54.
Table 54.
Table 54.
Table 54.
Table 54.
The day (D D D M M M t i t_i t i N G = 177 N_{\mathrm{G}} = 177 N G = 177 N S = 105 N_{\mathrm{S}} = 105 N S = 105 Table 51 in Jetsu et al. 2013 [11 ] . The ten first lines of all 282 lines are shown here for guidance regarding the contents of this ASCII file which can be downloaded on Dryad (http://dx.doi.org/10.5061/dryad.tj4qg).
Methods
Let us assume that time is a straight line, where events are equidistant dots with a separation of 2 π 2\pi 2 π d = 1 d = 1 d = 1 d ≠ 1 d \ne1 d = 1 P P P P P P
Analysis
If the Rayleigh method discovers the period P P P t = [ t 1 , t 2 , … , t n ] \mathbf{t} = [t_1,t_2,\ldots,t_n] t = [ t 1 , t 2 , … , t n ] t ∗ \mathbf{t}^{*} t ∗ n ∗ n^{*} n ∗ n n n t i t_i t i
ϕ i = FRAC [ ( t i − t 0 ) / P ] , (5) \phi_i = \operatorname{FRAC}[(t_i-t_0)/P], \tag*{(5)} ϕ i = FRAC [( t i − t 0 ) / P ] , (5) where t 0 t_0 t 0 ( t i − t 0 ) / P (t_i-t_0)/P ( t i − t 0 ) / P
r i = [ cos Θ i , sin Θ i ] , (6) \mathbf{r}_i = [\cos\Theta_i,\sin\Theta_i], \tag*{(6)} r i = [ cos Θ i , sin Θ i ] , (6) where Θ i = 360 ∘ ϕ i \Theta_i = 360^\circ\phi_i Θ i = 36 0 ∘ ϕ i
z = ∣ R ∣ 2 / n , (7) z = \lvert\mathbf{R}\rvert^2/n, \tag*{(7)} z = ∣ R ∣ 2 / n , (7) where vector R = ∑ i = 1 n r i \mathbf{R} = \sum_{i=1}^{n}\mathbf{r}_i R = ∑ i = 1 n r i Θ R = atan ( R y / R x ) \Theta_R = \operatorname{atan}(R_y/R_x) Θ R = atan ( R y / R x ) R x = ∑ i = 1 n cos Θ i R_x = \sum_{i=1}^{n}\cos\Theta_i R x = ∑ i = 1 n cos Θ i R y = ∑ i = 1 n sin Θ i R_y = \sum_{i=1}^{n}\sin\Theta_i R y = ∑ i = 1 n sin Θ i ϕ R = Θ R / ( 360 ∘ ) \phi_R = \Theta_R/(360^\circ) ϕ R = Θ R / ( 36 0 ∘ ) Θ i \Theta_i Θ i ∣ R ∣ = n \lvert\mathbf{R}\rvert= n ∣ R ∣ = n Θ i \Theta_i Θ i ∣ R ∣ ≈ 0 \lvert\mathbf{R}\rvert\approx0 ∣ R ∣ ≈ 0
Q z = e − z . (8) Q_z = e^{-z}. \tag*{(8)} Q z = e − z . (8) We use the ephemeris zero epoch
t E = t 0 + P ϕ R . (9) t_E = t_0 + P\phi_R. \tag*{(9)} t E = t 0 + P ϕ R . (9) The mutual directions of r i \mathbf{r}_i r i ∣ R ∣ \lvert\mathbf{R}\rvert ∣ R ∣ m t m_t m t t i t_i t i t 0 t_0 t 0 t E t_E t E t E t_E t E (9) , vector R \mathbf{R} R Θ = Θ R = 0 ∘ \Theta= \Theta_R = 0^\circ Θ = Θ R = 0 ∘ r i \mathbf{r}_i r i − 90 ∘ < Θ i < 90 ∘ -90^\circ< \Theta_i < 90^\circ − 9 0 ∘ < Θ i < 9 0 ∘ P P P r i \mathbf{r}_i r i z = R x 2 / n + R y 2 / n z = R_x^2/n + R_y^2/n z = R x 2 / n + R y 2 / n t 0 = t E t_0 = t_E t 0 = t E (5) and compute the “impact” of any subsample t ∗ \mathbf{t}^{*} t ∗ P P P
z x = ( R x / ∣ R x ∣ ) ( R x 2 / n ) , (10) z_x = (R_x/\lvert R_x\rvert)(R_x^2/n), \tag*{(10)} z x = ( R x / ∣ R x ∣) ( R x 2 / n ) , (10) where R x R_x R x n = n ∗ n = n^{*} n = n ∗ t ∗ \mathbf{t}^{*} t ∗ t ∗ \mathbf{t}^{*} t ∗ z x > 0 z_x > 0 z x > 0 z x < 0 z_x < 0 z x < 0 P P P z x ≈ 0 z_x \approx0 z x ≈ 0
Using the zero epoch t 0 = 0 t_0 = 0 t 0 = 0 n = 6 × 564 n = 6 \times564 n = 6 × 564 t i t_i t i Table 53 gives the t E t_E t E Table 55 for the P A P_A P A P M P_M P M (9) . These six large samples have t E = 0.53 ± 0.09 t_E = 0.53 \pm0.09 t E = 0.53 ± 0.09 P A P_A P A t E = 3.50 ± 0.09 t_E = 3.50 \pm0.09 t E = 3.50 ± 0.09 P M P_M P M
Table 55. Values of t E t_{\mathrm{E}} t E
Table 55.
Table 55.
Table 55.
Table 55.
Table 55.
Table 55.
Table 55.
Table 55.
t 0 = t E = 0.53 , P = P A = 2.85 (Algol) (11) t_0 = t_E = 0.53,\quad P = P_A = 2.85\quad\text{(Algol)} \tag*{(11)} t 0 = t E = 0.53 , P = P A = 2.85 (Algol) (11) t 0 = t E = 3.50 , P = P M = 29.6 (Moon) . (12) t_0 = t_E = 3.50,\quad P = P_M = 29.6\quad\text{(Moon)}. \tag*{(12)} t 0 = t E = 3.50 , P = P M = 29.6 (Moon) . (12) for computing the phases ϕ i \phi_i ϕ i (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 z z z x z_{\mathrm{x}} z x (7) and (8) for any particular SW, if the analysed t i t_i t i Q z ≤ 0.2 Q_z \le0.2 Q z ≤ 0.2 (11) or (12) . These periodicities are called weak if 0.05 < Q z ≤ 0.2 0.05 < Q_z \le0.2 0.05 < Q z ≤ 0.2
In our Figure 1 –Figure 13 , we project the t i t_i t i r i = [ cos Θ i , sin Θ i ] \mathbf{r}_i = [\cos\Theta_i,\sin\Theta_i] r i = [ cos Θ i , sin Θ i ] ϕ = 0 ≡ 0 ∘ \phi= 0 \equiv0^\circ ϕ = 0 ≡ 0 ∘ (11) . The next three points Ab, Ac and Ad are separated by Δ ϕ = 0.25 ≡ 90 ∘ \Delta\phi= 0.25 \equiv90^\circ Δ ϕ = 0.25 ≡ 9 0 ∘ r i \mathbf{r}_i r i ≡ − 90 ∘ \equiv-90^\circ ≡ − 9 0 ∘ ≡ + 90 ∘ \equiv+90^\circ ≡ + 9 0 ∘ z x > 0 z_{\mathrm{x}} > 0 z x > 0 P A P_{\mathrm{A}} P A 57 d / 20 57^{\mathrm{d}}/20 5 7 d /20 ϕ i \phi_i ϕ i t i t_i t i ϕ i \phi_i ϕ i Δ ϕ = 0.005 \Delta\phi= 0.005 Δ ϕ = 0.005 Figure 1 –Figure 13 . However, there are no such shifts in our computations. Our unambiguous terminology is:
Figure 1. Horus. Time runs in the counter clock-wise direction on these unit circles. We give the z z z Q z Q_{\mathrm{z}} Q z z x z_{\mathrm{x}} z x Q z ≤ 0.2 Q_{\mathrm{z}} \le0.2 Q z ≤ 0.2 Θ R \Theta_{\mathrm{R}} Θ R g i \mathbf{g}_{\mathrm{i}} g i ϕ = 0 ≡ 0 ∘ \phi= 0 \equiv0^\circ ϕ = 0 ≡ 0 ∘ ϕ = 0.5 ≡ 180 ∘ \phi= 0.5 \equiv180^\circ ϕ = 0.5 ≡ 18 0 ∘ Algol . (b) s i \mathbf{s}_{\mathrm{i}} s i g i \mathbf{g}_{\mathrm{i}} g i ϕ = 0 ≡ 0 ∘ \phi= 0 \equiv0^\circ ϕ = 0 ≡ 0 ∘ Full Moon phase. (d) s i \mathbf{s}_{\mathrm{i}} s i
Figure 13. Man. otherwise as in Fig 1
“Connected to the PA signal” ≡ t i \equiv t_i ≡ t i P A P_{\mathrm{A}} P A ≡ z x ≥ 1.0 \equiv z_{\mathrm{x}} \ge1.0 ≡ z x ≥ 1.0 Q z ≤ 0.2 Q_z \le0.2 Q z ≤ 0.2 (11) .
“Connected to Algol” ≡ t i \equiv t_i ≡ t i P A P_{\mathrm{A}} P A P A P_{\mathrm{A}} P A 0 ≤ z x < 1.0 0 \le z_{\mathrm{x}} < 1.0 0 ≤ z x < 1.0 z x < 0 ≡ z x < 1.0 z_{\mathrm{x}} < 0 \equiv z_{\mathrm{x}} < 1.0 z x < 0 ≡ z x < 1.0 Q z ≤ 0.2 Q_z \le0.2 Q z ≤ 0.2 (11) .
We use similar terminology for the Moon ((12) ), and Ma–Md points similar to Aa–Ad.
Our notations for the lucky and unlucky time points t i t_i t i g i g_i g i s i s_i s i r i \mathbf{r}_i r i (6) are g i g_i g i s i s_i s i Q z Q_z Q z n G n_G n G n S n_S n S g i g_i g i s i s_i s i g i g_i g i Table 54 : N G = 177 N_G = 177 N G = 177 s i s_i s i Table 54 : N S = 105 N_S = 105 N S = 105 Θ R \Theta_{\mathrm{R}} Θ R R \mathbf{R} R n 1 n_1 n 1 g i g_i g i s i s_i s i n 2 n_2 n 2 N G N_G N G N S N_S N S Θ R \Theta_{\mathrm{R}} Θ R
Q B = P ( n 1 , n 2 , N ) = ∑ i = n 1 n 2 ( n 2 i ) q B i ( 1 − q B ) n 2 − i , (13) Q_{\mathrm{B}} = P(n_1,n_2,N) = \sum_{i=n_1}^{n_2} \binom{n_2}{i} q_{\mathrm{B}}^i(1-q_{\mathrm{B}})^{n_2-i}, \tag*{(13)} Q B = P ( n 1 , n 2 , N ) = i = n 1 ∑ n 2 ( i n 2 ) q B i ( 1 − q B ) n 2 − i , (13) where N = N G N = N_G N = N G N S N_S N S q B = n G / N G q_{\mathrm{B}} = n_G/N_G q B = n G / N G n S / N S n_S/N_S n S / N S Q B Q_{\mathrm{B}} Q B n 1 n_1 n 1 n 2 n_2 n 2 Θ R \Theta_{\mathrm{R}} Θ R Q z Q_z Q z n G n_G n G n S n_S n S Q B Q_{\mathrm{B}} Q B Table 54 : N G = 177 N_G = 177 N G = 177 N S = 105 N_S = 105 N S = 105
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 Figure 1 –Figure 13 . We give four separate tables for each SW. The results for the lucky and unlucky prognoses with P A P_{\mathrm{A}} P A P M P_{\mathrm{M}} P M
Results
Algol in lucky prognoses
Of all 28 SWs, only the lucky prognoses of Horus , Re , Wedjat , Followers , Sakhmet and Ennead unambiguously strengthen the P A P_{\mathrm{A}} P A Algol , because they have an impact of z x ≥ 1.0 z_{\mathrm{x}} \ge1.0 z x ≥ 1.0 Q z ≤ 0.2 Q_{\mathrm{z}} \le0.2 Q z ≤ 0.2 Heliopolis and Enemy are connected to Algol (Q z ≤ 0.2 Q_{\mathrm{z}} \le0.2 Q z ≤ 0.2 P A P_{\mathrm{A}} P A z x < 1.0 z_{\mathrm{x}} < 1.0 z x < 1.0 P A P_{\mathrm{A}} P A z x z_{\mathrm{x}} z x
Horus This SW has the largest impact z x = + 3.5 z_{\mathrm{x}} = +3.5 z x = + 3.5 P A P_{\mathrm{A}} P A Q z = 0.03 Q_{\mathrm{z}} = 0.03 Q z = 0.03 n G = 14 n_{\mathrm{G}} = 14 n G = 14 g i \mathbf{g}_{\mathrm{i}} g i s i \mathbf{s}_{\mathrm{i}} s i ϕ = 0 ≡ 0 ∘ \phi= 0 \equiv0^\circ ϕ = 0 ≡ 0 ∘ Δ ϕ = 0.25 ≡ 90 ∘ \Delta\phi= 0.25 \equiv90^\circ Δ ϕ = 0.25 ≡ 9 0 ∘ g i \mathbf{g}_{\mathrm{i}} g i ≡ − 90 ∘ \equiv-90^\circ ≡ − 9 0 ∘ ≡ + 90 ∘ \equiv+90^\circ ≡ + 9 0 ∘ P A P_{\mathrm{A}} P A g i \mathbf{g}_{\mathrm{i}} g i Θ i \Theta_{\mathrm{i}} Θ i Θ R = 11 ∘ \Theta_{\mathrm{R}} = 11^\circ Θ R = 1 1 ∘ Q B = 0.006 Q_{\mathrm{B}} = 0.006 Q B = 0.006 n 1 = 4 n_1 = 4 n 1 = 4 n 2 = 10 n_2 = 10 n 2 = 10 N G = 177 N_{\mathrm{G}} = 177 N G = 177 g i \mathbf{g}_{\mathrm{i}} g i P A P_{\mathrm{A}} P A [13 ]
g i ( 14 , 2 ) ≡ + 6 ∘ \mathbf{g}_{\mathrm{i}}(14, 2) \equiv+6^\circ g i ( 14 , 2 ) ≡ + 6 ∘
The texts [12 , 13 ] for the next best g i \mathbf{g}_{\mathrm{i}} g i g i ( 19 , 12 ) ≡ + 13 ∘ g_i(19,12) \equiv+13^\circ g i ( 19 , 12 ) ≡ + 1 3 ∘
g i ( 27 , 1 ) ≡ + 19 ∘ g_i(27,1) \equiv+19^\circ g i ( 27 , 1 ) ≡ + 1 9 ∘
g i ( 24 , 3 ) ≡ + 19 ∘ g_i(24,3) \equiv+19^\circ g i ( 24 , 3 ) ≡ + 1 9 ∘
g i ( 1 , 7 ) ≡ + 32 ∘ g_i(1,7) \equiv+32^\circ g i ( 1 , 7 ) ≡ + 3 2 ∘
g i ( 15 , 11 ) ≡ + 38 ∘ g_i(15,11) \equiv+38^\circ g i ( 15 , 11 ) ≡ + 3 8 ∘
g i ( 27 , 3 ) ≡ + 38 ∘ g_i(27,3) \equiv+38^\circ g i ( 27 , 3 ) ≡ + 3 8 ∘
g i ( 18 , 1 ) ≡ − 38 ∘ g_i(18,1) \equiv-38^\circ g i ( 18 , 1 ) ≡ − 3 8 ∘
g i ( 1 , 9 ) ≡ + 51 ∘ g_i(1,9) \equiv+51^\circ g i ( 1 , 9 ) ≡ + 5 1 ∘
g i ( 23 , 7 ) ≡ − 69 ∘ g_i(23,7) \equiv-69^\circ g i ( 23 , 7 ) ≡ − 6 9 ∘
g i ( 29 , 3 ) ≡ − 69 ∘ g_i(29,3) \equiv-69^\circ g i ( 29 , 3 ) ≡ − 6 9 ∘
g i ( 7 , 9 ) ≡ + 88 ∘ g_i(7,9) \equiv+88^\circ g i ( 7 , 9 ) ≡ + 8 8 ∘
g i ( 1 , 10 ) ≡ − 120 ∘ g_i(1,10) \equiv-120^\circ g i ( 1 , 10 ) ≡ − 12 0 ∘
g i ( 28 , 3 ) ≡ + 164 ∘ g_i(28,3) \equiv+164^\circ g i ( 28 , 3 ) ≡ + 16 4 ∘
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 g i g_i g i g i g_i g i χ = + 3.5 \chi= +3.5 χ = + 3.5 g i g_i g i Figure 1 outlines Algol’s eclipse. In this case, the g i ( 7 , 9 ) ≡ + 88 ∘ g_i(7,9) \equiv+88^\circ g i ( 7 , 9 ) ≡ + 8 8 ∘ g i ( 28 , 3 ) ≡ + 164 ∘ g_i(28,3) \equiv+164^\circ g i ( 28 , 3 ) ≡ + 16 4 ∘
Three s i s_i s i fig:1b concentrate close to Ad and reach Q B = 0.07 Q_{\mathrm{B}} = 0.07 Q B = 0.07 n 1 = 3 n_1 = 3 n 1 = 3 n 2 = 25 n_2 = 25 n 2 = 25 N S = 105 N_{\mathrm{S}} = 105 N S = 105 s i s_i s i [13 ] are
s i ( 26 , 1 ) ≡ − 107 ∘ s_i(26,1) \equiv-107^\circ s i ( 26 , 1 ) ≡ − 10 7 ∘
s i ( 11 , 11 ) ≡ − 107 ∘ s_i(11,11) \equiv-107^\circ s i ( 11 , 11 ) ≡ − 10 7 ∘
s i ( 20 , 9 ) ≡ − 69 ∘ s_i(20,9) \equiv-69^\circ s i ( 20 , 9 ) ≡ − 6 9 ∘
s i ( 5 , 8 ) ≡ 6 ∘ s_i(5,8) \equiv6^\circ s i ( 5 , 8 ) ≡ 6 ∘
If the g i g_i g i s i s_i s i s i ( 5 , 8 ) s_i(5,8) s i ( 5 , 8 ) fig:7c ).
The g i g_i g i s i s_i s i Q Z > 0.2 Q_{\mathrm{Z}} > 0.2 Q Z > 0.2 g i g_i g i [12 ] also did, Mc ≡ 180 ∘ \equiv180^\circ ≡ 18 0 ∘ 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 ∘ \equiv-90^\circ ≡ − 9 0 ∘ fig:1c , i.e. g i ( 27 , 1 ) ≡ − 82 ∘ g_i(27,1) \equiv-82^\circ g i ( 27 , 1 ) ≡ − 8 2 ∘ g i ( 27 , 3 ) ≡ − 73 ∘ g_i(27,3) \equiv-73^\circ g i ( 27 , 3 ) ≡ − 7 3 ∘ g i ( 29 , 3 ) ≡ − 48 ∘ g_i(29,3) \equiv-48^\circ g i ( 29 , 3 ) ≡ − 4 8 ∘ 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 [14 ] (hereafter LE1) has inspired these descriptions. The text “White crown to Horus, and the red one to Seth ” in g i ( 29 , 3 ) g_i(29, 3) g i ( 29 , 3 ) Horus with Algol (Figure 1 a: Θ = − 69 ∘ \Theta=-69^\circ Θ = − 6 9 ∘ Seth (Figure 1 c: Θ = − 48 ∘ \Theta=-48^\circ Θ = − 4 8 ∘ 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.
Re The lucky prognoses reach Q z = 0.07 Q_z=0.07 Q z = 0.07 n G = 32 n_G=32 n G = 32 (11) and give the second largest impact z x = + 2.5 z_x=+2.5 z x = + 2.5 P A P_A P A Figure 2 a). Absence of small Q B Q_B Q B g i g_i g i Re (the Sun ) was casually following the undertakings of Horus . The s i s_i s i Re reach Q z = 0.2 Q_z=0.2 Q z = 0.2 n S = 26 n_S=26 n S = 26 (12) , and explicitly avoid Ma, the proposed Full Moon phase (Figure 2 d).
Figure 2. Re . otherwise as in Fig 1
Wedjat The lucky prognoses show weak periodicity (Q z = 0.1 Q_z = 0.1 Q z = 0.1 n G = 4 n_G = 4 n G = 4 (11) . They give the third largest impact z x = + 2.0 z_x = +2.0 z x = + 2.0 P A P_A P A P M P_M P M z x = + 2.9 z_x = +2.9 z x = + 2.9 Wedjat may represent Algol observed at its brightest close to the Full Moon. The g i g_i g i s i s_i s i Horus and Wedjat are similar (Figs fig:1a and fig:3a ) with the ephemeris of equation (11) . Wedjat is the Eye of Horus in Ancient Egyptian mythology.
Figure 3. Wedjat . otherwise as in Fig 1
Followers The lucky prognoses have an impact of z x = + 1.4 z_x = +1.4 z x = + 1.4 P A P_A P A fig:4a ). This periodicity is weak (Q z = 0.2 Q_z = 0.2 Q z = 0.2 n G = 15 n_G = 15 n G = 15 s i s_i s i Q z = 0.01 Q_z = 0.01 Q z = 0.01 fig:4b ). The five s i s_i s i Θ R \Theta_R Θ R Q B = 0.003 Q_B = 0.003 Q B = 0.003 n 1 = 5 n_1 = 5 n 1 = 5 n 2 = 18 n_2 = 18 n 2 = 18 N S = 105 N_S = 105 N S = 105 Algol . These s i s_i s i 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 g i ( 7 , 9 ) ≡ 88 ∘ g_i(7,9) \equiv88^\circ g i ( 7 , 9 ) ≡ 8 8 ∘ The crew follow Horus in the foreign land ” (Figs. 1a and 4a).
Figure 4. Followers . otherwise as in Fig 1
Sakhmet The g i g_i g i s i s_i s i Q Z = 0.06 Q_Z = 0.06 Q Z = 0.06 n G = 4 n_G = 4 n G = 4 0.05 0.05 0.05 n S = 3 n_S = 3 n S = 3 (11) . The impact of g i g_i g i P A P_A P A z x = + 1.3 z_x = +1.3 z x = + 1.3 fig:5a ). The three s i s_i s i Algol , because they reach the most extreme significance in this study, Q B = 0.0004 Q_B = 0.0004 Q B = 0.0004 n 1 = 3 n_1 = 3 n 1 = 3 n 2 = 6 n_2 = 6 n 2 = 6 N S = 105 N_S = 105 N S = 105 [13 ] are
s i ( 27 , 8 ) ≡ − 95 ∘ s_i(27, 8) \equiv-95^{\circ} s i ( 27 , 8 ) ≡ − 9 5 ∘ Re sets because the Majesty of the goddess Sakhmet is angry in the land of Temhu. ”
s i ( 13 , 6 ) ≡ − 82 ∘ s_i(13, 6) \equiv-82^{\circ} s i ( 13 , 6 ) ≡ − 8 2 ∘ It is the day of the proceeding of Sakhmet to Letopolis. Her great executioners passed by the offerings of Letopolis on this day. ”
s i ( 7 , 10 ) ≡ − 82 ∘ s_i(7, 10) \equiv-82^{\circ} s i ( 7 , 10 ) ≡ − 8 2 ∘ It is the day of the executioners of Sakhmet. ”
These three unlucky prognoses (fig:5b ) are immediately followed by lucky ones (fig:5a ). The g i g_i g i s i s_i s i Sakhmet (Figure 5 ) resemble those of Horus (Figure 1 ) with the ephemeris of (11) . The Eye of Horus (Wedjat ) was transformed into the vengeful goddess Sakhmet in the legend [14 ] of the Destruction of Mankind (hereafter LE2). The s i s_i s i Horus , Wedjat and Sakhmet point close to Ad which is after Algol ’s proposed eclipse at Ac (fig:1b , fig:3b and fig:5b ), and may refer to the abrupt pacification of enraged Sakhmet in LE2.
Figure 5. Sakhmet . otherwise as in Fig 1
Ennead The lucky prognoses show weak periodicity (Fig 6a: Q z = 0.1 , n G = 18 Q_{\mathrm{z}} = 0.1,n_{\mathrm{G}} = 18 Q z = 0.1 , n G = 18 z x = + 1.1 z_{\mathrm{x}} = +1.1 z x = + 1.1 P A P_{\mathrm{A}} P A (11) , as well as some concentration (Q B = 0.02 Q_{\mathrm{B}} = 0.02 Q B = 0.02 n 1 = 12 n_{1} = 12 n 1 = 12 n 2 = 63 n_{2} = 63 n 2 = 63 N G = 177 N_{\mathrm{G}} = 177 N G = 177 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).
Figure 6. Ennead. otherwise as in Fig 1
Heliopolis The lucky prognoses show weak periodicity with P A P_{\mathrm{A}} P A z x = + 0.2 z_{\mathrm{x}} = +0.2 z x = + 0.2 (11) .
Enemy These lucky prognoses weaken the P A P_{\mathrm{A}} P A z x = − 1.0 z_{\mathrm{x}} = -1.0 z x = − 1.0 (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 P A P_{\mathrm{A}} P A
The lucky prognoses of Earth , Heaven , Busiris , Rebel , Thoth and Onnophris are connected to the P M P_{\mathrm{M}} P M z x ≥ 1.0 z_{\mathrm{x}} \ge1.0 z x ≥ 1.0 Q z ≤ 0.2 Q_{\mathrm{z}} \le0.2 Q z ≤ 0.2 Nut are weakly connected to the Moon .
Earth These lucky prognoses reach the highest impact parameter value of this study, z x = + 5.3 z_{\mathrm{x}} = +5.3 z x = + 5.3 P M P_{\mathrm{M}} P M Q z = 0.001 Q_{\mathrm{z}} = 0.001 Q z = 0.001 n G = 19 n_{\mathrm{G}} = 19 n G = 19 Earth occurred before and during Ma, the proposed Full Moon phase (Figure 7 c). The unlucky prognoses also show a weak connection to Algol (Figure 7 b: Q z = 0.06 Q_{\mathrm{z}} = 0.06 Q z = 0.06 n S = 5 n_{\mathrm{S}} = 5 n S = 5 Moon (Figure 7 d: Q z = 0.2 Q_{\mathrm{z}} = 0.2 Q z = 0.2 n S = 5 n_{\mathrm{S}} = 5 n S = 5
Figure 7. Earth . otherwise as in Figure 1
Heaven The second largest impact z x = + 3.4 z_x = +3.4 z x = + 3.4 P M P_M P M Moon phase (as shown in fig:8c ). This is significant periodicity (Q z = 0.03 Q_z = 0.03 Q z = 0.03 n G = 19 n_G = 19 n G = 19 Q B = 0.002 Q_B = 0.002 Q B = 0.002 n 1 = 12 n_1 = 12 n 1 = 12 n 2 = 45 n_2 = 45 n 2 = 45 N G = 177 N_G = 177 N G = 177 Moon (fig:8d : Q z = 0.06 Q_z = 0.06 Q z = 0.06 n S = 4 n_S = 4 n S = 4
Figure 8. Heaven. otherwise as in Figure 1
Busiris The third largest impact on the P M P_{\mathrm M} P M z X = + 3.0 z_{\mathrm X}=+3.0 z X = + 3.0 Busiris . This periodicity reaches Q Z = 0.05 Q_{\mathrm Z}=0.05 Q Z = 0.05 n G = 4 n_{\mathrm G}=4 n G = 4 (12) . And again, the lucky prognoses are close to Ma, the proposed Full Moon phase (fig:9c ).
Figure 9. Busiris. otherwise as in Figure 1
Rebel The lucky prognoses show weak periodicity (Q Z = 0.2 Q_{\mathrm Z}=0.2 Q Z = 0.2 n G = 3 n_{\mathrm G}=3 n G = 3 (12) and have an impact of z X = 1.6 z_{\mathrm X}=1.6 z X = 1.6 P M P_{\mathrm M} P M
Thoth and Onnophris The lucky prognoses of these SW have a weaker impact on the P M P_{\mathrm M} P M 1.0 ≤ z X ≤ 1.3 1.0 \le z_{\mathrm X} \le1.3 1.0 ≤ z X ≤ 1.3
Nut The lucky prognoses show a weak connection to the Moon . They have no impact on P M P_{\mathrm M} P M z X = − 0.1 z_{\mathrm X}=-0.1 z X = − 0.1 (12) .
Algol in unlucky prognoses
The P A P_{\mathrm{A}} P A P M P_{\mathrm{M}} P M g i g_i g i [10 , 11 ] . It is therefore self-evident that the unlucky prognoses s i s_i s i s i s_i s i s i s_i s i z x < 0 z_x < 0 z x < 0 (11) or (12) . Man and Flame are the only exceptions to this general rule (z x ≥ 0 z_x \ge0 z x ≥ 0
Heart The unlucky prognoses have z x = − 3.1 z_x = -3.1 z x = − 3.1 (11) . They point towards Ac, the proposed eclipse phase of Algol (fig:10b ). This periodicity reaches a significance of Q z = 0.04 Q_z = 0.04 Q z = 0.04 n S = 5 n_S = 5 n S = 5 Q B = 0.04 Q_B = 0.04 Q B = 0.04 n 1 = 5 n_1 = 5 n 1 = 5 n 2 = 39 n_2 = 39 n 2 = 39 N S = 105 N_S = 105 N S = 105
Figure 10. Heart, otherwise as in Fig 1
Nun The three unlucky prognoses of this SW reach Q z = 0.06 Q_z = 0.06 Q z = 0.06 Q B = 0.003 Q_B = 0.003 Q B = 0.003 n 1 = 3 n_1 = 3 n 1 = 3 n 2 = 11 n_2 = 11 n 2 = 11 N S = 105 N_S = 105 N S = 105 (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 z x z_x z x (12) , and then the two exceptions of Man and Flame .
Seth “See you on the dark side of the Moon” sums up the unlucky prognoses of Seth (Figure 11 d). The significance is Q z = 0.05 Q_z = 0.05 Q z = 0.05 n S = 9 n_S = 9 n S = 9 (12) . Leitz [12 ] has argued that the following texts [13 ] at two consecutive days
Figure 11. Seth. otherwise as in Figure 1
s i ( 16 , 7 ) ≡ 173 ∘ s_i(16,7) \equiv173^\circ s i ( 16 , 7 ) ≡ 17 3 ∘ Do not look, darkness being on this day (or, do not see darkness on this day). ”
s i ( 17 , 7 ) ≡ 185 ∘ s_i(17,7) \equiv185^\circ s i ( 17 , 7 ) ≡ 18 5 ∘ Do not pronounce the name of Seth on this day. ”
take place during the New Moon . The s i s_i s i M c ≡ 180 ∘ M_c \equiv180^\circ M c ≡ 18 0 ∘ (12) and Leitz’ attribution [12 ] of the texts to the New Moon . We conclude that Seth is connected to the Moon and strongly suggest that M c M_c M c (12) is close to the New Moon . Hence, the Full Moon is close to M a M_a M a
Osiris 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 Q Z = 0.05 Q_Z = 0.05 Q Z = 0.05 n S = 4 n_S = 4 n S = 4 Q B = 0.02 Q_B = 0.02 Q B = 0.02 n 1 = 3 n_1 = 3 n 1 = 3 n 2 = 15 n_2 = 15 n 2 = 15 N S = 105 N_S = 105 N S = 105 (12) .
Figure 12. Osiris. otherwise as in Figure 1
Abydos and Lion These unlucky prognoses show a weak connection to the Moon .
Man The significance estimates for the unlucky prognoses are Q Z = 0.02 Q_Z = 0.02 Q Z = 0.02 n S = 6 n_S = 6 n S = 6 Q B = 0.009 Q_B = 0.009 Q B = 0.009 n 1 = 5 n_1 = 5 n 1 = 5 n 2 = 23 n_2 = 23 n 2 = 23 N S = 105 N_S = 105 N S = 105 Man concentrate on a few days after Ma, the proposed Full Moon phase (fig:13d ).
Flame The significance estimates for these unlucky prognoses are Q Z = 0.03 Q_Z = 0.03 Q Z = 0.03 n S = 4 n_S = 4 n S = 4 Q B = 0.003 Q_B = 0.003 Q B = 0.003 n 1 = 4 n_1 = 4 n 1 = 4 n 2 = 17 n_2 = 17 n 2 = 17 N S = 105 N_S = 105 N S = 105
No Algol or the Moon in lucky or unlucky prognoses
Eye, Fire, Majesty, Shu and Sobek These SWs are not connected to Algol or the Moon , because their g i g_i g i s i s_i s i Q Z > 0.2 Q_Z > 0.2 Q Z > 0.2
This concludes our analysis of 28 SWs. Numerous other [7 ] SWs in CC need to be analysed in the future. Combining the inverse relations of equations (3) and (4) to the ephemerides of equations (11) and (12) will have countless applications. For example, the first eclipse of Algol would have occurred on t ( 2.6 , 1 ) = 1.96 t(2.6, 1) = 1.96 t ( 2.6 , 1 ) = 1.96 D = 2.1 D = 2.1 D = 2.1 M = 1 M = 1 M = 1 Moon on t ( 14.6 , 12 ) = 343.9 t(14.6, 12) = 343.9 t ( 14.6 , 12 ) = 343.9 D = 14.6 D = 14.6 D = 14.6 M = 12 M = 12 M = 12 Algol (equation (11): ϕ = 0.5 \phi= 0.5 ϕ = 0.5 Moon (equation (12): ϕ = 0.5 \phi= 0.5 ϕ = 0.5
Discussion
Previously, we [11 ] applied four tests to the astrophysical hypothesis H 1 H_1 H 1 P e r i o d P A = 2.850 Period\ P_A = 2.850 P er i o d P A = 2.850 C C CC C C P o r b P_{\rm orb} P orb
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 [17 ] 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 H 1 H_1 H 1
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 [11 ] . This phenomenon is displayed in Figure 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 Figure 14 ). All D = 1 D = 1 D = 1 D = 20 D = 20 D = 20 Algol .
Figure 14. Eclipses of Algol with P A = 2.85 P_A = 2.85 P A = 2.85 T A 1 = 10 T_{A1} = 10 T A 1 = 10 T A 2 = 7 T_{A2} = 7 T A 2 = 7 T A 3 = 3 T_{A3} = 3 T A 3 = 3
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 T A 1 = 10 T_{A1} = 10 T A 1 = 10 0 . m 1 0.^{\mathrm m}1 0 . m 1 possible for T A 2 = 7 T_{A2} = 7 T A 2 = 7 Algol is more than 0 . m 1 0.^{\mathrm m}1 0 . m 1 γ \gamma γ Figure 14 : tilted open triangle limits). This detection could become certain for T A 3 = 3 T_{A3} = 3 T A 3 = 3 Algol is also at least 0 . m 1 0.^{\mathrm m}1 0 . m 1 ζ \zeta ζ ϵ \epsilon ϵ γ \gamma γ δ \delta δ β \beta β Figure 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 (Figure 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 (Figure 14 : open circles at 11th, 31st and 54th days).
Here, our statistical analysis of SWs giving the largest impact on the P A P_A P A 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 g i ( 7 , 9 ) g_i(7,9) g i ( 7 , 9 ) smote him who rebelled ”, as in LE1 or LE2. The “will is written ” for him in g i ( 28 , 3 ) g_i(28,3) g i ( 28 , 3 ) g i g_i g i 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 P A P_A P A
The two periods, P A P_A P A P M P_M P M 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 P A P_A P A P M P_M P M 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 Q z ≤ 0.2 Q_z \leq0.2 Q z ≤ 0.2 P A P_A P A P M P_M P M σ t ≈ ± 0.5 \sigma_t \approx\pm0.5 σ t ≈ ± 0.5 ± 1.5 \pm1.5 ± 1.5
Conclusions
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 [17 ] 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.
Acknowledgments
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.
Data Availability: The authors confirm that all data underlying these findings are fully available without restriction. Table 52 –Table 54 and the Python 3.0 program tableS1.py have been deposited to Dryad (http://dx.doi.org/10.5061/dryad.tj4qg).
Figure S1. Text of Cairo Calendar page rto VIII. Inside our superimposed rectangle is the hieratic writing for the word Horus . Reprinted from Leitz [12 ] under a CC BY license, with permission from Harrassowitz Verlag, original copyright [1994].
S1 Table
Analysis results for all SWs Day (D D D M M M g i g_i g i s i s_i s i ϕ i \phi_i ϕ i Θ i \Theta_i Θ i Θ R \Theta_R Θ R Δ Θ i = Θ i − Θ R \Delta\Theta_i = \Theta_i - \Theta_R Δ Θ i = Θ i − Θ R P A = 2.85 P_A = 2.85 P A = 2.85 P M = 29.6 P_M = 29.6 P M = 29.6 n 1 n_1 n 1 n 2 n_2 n 2 q B q_B q B Q B Q_B Q B Δ Θ i \Delta\Theta_i Δ Θ i n 1 n_1 n 1 n 2 n_2 n 2 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.
Table S1. Horus: n G = 14 n_G = 14 n G = 14 N G = 177 N_G = 177 N G = 177 q B = 0.0791 q_B = 0.0791 q B = 0.0791 P = 2.85 P = 2.85 P = 2.85 Θ R = 11 \Theta_R = 11 Θ R = 11 z = 3.6 z = 3.6 z = 3.6 Q z = 0.03 Q_z = \mathbf{0.03} Q z = 0.03 z x = 3.5 z_x = \mathbf{3.5} z x = 3.5
Table S1. P = 29.60 P = 29.60 P = 29.60 Θ R = − 75 \Theta_R = -75 Θ R = − 75
Table S1. Horus: n S = 4 n_S = 4 n S = 4 N S = 105 N_S = 105 N S = 105 q B = 0.0381 q_B = 0.0381 q B = 0.0381 P = 2.85 P = 2.85 P = 2.85 Θ R = − 75 \Theta_R = -75 Θ R = − 75 z = 2.0 z = 2.0 z = 2.0 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 0.1 z_x = 0.1 z x = 0.1
Table S1. P = 29.60 P = 29.60 P = 29.60 Θ R = − 153 \Theta_R = -153 Θ R = − 153
Re: n G = 31 n_G = 31 n G = 31 N G = 177 N_G = 177 N G = 177 q B = 0.1751 q_B = 0.1751 q B = 0.1751
P = 2.85 P = 2.85 P = 2.85 Θ R = 23 \Theta_R = 23 Θ R = 23
Periodicity: z = 2.9 z = 2.9 z = 2.9 Q z = 0.05 Q_z = 0.05 Q z = 0.05 z x = 2.5 z_x = 2.5 z x = 2.5
P = 29.60 P = 29.60 P = 29.60 Θ R = − 7 \Theta_R = -7 Θ R = − 7
No periodicity
Re: n S = 26 n_S = 26 n S = 26 N S = 105 N_S = 105 N S = 105 q B = 0.2476 q_B = 0.2476 q B = 0.2476
P = 2.85 P = 2.85 P = 2.85 Θ R = − 124 \Theta_R = -124 Θ R = − 124
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 177 \Theta_R = -177 Θ R = − 177
Weak periodicity: z = 1.7 z = 1.7 z = 1.7 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = − 1.7 z_x = -1.7 z x = − 1.7
Wedjat: n G = 4 n_G = 4 n G = 4 N G = 177 N_G = 177 N G = 177 q B = 0.0226 q_B = 0.0226 q B = 0.0226
P = 2.85 P = 2.85 P = 2.85 Θ R = − 8 \Theta_R = -8 Θ R = − 8
Weak periodicity: z = 2.1 z = 2.1 z = 2.1 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 2.0 z_x = \mathbf{2.0} z x = 2.0
Table 11.
P = 29.60 P = 29.60 P = 29.60 Θ R = 6 \Theta_R = 6 Θ R = 6
Periodicity: z = 2.9 z = 2.9 z = 2.9 Q z = 0.05 Q_z = 0.05 Q z = 0.05 z x = 2.9 z_x = \mathbf{2.9} z x = 2.9
Table 12.
Wedjat: n S = 1 n_S = 1 n S = 1 N S = 105 N_S = 105 N S = 105 q B = 0.0095 q_B = 0.0095 q B = 0.0095
P = 2.85 P = 2.85 P = 2.85 Θ R = − 101 \Theta_R = -101 Θ R = − 101
No periodicity
Table 13.
P = 29.60 P = 29.60 P = 29.60 Θ R = 95 \Theta_R = 95 Θ R = 95
No periodicity
Table 14.
Son: n G = 6 n_G = 6 n G = 6 N G = 177 N_G = 177 N G = 177 q B = 0.0339 q_B = 0.0339 q B = 0.0339
P = 2.85 P = 2.85 P = 2.85 Θ R = 18 \Theta_R = 18 Θ R = 18
Weak periodicity: z = 2.3 z = 2.3 z = 2.3 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 2.0 z_x = \mathbf{2.0} z x = 2.0
Table 15.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 82 \Theta_R = -82 Θ R = − 82
No periodicity
Table 16.
Son: n S = 4 n_S = 4 n S = 4 N S = 105 N_S = 105 N S = 105 q B = 0.0381 q_B = 0.0381 q B = 0.0381
P = 2.85 P = 2.85 P = 2.85 Θ R = 85 \Theta_R = 85 Θ R = 85
No periodicity
Table 17.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 167 \Theta_R = -167 Θ R = − 167
No periodicity
Table 18.
Sakhmet: n G = 3 n_G = 3 n G = 3 N G = 177 N_G = 177 N G = 177 q B = 0.0169 q_B = 0.0169 q B = 0.0169
P = 2.85 P = 2.85 P = 2.85 Θ R = − 33 \Theta_R = -33 Θ R = − 33
Weak periodicity: z = 2.2 z = 2.2 z = 2.2 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 1.6 z_x = 1.6 z x = 1.6
Table 19.
P = 29.60 P = 29.60 P = 29.60 Θ R = 1 \Theta_R = 1 Θ R = 1
No periodicity
Table 20.
Sakhmet: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = − 86 \Theta_R = -86 Θ R = − 86
Periodicity: z = 3.0 z = 3.0 z = 3.0 Q z = 0.05 Q_z = 0.05 Q z = 0.05 z x = 0.0 z_x = 0.0 z x = 0.0
Table 21.
P = 29.60 P = 29.60 P = 29.60 Θ R = 78 \Theta_R = 78 Θ R = 78
No periodicity
Table 22.
Followers: n G = 15 n_G = 15 n G = 15 N G = 177 N_G = 177 N G = 177 q B = 0.0847 q_B = 0.0847 q B = 0.0847
P = 2.85 P = 2.85 P = 2.85 Θ R = 22 \Theta_R = 22 Θ R = 22
Weak periodicity: z = 1.6 z = 1.6 z = 1.6 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = 1.4 z_x = 1.4 z x = 1.4
P = 29.60 P = 29.60 P = 29.60 Θ R = 57 \Theta_R = 57 Θ R = 57
No periodicity
Followers: n S = 6 n_S = 6 n S = 6 N S = 105 N_S = 105 N S = 105 q B = 0.0571 q_B = 0.0571 q B = 0.0571
P = 2.85 P = 2.85 P = 2.85 Θ R = 82 \Theta_R = 82 Θ R = 82
Periodicity: z = 4.3 z = 4.3 z = 4.3 Q z = 0.01 Q_z = 0.01 Q z = 0.01 z x = 0.1 z_x = 0.1 z x = 0.1
P = 29.60 P = 29.60 P = 29.60 Θ R = 134 \Theta_R = 134 Θ R = 134
Weak periodicity: z = 1.7 z = 1.7 z = 1.7 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = − 0.8 z_x = -0.8 z x = − 0.8
Ennead: n G = 18 n_G = 18 n G = 18 N G = 177 N_G = 177 N G = 177 q B = 0.1017 q_B = 0.1017 q B = 0.1017
P = 2.85 P = 2.85 P = 2.85 Θ R = 46 \Theta_R = 46 Θ R = 46
Weak periodicity: z = 2.2 z = 2.2 z = 2.2 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 1.1 z_x = 1.1 z x = 1.1
P = 29.60 P = 29.60 P = 29.60 Θ R = 64 \Theta_R = 64 Θ R = 64
No periodicity
Table 56.
Ennead: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = − 117 \Theta_R = -117 Θ R = − 117
No periodicity
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 171 \Theta_R = -171 Θ R = − 171
No periodicity
Table 58.
Heliopolis: n G = 4 n_G = 4 n G = 4 N G = 177 N_G = 177 N G = 177 q B = 0.0226 q_B = 0.0226 q B = 0.0226
P = 2.85 P = 2.85 P = 2.85 Θ R = − 70 \Theta_R = -70 Θ R = − 70
Weak periodicity: z = 2.1 z = 2.1 z = 2.1 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 0.2 z_x = \mathbf{0.2} z x = 0.2
P = 29.60 P = 29.60 P = 29.60 Θ R = 138 \Theta_R = 138 Θ R = 138
No periodicity
Heliopolis: n S = 4 n_S = 4 n S = 4 N S = 105 N_S = 105 N S = 105 q B = 0.0381 q_B = 0.0381 q B = 0.0381
P = 2.85 P = 2.85 P = 2.85 Θ R = − 168 \Theta_R = -168 Θ R = − 168
Weak periodicity: z = 2.3 z = 2.3 z = 2.3 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = − 2.2 z_x = -2.2 z x = − 2.2
P = 29.60 P = 29.60 P = 29.60 Θ R = 132 \Theta_R = 132 Θ R = 132
No periodicity
Enemy: n G = 6 n_G = 6 n G = 6 N G = 177 N_G = 177 N G = 177 q B = 0.0339 q_B = 0.0339 q B = 0.0339
P = 2.85 P = 2.85 P = 2.85 Θ R = 140 \Theta_R = 140 Θ R = 140
Weak periodicity: z = 1.7 z = 1.7 z = 1.7 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = − 1.0 z_x = -1.0 z x = − 1.0
P = 29.60 P = 29.60 P = 29.60 Θ R = 77 \Theta_R = 77 Θ R = 77
No periodicity
Table 56.
Enemy: n S = 5 n_S = 5 n S = 5 N S = 105 N_S = 105 N S = 105 q B = 0.0476 q_B = 0.0476 q B = 0.0476
P = 2.85 P = 2.85 P = 2.85 Θ R = − 121 \Theta_R = -121 Θ R = − 121
No periodicity
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 146 \Theta_R = -146 Θ R = − 146
No periodicity
Table 58.
Earth: n G = 19 n_G = 19 n G = 19 N G = 177 N_G = 177 N G = 177 q B = 0.1073 q_B = 0.1073 q B = 0.1073
P = 2.85 P = 2.85 P = 2.85 Θ R = − 61 \Theta_R = -61 Θ R = − 61
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 27 \Theta_R = -27 Θ R = − 27
Periodicity: z = 6.6 z = 6.6 z = 6.6 Q z = 0.001 Q_z = 0.001 Q z = 0.001 z x = 5.3 z_x = 5.3 z x = 5.3
Earth: n S = 5 n_S = 5 n S = 5 N S = 105 N_S = 105 N S = 105 q B = 0.0476 q_B = 0.0476 q B = 0.0476
P = 2.85 P = 2.85 P = 2.85 Θ R = − 75 \Theta_R = -75 Θ R = − 75
Weak periodicity: z = 2.9 z = 2.9 z = 2.9 Q z = 0.06 Q_z = 0.06 Q z = 0.06 z x = 0.2 z_x = 0.2 z x = 0.2
P = 29.60 P = 29.60 P = 29.60 Θ R = − 119 \Theta_R = -119 Θ R = − 119
Weak periodicity: z = 1.8 z = 1.8 z = 1.8 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = − 0.4 z_x = -0.4 z x = − 0.4
Heaven: n G = 19 n_G = 19 n G = 19 N G = 177 N_G = 177 N G = 177 q B = 0.1073 q_B = 0.1073 q B = 0.1073
P = 2.85 P = 2.85 P = 2.85 Θ R = 10 \Theta_R = 10 Θ R = 10 P = 29.60 P = 29.60 P = 29.60 Θ R = 0 \Theta_R = 0 Θ R = 0
No periodicity Periodicity: z = 3.4 z = 3.4 z = 3.4 Q z = 0.03 Q_z = 0.03 Q z = 0.03 z x = 3.4 z_x = 3.4 z x = 3.4
Heaven: n S = 4 n_S = 4 n S = 4 N S = 105 N_S = 105 N S = 105 q B = 0.0381 q_B = 0.0381 q B = 0.0381
P = 2.85 P = 2.85 P = 2.85 Θ R = 103 \Theta_R = 103 Θ R = 103 P = 29.60 P = 29.60 P = 29.60 Θ R = − 132 \Theta_R = -132 Θ R = − 132
No periodicity Weak periodicity: z = 2.9 z = 2.9 z = 2.9 Q z = 0.06 Q_z = 0.06 Q z = 0.06 z x = − 1.3 z_x = -1.3 z x = − 1.3
Busiris: n G = 4 n_G = 4 n G = 4 N G = 177 N_G = 177 N G = 177 q B = 0.0226 q_B = 0.0226 q B = 0.0226
P = 2.85 P = 2.85 P = 2.85 Θ R = 19 \Theta_R = 19 Θ R = 19 P = 29.60 P = 29.60 P = 29.60 Θ R = 3 \Theta_R = 3 Θ R = 3
No periodicity Periodicity: z = 3.0 z = 3.0 z = 3.0 Q z = 0.05 Q_z = 0.05 Q z = 0.05 z x = 3.0 z_x = 3.0 z x = 3.0
Table 56.
Busiris: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = − 123 \Theta_R = -123 Θ R = − 123 P = 29.60 P = 29.60 P = 29.60 Θ R = − 111 \Theta_R = -111 Θ R = − 111
No periodicity No periodicity
Table 57.
Table 58.
Rebel: n G = 3 n_G = 3 n G = 3 N G = 177 N_G = 177 N G = 177 q B = 0.0169 q_B = 0.0169 q B = 0.0169
P = 2.85 P = 2.85 P = 2.85 Θ R = 125 \Theta_R = 125 Θ R = 125
No periodicity
Table 31.
P = 29.60 P = 29.60 P = 29.60 Θ R = 11 \Theta_R = 11 Θ R = 11
Weak periodicity: z = 1.6 z = 1.6 z = 1.6 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = 1.6 z_x = 1.6 z x = 1.6
Table 32.
Rebel: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = 107 \Theta_R = 107 Θ R = 107
No periodicity
Table 33.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 162 \Theta_R = -162 Θ R = − 162
No periodicity
Table 34.
Thoth: n G = 10 n_G = 10 n G = 10 N G = 177 N_G = 177 N G = 177 q B = 0.0565 q_B = 0.0565 q B = 0.0565
P = 2.85 P = 2.85 P = 2.85 Θ R = 115 \Theta_R = 115 Θ R = 115
No periodicity
Table 35.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 39 \Theta_R = -39 Θ R = − 39
Weak periodicity: z = 2.1 z = 2.1 z = 2.1 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 1.3 z_x = 1.3 z x = 1.3
Table 36.
Thoth: n S = 2 n_S = 2 n S = 2 N S = 105 N_S = 105 N S = 105 q B = 0.0190 q_B = 0.0190 q B = 0.0190
P = 2.85 P = 2.85 P = 2.85 Θ R = 66 \Theta_R = 66 Θ R = 66
Weak periodicity: z = 1.7 z = 1.7 z = 1.7 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = 0.3 z_x = 0.3 z x = 0.3
Table 37.
P = 29.60 P = 29.60 P = 29.60 Θ R = 149 \Theta_R = 149 Θ R = 149
No periodicity
Table 38.
Onnophris: n G = 7 n_G = 7 n G = 7 N G = 177 N_G = 177 N G = 177 q B = 0.0395 q_B = 0.0395 q B = 0.0395
P = 2.85 P = 2.85 P = 2.85 Θ R = − 138 \Theta_R = -138 Θ R = − 138
No periodicity
Table 39.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 47 \Theta_R = -47 Θ R = − 47
Weak periodicity: z = 2.1 z = 2.1 z = 2.1 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = 1.0 z_x = 1.0 z x = 1.0
Table 40.
Onnophris: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = − 32 \Theta_R = -32 Θ R = − 32
No periodicity
Table 41.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 102 \Theta_R = -102 Θ R = − 102
No periodicity
Table 42.
Nut: n G = 3 n_G = 3 n G = 3 N G = 177 N_G = 177 N G = 177 q B = 0.0169 q_B = 0.0169 q B = 0.0169
P = 2.85 P = 2.85 P = 2.85 Θ R = 55 \Theta_R = 55 Θ R = 55
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 105 \Theta_R = -105 Θ R = − 105
Weak periodicity: z = 1.8 z = 1.8 z = 1.8 Q z = 0.2 Q_z = 0.2 Q z = 0.2 z x = − 0.1 z_x = -0.1 z x = − 0.1
Nut: n S = 4 n_S = 4 n S = 4 N S = 105 N_S = 105 N S = 105 q B = 0.0381 q_B = 0.0381 q B = 0.0381
P = 2.85 P = 2.85 P = 2.85 Θ R = 81 \Theta_R = 81 Θ R = 81
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 139 \Theta_R = 139 Θ R = 139
No periodicity
Heart: n G = 27 n_G = 27 n G = 27 N G = 177 N_G = 177 N G = 177 q B = 0.1525 q_B = 0.1525 q B = 0.1525
P = 2.85 P = 2.85 P = 2.85 Θ R = − 91 \Theta_R = -91 Θ R = − 91
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 47 \Theta_R = 47 Θ R = 47
No periodicity
Table 56.
Heart: n S = 5 n_S = 5 n S = 5 N S = 105 N_S = 105 N S = 105 q B = 0.0476 q_B = 0.0476 q B = 0.0476
P = 2.85 P = 2.85 P = 2.85 Θ R = − 172 \Theta_R = -172 Θ R = − 172
Periodicity: z = 3.2 z = 3.2 z = 3.2 Q z = 0.04 Q_z = 0.04 Q z = 0.04 z x = − 3.1 z_x = -3.1 z x = − 3.1
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = 154 \Theta_R = 154 Θ R = 154
No periodicity
Table 58.
Nun: n G = 7 n_G = 7 n G = 7 N G = 177 N_G = 177 N G = 177 q B = 0.0395 q_B = 0.0395 q B = 0.0395
P = 2.85 P = 2.85 P = 2.85 Θ R = − 38 \Theta_R = -38 Θ R = − 38
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 52 \Theta_R = -52 Θ R = − 52
No periodicity
Nun: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = − 122 \Theta_R = -122 Θ R = − 122
Weak periodicity: z = 2.9 z = 2.9 z = 2.9 Q z = 0.06 Q_z = 0.06 Q z = 0.06 z x = − 0.8 z_x = -0.8 z x = − 0.8
P = 29.60 P = 29.60 P = 29.60 Θ R = − 161 \Theta_R = -161 Θ R = − 161
Weak periodicity: z = 2.4 z = 2.4 z = 2.4 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = − 2.1 z_x = -2.1 z x = − 2.1
Seth: n G = 5 n_G = 5 n G = 5 N G = 177 N_G = 177 N G = 177 q B = 0.0282 q_B = 0.0282 q B = 0.0282
P = 2.85 P = 2.85 P = 2.85 Θ R = 11 \Theta_R = 11 Θ R = 11
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 73 \Theta_R = -73 Θ R = − 73
No periodicity
Table 56.
Seth: n S = 9 n_S = 9 n S = 9 N S = 105 N_S = 105 N S = 105 q B = 0.0857 q_B = 0.0857 q B = 0.0857
P = 2.85 P = 2.85 P = 2.85 Θ R = 57 \Theta_R = 57 Θ R = 57
No periodicity
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 180 \Theta_R = -180 Θ R = − 180
Periodicity: z = 3.1 z = 3.1 z = 3.1 Q z = 0.05 Q_z = 0.05 Q z = 0.05 z x = − 3.1 z_x = -3.1 z x = − 3.1
Table 58.
Table Osiris: $n_G = 8$, $N_G = 177$, $q_B = 0.0452$. P = 2.85 P = 2.85 P = 2.85 Θ R = − 82 \Theta_R = -82 Θ R = − 82
Table Osiris: $n_G = 8$, $N_G = 177$, $q_B = 0.0452$. P = 29.60 P = 29.60 P = 29.60 Θ R = 93 \Theta_R = 93 Θ R = 93
Table Osiris: $n_S = 4$, $N_S = 105$, $q_B = 0.0381$. P = 2.85 P = 2.85 P = 2.85 Θ R = − 141 \Theta_R = -141 Θ R = − 141
Table Osiris: $n_S = 4$, $N_S = 105$, $q_B = 0.0381$. P = 29.60 P = 29.60 P = 29.60 Θ R = 144 \Theta_R = 144 Θ R = 144 z = 3.0 z = 3.0 z = 3.0 Q z = 0.05 Q_z = 0.05 Q z = 0.05 z x = − 2.0 z_x = -2.0 z x = − 2.0
Table Abydos: $n_G = 3$, $N_G = 177$, $q_B = 0.0169$. P = 2.85 P = 2.85 P = 2.85 Θ R = − 14 \Theta_R = -14 Θ R = − 14
Table Abydos: $n_G = 3$, $N_G = 177$, $q_B = 0.0169$. P = 29.60 P = 29.60 P = 29.60 Θ R = 151 \Theta_R = 151 Θ R = 151
Table Abydos: $n_S = 2$, $N_S = 105$, $q_B = 0.0190$. P = 2.85 P = 2.85 P = 2.85 Θ R = 3 \Theta_R = 3 Θ R = 3
Table Abydos: $n_S = 2$, $N_S = 105$, $q_B = 0.0190$. P = 29.60 P = 29.60 P = 29.60 Θ R = 129 \Theta_R = 129 Θ R = 129 z = 1.9 z = 1.9 z = 1.9 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = − 0.8 z_x = -0.8 z x = − 0.8
Table Lion: $n_G = 1$, $N_G = 177$, $q_B = 0.0056$. P = 2.85 P = 2.85 P = 2.85 Θ R = − 177 \Theta_R = -177 Θ R = − 177
Table Lion: $n_G = 1$, $N_G = 177$, $q_B = 0.0056$. P = 29.60 P = 29.60 P = 29.60 Θ R = − 24 \Theta_R = -24 Θ R = − 24
Table Lion: $n_S = 6$, $N_S = 105$, $q_B = 0.0571$. P = 2.85 P = 2.85 P = 2.85 Θ R = − 162 \Theta_R = -162 Θ R = − 162
Table Lion: $n_S = 6$, $N_S = 105$, $q_B = 0.0571$. P = 29.60 P = 29.60 P = 29.60 Θ R = − 100 \Theta_R = -100 Θ R = − 100 z = 2.2 z = 2.2 z = 2.2 Q z = 0.1 Q_z = 0.1 Q z = 0.1 z x = − 0.1 z_x = -0.1 z x = − 0.1
Man: n G = 5 n_G = 5 n G = 5 N G = 177 N_G = 177 N G = 177 q B = 0.0282 q_B = 0.0282 q B = 0.0282
P = 2.85 P = 2.85 P = 2.85 Θ R = − 27 \Theta_R = -27 Θ R = − 27
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 3 \Theta_R = 3 Θ R = 3
No periodicity
Man: n S = 6 n_S = 6 n S = 6 N S = 105 N_S = 105 N S = 105 q B = 0.0571 q_B = 0.0571 q B = 0.0571
P = 2.85 P = 2.85 P = 2.85 Θ R = − 165 \Theta_R = -165 Θ R = − 165
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 54 \Theta_R = 54 Θ R = 54
Periodicity: z = 3.9 z = 3.9 z = 3.9 Q z = 0.02 Q_z = 0.02 Q z = 0.02 z x = 1.4 z_x = 1.4 z x = 1.4
Flame: n S = 4 n_S = 4 n S = 4 N S = 105 N_S = 105 N S = 105 q B = 0.0381 q_B = 0.0381 q B = 0.0381
P = 2.85 P = 2.85 P = 2.85 Θ R = 123 \Theta_R = 123 Θ R = 123
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 85 \Theta_R = 85 Θ R = 85
Periodicity: z = 3.6 z = 3.6 z = 3.6 Q z = 0.03 Q_z = 0.03 Q z = 0.03 z x = 0.0 z_x = 0.0 z x = 0.0
Table 56.
Eye: n G = 4 n_G = 4 n G = 4 N G = 177 N_G = 177 N G = 177 q B = 0.0226 q_B = 0.0226 q B = 0.0226
P = 2.85 P = 2.85 P = 2.85 Θ R = − 24 \Theta_R = -24 Θ R = − 24
No periodicity
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = 115 \Theta_R = 115 Θ R = 115
No periodicity
Table 58.
Eye: n S = 5 n_S = 5 n S = 5 N S = 105 N_S = 105 N S = 105 q B = 0.0476 q_B = 0.0476 q B = 0.0476
P = 2.85 P = 2.85 P = 2.85 Θ R = 81 \Theta_R = 81 Θ R = 81
No periodicity
Table 9.
P = 29.60 P = 29.60 P = 29.60 Θ R = 124 \Theta_R = 124 Θ R = 124
No periodicity
Table 10.
Fire: n G = 4 n_G = 4 n G = 4 N G = 177 N_G = 177 N G = 177 q B = 0.0226 q_B = 0.0226 q B = 0.0226
P = 2.85 P = 2.85 P = 2.85 Θ R = 122 \Theta_R = 122 Θ R = 122
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 4 \Theta_R = 4 Θ R = 4
No periodicity
Fire: n S = 7 n_S = 7 n S = 7 N S = 105 N_S = 105 N S = 105 q B = 0.0667 q_B = 0.0667 q B = 0.0667
P = 2.85 P = 2.85 P = 2.85 Θ R = 171 \Theta_R = 171 Θ R = 171
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 143 \Theta_R = 143 Θ R = 143
No periodicity
Majesty: n G = 13 n_G = 13 n G = 13 N G = 177 N_G = 177 N G = 177 q B = 0.0734 q_B = 0.0734 q B = 0.0734
P = 2.85 P = 2.85 P = 2.85 Θ R = − 64 \Theta_R = -64 Θ R = − 64
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 10 \Theta_R = 10 Θ R = 10
No periodicity
Table 56.
Majesty: n S = 7 n_S = 7 n S = 7 N S = 105 N_S = 105 N S = 105 q B = 0.0667 q_B = 0.0667 q B = 0.0667
P = 2.85 P = 2.85 P = 2.85 Θ R = − 138 \Theta_R = -138 Θ R = − 138
No periodicity
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 169 \Theta_R = -169 Θ R = − 169
No periodicity
Table 58.
Shu: n G = 3 n_G = 3 n G = 3 N G = 177 N_G = 177 N G = 177 q B = 0.0169 q_B = 0.0169 q B = 0.0169
P = 2.85 P = 2.85 P = 2.85 Θ R = 5 \Theta_R = 5 Θ R = 5
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 158 \Theta_R = -158 Θ R = − 158
No periodicity
Shu: n S = 1 n_S = 1 n S = 1 N S = 105 N_S = 105 N S = 105 q B = 0.0095 q_B = 0.0095 q B = 0.0095
P = 2.85 P = 2.85 P = 2.85 Θ R = 13 \Theta_R = 13 Θ R = 13
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = − 99 \Theta_R = -99 Θ R = − 99
No periodicity
Sobek: n G = 1 n_G = 1 n G = 1 N G = 177 N_G = 177 N G = 177 q B = 0.0056 q_B = 0.0056 q B = 0.0056
P = 2.85 P = 2.85 P = 2.85 Θ R = 25 \Theta_R = 25 Θ R = 25
No periodicity
P = 29.60 P = 29.60 P = 29.60 Θ R = 107 \Theta_R = 107 Θ R = 107
No periodicity
Table 56.
Sobek: n S = 3 n_S = 3 n S = 3 N S = 105 N_S = 105 N S = 105 q B = 0.0286 q_B = 0.0286 q B = 0.0286
P = 2.85 P = 2.85 P = 2.85 Θ R = 161 \Theta_R = 161 Θ R = 161
No periodicity
Table 57.
P = 29.60 P = 29.60 P = 29.60 Θ R = − 89 \Theta_R = -89 Θ R = − 89
No periodicity
Table 58.
