first-style=short  165(33,38) 165(47,283) Open Access. © 2018 S. Porceddu et
al., published by De Gruyter. This work is licensed under the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 License

[1]Sebastian Porceddu

# Algol as Horus in the Cairo Calendar: the possible means and the motives of
the observations

Lauri Jetsu  Tapio Markkanen  Joonas Lyytinen  Perttu Kajatkari  Jyri Lehtinen
Jaana Toivari-Viitala

###### Abstract

An ancient Egyptian Calendar of Lucky and Unlucky Days, the Cairo Calendar
(CC), assigns luck with the period of 2.850 days. Previous astronomical,
astrophysical and statistical analyses of CC support the idea that this was
the period of the eclipsing binary Algol three millennia ago. However, next to
nothing is known about who recorded Algol's period into CC and especially how.
Here, we show that the ancient Egyptian scribes had the possible means and the
motives for such astronomical observations. Their principles of describing
celestial phenomena as activity of gods reveal why Algol received the title of
Horus

00footnotetext: Corresponding Author: Sebastian Porceddu: Department of
Physics, University of Helsinki, Finland;
Email<EMAIL_ADDRESS>Lauri Jetsu: Department
of Physics, University of Helsinki, Finland00footnotetext: Tapio Markkanen:
Department of Physics, University of Helsinki, Finland00footnotetext: Joonas
Lyytinen: Department of Physics, University of Helsinki,
Finland00footnotetext: Perttu Kajatkari: Department of Physics, University of
Helsinki, Finland00footnotetext: Jyri Lehtinen: Max-Planck-Institut für
Sonnensystemforschung, Göttingen, Germany00footnotetext: Jaana Toivari-
Viitala: Department of World Cultures, University of Helsinki, Finland

##  1 Introduction

The ancient Egyptian texts known as the Calendars of Lucky and Unlucky Days,
or hemerologies, are literary works that assign prognoses to each day of the
Egyptian year (Wells 2001a, p117-118), (Leitz 1994, p1-2) (Bacs 1990, p41-45)
(Troy 1989, p127-147) and (Helck et al. 1975–1992, p156). These prognoses
denote whether the day, or a part of the day, is considered ``good'' ▲or
``bad'' ▲. 111 We use the symbol “ ▲” to denote the words and phrases
translated into Ancient Egyptian language in the list of Appendix A. Nine such
texts have been found (Troy 1989, p140-143), (Leitz 1994, p-2) and (Porceddu
et al. 2008, p328). Here, we study the best preserved one of these nine texts,
CC, dated to 1271-1163 B.C. (Bakir 1966, p2-5), (Van Walsem 1982, p233) and
(Helck et al. 1975–1992, p156), and published by Abd el-Mohsen Bakir. As in
all our three previous studies (Porceddu et al. 2008; Jetsu et al. 2013; Jetsu
and Porceddu 2015), we use only the best preserved continuous calendar which
is found on pages recto III-XXX and verso I-IX of papyrus Cairo 86637. The
other texts and fragments contained in the same papyrus are ignored from this
analysis because the connection of these fragments to the main calendar is not
apparent and we do not know what year they describe, so combining any data
points from these sources to the dataset created from the long Cairo Calendar
would introduce a random noise component to the analysis. All CC text passages
we quote in this article have been translated by us from the hieroglyphic
transcription of Leitz (1994), assisted by the translations of Bakir (1966, in
English) and Leitz (1994, in German).

The synodic period of the Moon was discovered in CC with a statistical method
called the Rayleigh test, as well as a few other periods (Porceddu et al.
2008, p334). In a footnote of that study, it was noted that one seemingly less
significant period, 2.850 days, was rather close to the current 2.867 days
period of Algol ($\beta$ Persei). This star is a prototype of a class of stars
called eclipsing binaries. The two stars, Algol A and Algol B, orbit around a
common centre of mass with a period of 2.867 days. Algol A is brighter than
Algol B. However, Algol B has a larger radius than Algol A. Our line of sight
nearly coincides with the orbital plane of this double star system. Therefore,
these stars eclipse each other during every orbital round. In a primary
eclipse, the dimmer Algol B partly eclipses the brighter Algol A. This primary
eclipse can be observed with naked eye. In a secondary eclipse, the brighter
Algol A partly eclipses the dimmer Algol B, but the decrease in total
brightness of this binary system is so small that this secondary eclipse event
can not be observed with naked eye. Hence, the brightness of Algol appears to
remain constant for a naked eye observer, except during the primary eclipses.
These primary eclipses last about ten hours. For most of the time, Algol is
brighter than its six close-by bright comparison stars (Jetsu et al. 2013,
their Figure 5a). During a primary eclipse, Algol first becomes dimmer for
five hours and then regains its brightness in another five hours. For a few
hours, Algol appears visibly dimmer than all its six comparison stars. A naked
eye observer can easily notice this as a clear change in Algol's constellation
pattern.

The normalized Rayleigh test of the CC data confirmed the high significance of
the 2.850 days period (Jetsu et al. 2013). The period increase from 2.850 to
2.867 days during the past three millennia gave a mass transfer rate estimate
from Algol B to Algol A. This estimate of Jetsu et al. (2013) agreed with the
one predicted by the best evolutionary model of Algol (Sarna 1993, p540). A
sequence of eight astronomical criteria was also presented which proved that
the ancient Egyptians could have discovered Algol's periodic variability with
naked eyes (Jetsu et al. 2013, p9-10), i.e. it is the star where it is easiest
to discover regular short-term variability without the aid of a telescope.

In the Hellenistic tradition, Algol was called ``the head of Gorgon''. Similar
tradition was continued in the Arabic name ``Demon's Head''. The name Algol is
derived from the Arabic word, head of the Ghoul (ra's al-ghūl) (Davis 1957).
These names seem to indicate that some exotic or foreboding feature or
mutability was known in the folklore of the ancient peoples. All the way to
medieval astrology, the ill omens associated with the ``evil eye'' of Algol
were known, so it is actually surprising that it is so difficult to find any
direct reference to Algol's variability in old astronomical texts (Davis
1957). The list of ill-omened names is so impressive (Allen 1899, p332-333)
that it is unlikely that the variability would have gone undetected through
millennia of practical star observing by the ancient Egyptians.

Of the modern astronomers, Fabricius discovered the first variable star, Mira,
in 1596. The second variable star, Algol, was discovered by Montanari in 1669.
Goodricke (1783) determined the 2.867 days period of Algol in 1783. A close
friend and tutor of John Goodricke, Edward Pigott, also discovered several new
variable stars (Hoskin 1979). In his last paper, Pigott (1805, p152) argued
that the brightness of Algol must have been constant in Antiquity, because the
variability that he observed was so easy to notice with naked eyes. Kopal
(1946, p3) suggested that those ancient discoveries ``may have been buried in
the ashes of the Library of Alexandria''. More recently, Wilk (2000) has
presented the theory that classical mythology contains knowledge of the
variability of various stars, including Algol. This star also seems to belong
to the constellation called ``Elk'' by the Siberian shamans of the Khanty
tribe, who have noticed that this animal sometimes loses one pair of legs
(Pentikäinen 1997, p58-65).

A statistical analysis of 28 selected words (hereafter SWs) of the
mythological narratives of CC was performed to find traces of the Egyptians'
symbolism for Algol (Jetsu and Porceddu 2015). We notate the SWs of that
particular study for example ``Horus'' or ``Seth'' to distinguish them from
other Egyptian deities such as Isis and Nephthys. Out of all 28 SWs, the word
``Horus'' had the strongest connection to the 2.850 days periodicity (Jetsu
and Porceddu 2015). ``Horus'', etymologically ``the distant one'', was one of
the earliest attested Egyptian deities. Predominantly a sky god or stellar
god, the living king was identified as an earthly ``Horus'' (Roeder 1994,
p42-43) and (Meltzer 2001, p119-122). Horus is described as a star in the
oldest ancient Egyptian texts (Krauss 2016). Another deity, ``Seth'', the
adversary of ``Horus'', was shown to be connected to the period of the Moon
(Jetsu and Porceddu 2015).

Statistical analyses have confirmed the ancient Egyptian discovery of Algol's
period (Porceddu et al. 2008; Jetsu et al. 2013; Jetsu and Porceddu 2015).
Here, our aim is to connect this astonishing ancient discovery to its
contemporary cultural and historical background by presenting ten general
arguments about CC (Sects 4.1-4.10). These arguments strongly support the idea
that the ancient Egyptian scribes had the possible means and the motives to
record Algol's period into CC. The connection of CC mythological texts to the
perceived behaviour of the Moon and Algol is verified in Sects 4.7 and 4.8.

##  2 Materials and Methods

###  2.1 Materials

We use statistical methods to discover the principles of describing celestial
phenomena in CC, thus no other Egyptian texts are used as material in the core
analysis. We begin with a general description of CC.

This document is one of the texts known as Calendars of Lucky and Unlucky
Days. In these Calendars the days of the year are assigned good and bad
prognoses. Nine full and partial Calendars of Lucky and Unlucky Days have been
discovered (Leitz 1994, p1-2), (Troy 1989, p140-143) and (Helck et al.
1975–1992, p156). Eight of them date to the New Kingdom, ca. 1550-1069 B.C.,
while one of them is from the Middle Kingdom, ca. 2030-1640 B.C. Papyrus Cairo
86637, the source of CC, was originally dated to the ninth regnal year of
Ramses II (Brunner-Traut 1970), around 1271-1270 B.C. according to the
generally accepted chronology (Shaw 2000, p480-490) which has been disputed
(Huber 2011). However, the date is nowhere to be explicitly found (Leitz 1994,
p1-2). Van Walsem (1982, p233) revised the date of the papyrus to the early
20th dynasty, around 1185-1176 B.C. We have also checked the paleographical
correspondences of plentifully recurring signs, such as F35, G17, N5, O1 and
R8 (Wimmer 1995, p118,129,194,246,274), and these seem to support the
conclusion of dating the manuscript to the latter half of the 19th dynasty or
the beginning of the 20th, i.e. 1244-1163 B.C. A compromise date 1224 B.C. was
used in the astrophysical and astronomical computations (Jetsu et al. 2013,
p1), as well as in the SW analysis (Jetsu and Porceddu 2015, p1). The results
of both of those studies did not depend on the exact dating of CC.

CC is a calendar for the entire year. We use the daily prognoses of CC
published in Table 1 of Jetsu et al. (2013), where the German notations by
Leitz (1994, p480-482) were used (G=Gut= ``good'', S=Schlecht=``bad''). CC is
based on the Civil Calendar of 12 months of 30 days each plus five additional
epagomenal days for which no prognoses are given. The months were arranged
into three seasons of four months each. These seasons were Akhet (flood
season) ▲, Peret (winter season) ▲and Shemu (harvest season) ▲. The
conventionally given format for a calendar date is for example I Akhet 27 for
the 27th day of the first month of the Akhet season.

The CC texts systematically give a date, inscribed in red colour, and then
three prognoses for that date (FAQ 1) 222Some frequently asked questions (FAQ)
about our research have been collected into Appendix B, where we give short
answers those questions, as well as indicate the sections of this manuscript
where the more detailed answers can be found.. For example, the GGG prognosis
combination for the date I Akhet 27 means that all the three parts of the day
are lucky. This fully positive prognosis is the most common for any day. Kemp
and Rose (1991) noted that the ratio of good and bad prognoses in CC is close
to the value of the so-called Golden Section, in accordance with modern
psychological experiments regarding positive and negative judgements.

Generally speaking, on SSS days people were under a special threat to suffer
from hunger, thirst and various illnesses. The prognoses of such days were
attributed to mostly negative mythological events and children born on such a
date might have been foretold to die of illness. On the other hand, those born
on GGG days would live a long life. Such days were in general supposed to
consist of joy, success, freedom, health and various feasts. While on SSS days
some restrictions were suggested on journeying and consumption of foods, on
GGG days it was recommended to give offerings and feasts to the gods (Troy
1989, p138).

In the longer and better preserved texts, especially in CC, there are
descriptions of mythological events relating to the date and also some
instructions on suggested behaviour during the day (FAQ 1). For example,
regarding the day I Akhet 27 the description in CC, page recto VIII, reads
that the god ``Horus'' and his enemy ``Seth'' are resting from their perpetual
struggle. It is recommended not to kill any ``snakes'' ▲during the day. The
practical influence of the Calendars of Lucky and Unlucky Days on the life of
ancient Egyptians is not exactly known. The various instructions and
restrictions such as ``make offering to the gods of your city'' (Leitz 1994,
p82) or ``do not go out of your house to any road on this day'' (Leitz 1994,
p238) seem to be presented in the context of the everyday life of a worker. It
was suggested that the Calendars of Lucky and Unlucky Days would have
determined the rest days for the workers (Helck et al. 1975–1992, p153-155),
but no correlation of the Lucky and Unlucky Days was found with days of kings'
ascensions to throne, official building works, battles, journeys, court trials
or working days, except when the day was also a regular feast date (Drenkhahn
1972, p87-94). In CC, the prognosis of the first day of each month is always
GGG and the day is called ``feast'' ▲. On the other hand, the prognosis of the
20th day of each month is always SSS.

In most cases, the prognosis is homogeneous for the whole day (i.e. GGG or
SSS). There are only 29 heterogeneous prognoses in CC. These days provide a
glimpse into the logic behind the day division. Generally speaking,
arrangement into morning, mid-day and evening is obvious, but these can be
defined in multiple ways. For example, the prognosis for the date I Akhet 8 in
CC is GGS. The text advises one not to go out during the ``night'' ▲. The
prognosis for I Akhet 25 is also GGS but the text advises one not to go out
during the ``evening'' ▲. Thus it remains unclear if the third part of the day
comprises night hours as well. Jetsu et al. (2013, p2-7) showed that the
period analysis results for CC did not depend on how the three prognoses were
distributed within each day.

The practice of assigning good and bad omens to days of the year seems rather
close to astrology and reading predictions from the stars, and indeed the
Calendar of Lucky and Unlucky Days was mixed with Babylonian based astrology
in the Greek and Roman times (Leitz and Thissen 1995, p38-55). But it is to be
noted that celestial matters did not fully determine the prognoses in the
Calendar of Lucky and Unlucky days, but played a part in it alongside natural
cycles such as the floods of the Nile, or the seasonal dangers presented by
winds, wild animals and illnesses. There is also plenty of evidence for
various kinds of ritual recurrence, such as that III Akhet 26 is described the
strengthening of ``the djed-pillar'' ▲ (Helck et al. 1975–1992, a ritual
object whose raising is connected to the myth of the resurrection of the god
Osiris who was killed by Seth) and II Peret 6 is described the erection of
``the djed-pillar'', with a separation of exactly 70 days, the approximate
interval between Sirius' heliacal setting and Sirius' heliacal rising. It was
considered the ideal duration of funerary ceremonies because the star was
believed to spend this time in the underworld, undergoing rituals of
purification. CC also makes explicit references to the heliacal rising,
culmination and heliacal setting of certain hour-stars (Hardy 2003). All
prognoses based on this type of aperiodic events induce statistical noise into
CC. When applying period analysis to the CC data, this noise interferes with
the detection of any periodic signal.

When searching for regular periodic astronomical phenomena in CC, one should
realize that only a few events relating to celestial objects, however
important they were considered to be, could have determined an extensive and
significant set of periodic prognoses. For example, the heliacal rising of a
star is a yearly event, and may affect the prognosis of one day. Thus, it can
not be discovered from the calendars by period analysis. The synodic periods
of planets, because of their length, are also out of the question (Jetsu et
al. 2013, p13). Except for the Moon, the only other detectable astronomical
periods in CC are those of the regular brightness changes of a variable star
(Jetsu et al. 2013, p9).

###  2.2 Methods

Table 1: ``GGG'' prognosis texts mentioning Horus, Wedjat or Sakhmet. The columns are SW (Selected word), ancient Egyptian month (``Month''), day ($D$), numerical month value ($M$), time point ($g(D,M)$) and the phase angles ($\Theta_{\mathrm{Algol}}$ and $\Theta_{\mathrm{Moon}}$). All values are in the order of increasing $\Theta_{\mathrm{Algol}}$, because this allows an easy comparison with the results shown in List 1 and Figure 1. SW |  Month |  $D$ |  $M$ |  $g(D,M)$ |  $\Theta_{\mathrm{Algol}}$ |  $\Theta_{\mathrm{Moon}}$
---|---|---|---|---|---|---
Horus |  II Akhet |  14 |  2 |  43.33 |  6 |  124
Wedjat |  I Peret |  1 |  5 |  120.33 |  13 |  341
Sakhmet |  I Peret |  1 |  5 |  120.33 |  13 |  341
Horus |  IV Shemu |  19 |  12 |  348.33 |  13 |  234
Horus |  I Akhet |  27 |  1 |  26.33 |  19 |  278
Horus |  III Akhet |  24 |  3 |  83.33 |  19 |  251
Horus |  III Peret |  1 |  7 |  180.33 |  32 |  351
Horus |  III Akhet |  27 |  3 |  86.33 |  38 |  287
Horus |  III Shemu |  15 |  11 |  314.33 |  38 |  180
Horus |  I Shemu |  1 |  9 |  240.33 |  51 |  0
Wedjat |  II Akhet |  3 |  2 |  32.33 |  57 |  351
Horus |  I Shemu |  7 |  9 |  246.33 |  88 |  73
Horus |  III Akhet |  28 |  3 |  87.33 |  164 |  300
Horus |  II Shemu |  1 |  10 |  270.33 |  240 |  5
Sakhmet |  IV Akhet |  16 |  4 |  105.33 |  278 |  158
Horus |  III Peret |  23 |  7 |  202.33 |  291 |  258
Horus |  III Akhet |  29 |  3 |  88.33 |  291 |  312
Sakhmet |  I Peret |  9 |  5 |  128.33 |  303 |  78
Wedjat |  II Shemu |  30 |  10 |  299.33 |  303 |  358
Sakhmet |  I Peret |  29 |  5 |  148.33 |  309 |  321
Horus |  I Akhet |  18 |  1 |  17.33 |  322 |  168
Wedjat |  I Shemu |  6 |  9 |  245.33 |  322 |  61
Table 2:  ``SSS'' prognosis texts mentioning Horus, Wedjat or Sakhmet. Notations are as in Table 1. SW |  Month |  $D$ |  $M$ |  $s(D,M)$ |  $\Theta_{\mathrm{Algol}}$ |  $\Theta_{\mathrm{Moon}}$
---|---|---|---|---|---|---
Horus |  IV Peret |  5 |  8 |  214.33 |  6 |  44
Horus |  I Akhet |  26 |  1 |  25.33 |  253 |  265
Horus |  III Shemu |  11 |  11 |  310.33 |  253 |  132
Wedjat |  II Peret |  10 |  6 |  159.33 |  259 |  95
Sakhmet |  IV Peret |  27 |  8 |  236.33 |  265 |  312
Sakhmet |  II Peret |  13 |  6 |  162.33 |  278 |  132
Sakhmet |  II Shemu |  7 |  10 |  276.33 |  278 |  78
Horus |  I Shemu |  20 |  9 |  259.33 |  291 |  231
Table 3: ``GGG'' prognosis texts mentioning Horus, Seth or Osiris. Notations are as in Table 1, except that all values are in the order of increasing $\Theta_{\mathrm{Moon}}$. SW |  Month |  $D$ |  $M$ |  $g(D,M)$ |  $\Theta_{\mathrm{Algol}}$ |  $\Theta_{\mathrm{Moon}}$
---|---|---|---|---|---|---
Horus |  I Shemu |  1 |  9 |  240.33 |  51 |  0
Horus |  II Shemu |  1 |  10 |  270.33 |  240 |  5
Osiris |  II Shemu |  1 |  10 |  270.33 |  240 |  5
Osiris |  III Peret |  6 |  7 |  185.33 |  303 |  51
Seth |  IV Akhet |  9 |  4 |  98.33 |  114 |  73
Horus |  I Shemu |  7 |  9 |  246.33 |  88 |  73
Osiris |  IV Akhet |  11 |  4 |  100.33 |  6 |  98
Horus |  II Akhet |  14 |  2 |  43.33 |  6 |  124
Osiris |  II Akhet |  16 |  2 |  45.33 |  259 |  149
Seth |  IV Shemu |  13 |  12 |  342.33 |  335 |  161
Osiris |  IV Shemu |  13 |  12 |  342.33 |  335 |  161
Horus |  I Akhet |  18 |  1 |  17.33 |  322 |  168
Horus |  III Shemu |  15 |  11 |  314.33 |  38 |  180
Osiris |  II Peret |  17 |  6 |  166.33 |  63 |  180
Horus |  IV Shemu |  19 |  12 |  348.33 |  13 |  234
Horus |  III Akhet |  24 |  3 |  83.33 |  19 |  251
Horus |  III Peret |  23 |  7 |  202.33 |  291 |  258
Horus |  I Akhet |  27 |  1 |  26.33 |  19 |  278
Seth |  I Akhet |  27 |  1 |  26.33 |  19 |  278
Horus |  III Akhet |  27 |  3 |  86.33 |  38 |  287
Seth |  III Akhet |  27 |  3 |  86.33 |  38 |  287
Horus |  III Akhet |  28 |  3 |  87.33 |  164 |  300
Osiris |  III Akhet |  28 |  3 |  87.33 |  164 |  300
Horus |  III Akhet |  29 |  3 |  88.33 |  291 |  312
Seth |  III Akhet |  29 |  3 |  88.33 |  291 |  312
Osiris |  III Peret |  28 |  7 |  207.33 |  202 |  319
Horus |  III Peret |  1 |  7 |  180.33 |  32 |  351

We relate CC texts to astronomical events by the phase angles calculated from
the days that the texts refer to (Eqs. 1-4). We select four samples from CC
(Tables 1-4) which give us two lists of CC text passages (Lists 1 and 2).

Recently, a statistical study was made of the occurrence of 28 different SWs
in the CC prognosis texts (Jetsu and Porceddu 2015, p3). The occurrences of
individual SWs were studied separately. The lucky prognosis texts mentioning
``Horus'' were studied in greater detail, and a few unlucky texts mentioning
``Sakhmet'' or ``Seth''. Those texts were taken as such from the CC
translation of Bakir (1966).

We downloaded these SW data (Jetsu and Porceddu 2015, p1) from the Dryad
database333http://dx.doi.org/10.5061/dryad.tj4qg, where the respective ASCII
file-name is data2.txt. This gave us the dates when any particular SW is
mentioned in CC. Here, we concentrate on the following five particular SWs:
``Horus'', ``Wedjat'', ``Sakhmet'', ``Seth'' and ``Osiris''. These five
deities are the most relevant ones regarding the two prominent myths ``The
Destruction of Mankind'' and ``The Contendings of Horus and Seth'' that will
be described in Sect. 4.7.

``Horus'', etymologically the distant one, was a sky god or stellar god
associated with kingship and order. Krauss (2016, p137-141) suggests that
Horus was originally a stellar god who later became subordinated to solar
mythology. Already the earliest texts regarding Horus describe him as the
``Foremost star of the sky''. In the Pyramid Texts, a younger Horus is called
Horus-son-of-Isis and is distinguished from the elder Horus (Haroeris).
According to Krauss these would be Venus as the morning star and the evening
star. On one hand the connection of the planet Venus, usually considered
feminine, with the king would be unique to Egypt. Yet Venus was certainly
associated with Benu, the divinity who in the creation myth laid the first
stone benben, which became Earth, upon the primal sea. Horus' rival god Seth
was the embodiment of disorder, identified in some sources with the planet
Mercury, the other inner planet besides Venus. In the most commonly known
mythological narrative Osiris, the father of Horus, was killed by Seth. Horus
avenged his father and the resurrected Osiris was to be considered the patron
of the dead, especially the dead king (Meltzer 2001, p119-120). Sakhmet was a
lion goddess associated with the scorching, destructive power of the Sun, also
called Wadjet or the Eye of Horus (Leitz and Budde 2003, p361) and (Jong 2001,
p512-513).

While

Table 4: ``SSS'' prognosis texts mentioning Horus, Seth or Osiris. Notations are as in Table 3. SW |  Month |  $D$ |  $M$ |  $s(D,M)$ |  $\Theta_{\mathrm{Algol}}$ |  $\Theta_{\mathrm{Moon}}$
---|---|---|---|---|---|---
Horus |  IV Peret |  5 |  8 |  214.33 |  6 |  44
Seth |  II Akhet |  12 |  2 |  41.33 |  114 |  100
Seth |  III Akhet |  13 |  3 |  72.33 |  69 |  117
Osiris |  III Akhet |  13 |  3 |  72.33 |  69 |  117
Osiris |  III Akhet |  14 |  3 |  73.33 |  196 |  129
Horus |  III Shemu |  11 |  11 |  310.33 |  253 |  132
Seth |  IV Shemu |  11 |  12 |  340.33 |  82 |  137
Osiris |  I Peret |  14 |  5 |  133.33 |  215 |  139
Seth |  III Akhet |  18 |  3 |  77.33 |  341 |  178
Seth |  III Peret |  17 |  7 |  196.33 |  253 |  185
Seth |  IV Peret |  17 |  8 |  226.33 |  82 |  190
Osiris |  IV Akhet |  19 |  4 |  108.33 |  297 |  195
Seth |  II Akhet |  20 |  2 |  49.33 |  44 |  197
Horus |  I Shemu |  20 |  9 |  259.33 |  291 |  231
Horus |  I Akhet |  26 |  1 |  25.33 |  253 |  266
Seth |  I Akhet |  26 |  1 |  25.33 |  253 |  266
Seth |  IV Peret |  24 |  8 |  233.33 |  246 |  275

Jetsu and Porceddu (2015) studied some texts mentioning ``Horus'', ``Sakhmet''
or ``Seth'', and used the text from the CC translation of Bakir (1966), we
study all prognosis texts mentioning any of the above mentioned five deities,
and we use our own translations of these CC texts.

We calculate the ``Egyptian days'' for these SWs from

$\displaystyle N_{\mathrm{E}}=30(M-1)+D,$ (1)

where $M$ is the month and $D$ is the day of the date in CC (Jetsu et al.
2013, Table 1). The SW dates are transformed (Jetsu and Porceddu 2015) into
time points with the relation

$\displaystyle t=t(D,M)=N_{\mathrm{E}}-1+0.33.$ (2)

We use the notations $g=g(D,M)=t(D,M)$ and $s=s(D,M)=t(D,M)$ for the time
points of GGG and SSS prognosis days, because these two prognosis samples were
analysed and studied separately (Jetsu et al. 2013; Jetsu and Porceddu 2015).

For any period value $P$, the phases of $t$ are

$\displaystyle\phi=\mathrm{FRAC}[(t-t_{0})/P],$ (3)

where $\mathrm{FRAC}$ removes the integer part of $(t-t_{0})/P$ and $t_{0}$ is
the zero epoch. In other words, $\mathrm{FRAC}$ removes the number of full $P$
rounds completed after the zero epoch $t_{0}$. The phase angles are

$\displaystyle\Theta=360^{\mathrm{o}}\phi.$ (4)

Jetsu et al. (2013) discovered two significant periods,
$P_{\mathrm{Algol}}=2.850$ and $P_{\mathrm{Moon}}=29.6$ days, in the lucky
prognoses of CC. In their next study, they determined these two ephemerides
(Jetsu and Porceddu 2015) for the phases of Eq. 3

$\displaystyle t_{0}=0.53,$ $\displaystyle P=P_{\mathrm{Algol}}=2.850$ (5)
$\displaystyle t_{0}=3.50,$ $\displaystyle P=P_{\mathrm{Moon}}=29.6$ (6)

The phase angles $\Theta$ (Eq. 4) computed with the ephemerides of Eqs. 5 and
6 are hereafter denoted with $\Theta_{\mathrm{Algol}}$ and
$\Theta_{\mathrm{Moon}}$, respectively. We also use their eight abbreviations
(Jetsu and Porceddu 2015, p6-7)

  * Aa

$\equiv\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$ = Mid-epoch of Algol's
secondary eclipse

  * Ab

$\equiv\Theta_{\mathrm{Algol}}=90^{\mathrm{o}}$ = Between mid-epochs of
secondary and primary eclipse

  * Ac

$\equiv\Theta_{\mathrm{Algol}}=180^{\mathrm{o}}$ = Mid-epoch of Algol's
primary eclipse

  * Ad

$\equiv\Theta_{\mathrm{Algol}}=270^{\mathrm{o}}$ = Between mid-epochs of
primary and secondary eclipse

  * Ma

$\equiv\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}$ = Full Moon

  * Mb

$\equiv\Theta_{\mathrm{Moon}}=90^{\mathrm{o}}$ = Between Full and New Moon

  * Mc

$\equiv\Theta_{\mathrm{Moon}}=180^{\mathrm{o}}$ = New Moon

  * Md

$\equiv\Theta_{\mathrm{Moon}}=270^{\mathrm{o}}$ = Between New and Full Moon

All $D$, $M$, $g(D,M)$, $s(D,M)$, $\Theta_{\mathrm{Moon}}$ and
$\Theta_{\mathrm{Algol}}$ values of ``Horus'', ``Wedjat'', ``Sakhmet'',
``Seth'' and ``Osiris'' are given in Tables 1-4.

We study all CC passages mentioning ``Horus'', ``Wedjat'', ``Sakhmet'',
``Seth'' and ``Osiris''. These passages give the date first, inscribed in red
colour. Then follow the daily prognoses, and the descriptive prognosis text.
The time point for every date is unambiguous, because the structure of CC is
regular, $12\times 30$ days (Eqs. 1 and 2). Hence, the time points for the
prognoses, the SWs and the prognosis texts describing the actions of deities
are also unambiguous. 444 The calendar dates are fixed and known even if the
texts or the deities mentioned in these texts were not studied at all. The
deities mentioned in these texts do not determine the dates (FAQ 1). The exact
dating of the CC, as a historical document, is irrelevant in the current
analysis, like it also was in the previous statistical studies (Jetsu et al.
2013; Jetsu and Porceddu 2015). Adding any arbitrary constant to the time
points of Eq. 2 shifts all phase angles $\Theta$ with the same amount, e.g.
the $\Theta_{\mathrm{Algol}}$ values of all passages of List 1. This is the
reason why our results based on Lists 1 and 2 do not depend on such shifts
(FAQ 2).

The number of passages mentioning some SWs is very small (e.g. $n=3$ for
``Sakhmet'' in Table 2). It is therefore necessary to explain how can we draw
reliable statistical conclusions from the analysis of such data (FAQ 3).
Firstly, the periods $P_{\mathrm{Algol}}$ and $P_{\mathrm{Moon}}$ were
detected from large samples of over five hundred time points and these
periodicities were extremely significant (Jetsu et al. 2013). For example, the
period $P_{\mathrm{Algol}}$ reached critical levels $Q^{*}<0.0001$ (Jetsu et
al. 2013, Table 7), i.e. the probability for this period being real was
$1-Q^{*}>0.9999$. Secondly, the ephemerides of Eqs. 5 and 6 are also very
reliable, because they were determined from the same large data samples (Jetsu
and Porceddu 2015). Thirdly, although the Rayleigh test significance estimates
computed by Jetsu and Porceddu (2015, their Eq. 8: $Q_{z}$) for some smaller
samples were not reliable, the binomial distribution significance estimates
for the very same samples were certainly reliable (their Eq. 13:
$Q_{\mathrm{B}}$). Fourthly, the order of the passages in Lists 1 and 2 is the
same (i.e. unambiguous) for any arbitrary epoch $t_{0}$ in Eqs. 5 and 6. For
these four reasons, the phase angles computed from the ephemerides of Eqs. 5
and 6 can be used just like the time given by an accurate modern clock. For
example, such a clock shows that most people go to sleep before midnight. It
is irrelevant if only a few (small $n$), or many (large $n$), people go to
sleep. Rearranging the texts of CC into the increasing order of
$\Theta_{\mathrm{Algol}}$ may show what the authors of CC wrote about
``Horus'' at different phases of the cycle (FAQ 4).

Figure 1: $\Theta_{\mathrm{Algol}}$ phase angles of lucky time points of Table
1. Time runs in the counter-clockwise direction on this circle. Epochs Aa, Ab,
Ac and Ad are separated by 90 degrees and they are denoted with dotted
straight lines. The relative locations of Algol A (white disk) and Algol B
(black disk) at these four epochs are shown in the small boxes. Primary and
secondary eclipses of Algol occur at Ac and Aa, respectively. The thick curved
line centered at Ac outlines the phase angles of the 10 hour primary eclipse
of Algol at $153.7^{\mathrm{o}}<$ $\Theta_{\mathrm{Algol}}<$
$206.3^{\mathrm{o}}$. The phase angle values $\Theta_{\mathrm{Algol}}$ of
``Horus'' (closed squares), ``Wedjat'' (open squares) and ``Sakhmet'' (closed
triangles) are denoted with continuous straight lines.  Figure 2:
$\Theta_{\mathrm{Algol}}$ of unlucky time points of Table 2. Notations as in
Figure 1.

It is possible to dispute our translations of the texts of Lists 1 and 2 (FAQ
5), but these translations are relevant only for the two Arguments VII and
VIII, i.e. the validity of the eight other arguments in Sects. 4.1-4.10 does
not depend on these translations. In these translations, we have used the
English translation by Bakir (1966) and the German translation by Leitz
(1994), and his transcript of the original papyrus, as well as photos of the
original papyrus. For example, all 460 SW identifications by Jetsu and
Porceddu (2015) and Leitz (1994) were identical, and here we use the same SW
list. Some words or sentences could be translated differently, but that would
not change the general description of the course of events in the translated
passages of Lists 1 and 2. Also the prognoses, which were taken as such from
Leitz (1994), are independent of any translation nuances.

A non-parametric method, the Rayleigh test, has been applied to the series of
$n$ time points $t_{1},t_{2},...t_{n}$ of CC (Jetsu et al. 2013; Jetsu and
Porceddu 2015). Here we study the SWs and the CC texts of these time points.
These time points are circular data and a ``non-parametric'' method means that
there is no model. It has been suggested that we should apply a
$\chi^{2}$-test to our data (FAQ 6). This ``parametric'' test could be applied
if the format of our data were $y(t_{1}),y(t_{2}),...y(t_{n})$, i.e. a time
series, like magnitudes of a star as function of time. The value of this test
statistic is $\chi^{2}=\sum_{i}^{n}[y(t_{i})-g(t_{i})]^{2}/\sigma_{i}^{2},$
where $g(t_{i})$ is the value of the model at $t_{i}$ and $\sigma_{i}$ is the
error of $y(t_{i})$. However, we can not apply this $\chi^{2}$-test, because
we have no time series, no model and no errors.

##  3 Results

###  3.1 Algol: Horus, Wedjat and Sakhmet passages of List 1

The lucky and unlucky days of CC texts mentioning ``Horus'', ``Wedjat'' or
``Sakhmet'' are given in Tables 1 and 2, respectively. We rearrange our
translations of CC passages of ``Horus'', ``Wedjat'' and ``Sakhmet'' into the
order of increasing $\Theta_{\mathrm{Algol}}$ in List 1 of Appendix C. These
passages are discussed in Sect. 4.8. We highlight the unlucky CC prognosis
text passages with red colour, to visually distinguish these unlucky texts
from the lucky ones, since red colour is also used in CC for writing the
prognosis ``bad'' and the name of the feared serpent creature Apep. Other
texts in papyrus Cairo 86637 display an even more varied use of red colour for
the sake of emphasis and captioning (Bakir 1966, p7).

The $\Theta_{\mathrm{Algol}}$ values of the $g(D,M)$ and $s(D,M)$ time points
of List 1 are shown separately in Figures 1 and 2. The relative positions of
Algol A (white disk) and Algol B (black disk) at points Aa, Ab, Ac and Ad are
shown in four small boxes of Figures 1 and 2. For a naked eye observer,
Algol's brightness appears constant, except for the 10 hour dimming during
$\Theta_{\mathrm{Algol}}$ values marked with a thick curved line centered at
Ac. These eclipse phase angles are in the interval
$153.7^{\mathrm{o}}\leq\Theta_{\mathrm{Algol}}\leq 206.3^{\mathrm{o}}$. Time
runs in the counter-clockwise direction. One complete orbital round
$P_{\mathrm{Algol}}$ is Aa $\rightarrow$ Ab $\rightarrow$ Ac $\rightarrow$ Ad
$\rightarrow$ Aa.

The lucky time points $g(D,M)$ of SWs having
$-90^{\mathrm{o}}<\Theta_{\mathrm{Algol}}<90^{\mathrm{o}}$ amplify the
$P_{\mathrm{Algol}}=2.^{\mathrm{d}}85$ signal (Jetsu and Porceddu 2015). The
closer a $\Theta_{\mathrm{Algol}}$ value of some $g(D,M)$ is to the point Aa
at $\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$ in Figure 1, the greater is the
amplifying impact of this $g(D,M)$ value on the $P_{\mathrm{Algol}}$ signal. A
previous study by Jetsu and Porceddu (2015) showed that of all their 28 SWs,
``Horus'' had the strongest impact on the $P_{\mathrm{Algol}}$ signal. The
other remaining SWs having an impact on the $P_{\mathrm{Algol}}$ signal were
``Re'', ``Wedjat'', ``Followers'', ``Sakhmet'' and ``Ennead''.

###  3.2 Moon: Horus, Seth and Osiris passages of List 2

Figure 3:  $\Theta_{\mathrm{Moon}}$ of the lucky time points of Table 3. Time
runs in the counter-clockwise direction on this circle. Epochs Ma, Mb, Mc and
Md are separated by 90 degrees and they are denoted with dotted straight
lines. The phases of the Moon are shown in the small boxes. The Full and the
New Moon occur at Ma and Mc, respectively. The phase angle values
$\Theta_{\mathrm{Moon}}$ of ``Horus'' (closed squares) ``Seth'' (open
triangles) and ``Osiris'' (closed circles) are denoted with continuous
straight lines.  Figure 4: $\Theta_{\mathrm{Moon}}$ of the unlucky time points
of Table 4. Notations as in Figure 3.

The lucky and unlucky days of CC texts mentioning ``Horus'', ``Seth'' or
``Osiris'' are given in Tables 3 and 4. Our translations of CC passages
mentioning ``Horus'', ``Seth'' and ``Osiris'' are rearranged into the order of
increasing $\Theta_{\mathrm{Moon}}$ in List 2 of Appendix D. We discuss these
passages in Sect. 4.8.

The $\Theta_{\mathrm{Moon}}$ values of the $g(D,M)$ and $s(D,M)$ time points
of List 2 are shown in Figs 3 and 4. The appearance of the lunar disk at
points Ma, Mb, Mc and Md is illustrated in four small boxes of Figures 3 and
4. Again, time runs in the counter-clockwise direction, where one complete
synodic lunar month $P_{\mathrm{Moon}}$ is Ma $\rightarrow$ Mb $\rightarrow$
Mc $\rightarrow$ Md $\rightarrow$ Ma.

The time points $g(D,M)$ of SWs with phase angles $\Theta_{\mathrm{Moon}}$
close to $\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}$ amplify the
$P_{\mathrm{Moon}}=29.^{\mathrm{d}}6$ signal. Jetsu and Porceddu (2015) showed
that of all their 28 SWs, ``Earth'' and ``Heaven'' had the strongest impact on
the $P_{\mathrm{Moon}}$ signal, i.e. their lucky prognoses were close to the
Ma point. This is natural because lunar feast dates where often described as
feasts in ``Earth'' and in ``Heaven'' (FAQ 4). The other SWs connected to
$P_{\mathrm{Moon}}$ were ``Busiris'', ``Rebel'', ``Thoth'' and ``Onnophris''
(Jetsu and Porceddu 2015). The unlucky time points $s(D,M)$ of two SWs,
``Seth'' and ``Osiris'', pointed to the opposite direction,
$\Theta_{\mathrm{Moon}}=180^{\mathrm{o}}$. The CC texts of ``Seth'' strongly
indicated that $\Theta_{\mathrm{Moon}}=180^{\mathrm{o}}$ coincided with the
New Moon. Hence, it was concluded that
$\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}\equiv$ Ma represented the Full Moon
(Jetsu and Porceddu 2015). These connections are hardly surprising either
because ``Osiris'', also called ``Onnophris'', was identified with the Moon
during the New Kingdom (Kaper 2001, p480-482). ``Thoth'' was another known
lunar god (Leitz and Budde 2003). ``Busiris'' was the place of origin for
``Osiris''. ``Rebel'' is often synonymous for ``Seth'' (Leitz 1994, p91).

Note that the texts mentioning ``Horus'' are included in both Lists 1 and 2,
because the name ``Horus'' appears in both mythical narratives of Sects. 4.7.1
and 4.7.2. However, the order of these ``Horus'' texts is different when
rearranged in the order of increasing $\Theta_{\mathrm{Algol}}$ or
$\Theta_{\mathrm{Moon}}$.

##  4 Discussion

We present one argument about CC in the end of each Sect. 4.1-4.10.

###  4.1 Measuring night-time with hour-stars

At the night-time in ancient Egypt, time was traditionally measured from the
positions of hour-stars.

The ancient Egyptian day was split into daytime and night-time, both with 12
hours. Time was counted using shadow clocks by day, and star clocks or water
clocks by night. The Egyptian ``hour-watcher'' ▲was a specialized scribe whose
job was to observe various hour-stars, i.e. clock stars whose positions began
and ended the night hours (Figure 5). More specifically, a text previously
known as ``The Cosmology of Nut'' whose title was deciphered by Lieven (2007)
to be ``The Fundamentals of the Course of the Stars'' informs us that stars
were traditionally observed when in positions of the ``culmination'' ▲upon the
first hour of the night (transit of the meridian between the eastern half of
the sky and the western half of the sky), ``heliacal setting'' ▲ and
``heliacal rising'' ▲(Clagett 1995, p56-65). Tabulated positions of hour-stars
marked the closing of each night hour. One approximation for the length of an
Egyptian night hour was 40 minutes (Leitz and Thissen 1995, p133).

The Ramesside Star Clocks from the tombs of Ramses VI, Ramses VII and Ramses
IX show an even more complex arrangement of hour-stars (Clagett 1995, p56-65).
Each table consists of thirteen rows of stars. The first row stands for the
opening of the first night hour and the other twelve rows stand for the
closing of each of the twelve night hours. In the rows, a star is positioned
in respect to a sitting human figure. Possible positions are ``upon the right
shoulder'' ▲, ``upon the right ear'' ▲, ``upon the right eye'' ▲, ``opposite
the heart'' ▲, ``upon the left eye'' ▲, ``upon the left ear'' ▲, and ``upon
the left shoulder'' ▲ (Figure 5). The system was originally interpreted as two
hour-watchers opposite to each other on the roof of the temple precisely
aligned to the line of the meridian. These timing observations would have been
necessarily made towards south, because of the positions of the hour-stars.
The hour-watcher facing south might have utilized a plumb and a sighting
device to determine when the given star is in exactly the right position and
announce the closing of the hour. If the hour-watchers were positioned 2-3
meters away from each other, the slow rotation of the night sky would have
provided a 10-15 minute difference between the marked positions (Leitz and
Thissen 1995, p33). According to Lull and Belmonte (2009, p165) it is more
likely that the figure that had been interpreted as an hour-priest would
rather be a divinity associated with time-keeping, leading to revised ideas
regarding the direction of the observations: even some constellations of the
northern half of the sky could have been used in this method of time-keeping.
The references to the observational practices of the hour-watchers are scarce
and known mostly from late period sources such as the inscription on a statue
depicting the astronomer Harkhebi and a sighting instrument with inscriptions
mentioning an astronomer named Hor (Clagett 1995, p489-496) and (Pries 2010,
p10-26). However, it is safe to say that such practices existed throughout
Pharaonic history. The observing conditions of the hour-watchers were rather
ideal, with about 300 clear nights each year (Mikhail and Haubold 1995, pD7).

Figure 5: Hour-watcher with a star chart from the tomb of Ramesses VI, 12th
century BC. ``Meridian'' is ``opposite the heart'' mentioned in Sect. 4.1.
Reprinted under a CC BY license @eng.wikipedia.org.

Algol is the 60th brightest star in the sky (Hoffleit and Jaschek 1991). At
the latitude of Middle Egypt, $\phi_{\mathrm{Earth}}=26.6^{\mathrm{o}}$, the
never setting circumpolar stars have declinations
$\delta_{\mathrm{Star}}>90^{\mathrm{o}}-\phi_{\mathrm{Earth}}=63.4^{\mathrm{o}}$.
The declinations of stars that never rise above horizon are
$\delta_{\mathrm{Star}}<\phi_{\mathrm{Earth}}-90^{\mathrm{o}}=-63.4^{\mathrm{o}}$.
Circumpolar stars are not ideal hour-stars, because they never rise or set.
Furthermore, their angular motion within a limited area around the celestial
pole is not ideal for measuring time. It is not easy to obtain accurate timing
from the minor changes in their positions. Stars below horizon can certainly
not be used as hour-stars. If the circumpolar stars and the stars below
horizon are excluded, Algol was the 56th brightest star at
$\phi_{\mathrm{Earth}}=26.6^{\mathrm{o}}$ in 1224 B.C. However, if a star
culminates in the south below an altitude of $a=10^{\mathrm{o}}$, its
brightness decreases about one magnitude due to atmospheric extinction and it
can be observed only for a short time even in ideal observing conditions. Such
a star is not a suitable hour-star. A star that sets $10^{\mathrm{o}}$ (or
less) below horizon in the north is neither an ideal hour-star, because it
rises and sets nearly at the same location, and its brightness decreases close
to the horizon due to extinction. Using the limits
$-53.4^{\mathrm{o}}<\delta_{\mathrm{Star}}<53.4^{\mathrm{o}}$, makes Algol the
51st brightest star in the Middle Egyptian sky in 1224 B.C. These declination
limits of ours are conservative, because many other stars culminating in the
north are useless for time keeping, and extinction influences the comparison
of the brightness levels between Algol and the stars always remaining close to
the horizon. This raises Algol much higher than the 51st best in the list of
suitable bright hour-stars (FAQ 7).

Algol's equatorial coordinates were right ascension
$\alpha_{\mathrm{Algol}}=1^{\mathrm{o}}$ and declination
$\delta_{\mathrm{Algol}}=25^{\mathrm{o}}$ in 1224 B.C. which gives the
following ecliptic coordinates, longitude
$\lambda_{\mathrm{Algol}}=11^{\mathrm{o}}$ and latitude
$\beta_{\mathrm{Algol}}=22^{\mathrm{o}}$. The ancient cultures used the latter
coordinate system based on the yearly motion of the Sun. Algol was located
very close to the vernal equinox and this ecliptic plane. If the timing
observations were made towards south, then the bright stars in the ecliptic
plane were the most suitable hour-star candidates (Clagett 1995, p2-56),
(Wells 2001b, p145-150) and (Böker 1984). This location of Algol in the sky
raises it very high in the list of suitable bright hour-stars (FAQ 7).
Furthermore, Algol culminated at the altitude of $a=88^{\mathrm{o}}$, and this
made it an ideal star for measuring time.

One modern hour equals 15 degrees in the equatorial plane, and therefore the
required minimum number of hour-stars for covering the entire sky is at least
24. The earliest known star clock scheme, the so-called diagonal star clock,
describes 36 decans (Leitz and Thissen 1995, p63). For a ten-day week, 12 of
the decans are tabulated marking the beginnings and ends of the hours. Because
the sidereal day is about four minutes shorter than the solar day, these stars
reach their positions four minutes earlier every consecutive night. After ten
days the stars occupy their positions about 39 minutes earlier, so in the
tabulation of decans for the next week each of the stars "works" one hour
earlier. The concept of the hour was relative as its beginning was allowed to
fluctuate about 35 minutes by the end of each week. The later Ramesside star
clocks are comprised of a system of 46 individual stars observed in positions
such as ``upon the right shoulder'', ``upon the right ear'', etc. as in our
Figure 5 (Leitz and Thissen 1995, p120). These positions were intended for
better accuracy but the known tables were given for 15-day periods, where the
timing of the hours fluctuated even 55 minutes. However, this model seems to
have been outdated by the time it was used in the decoration of Ramesside
tombs, so the later developments of the system remain unknown (Leitz and
Thissen 1995, p132).

No-one can recognize a single hour-star in the sky without comparing its
position to the positions of other bright stars in its vicinity. For the sake
of consistency, we introduce our own precise concept: ``hour-star pattern''.
Such a pattern contains one hour-star used by the scribes, and all bright
stars that they used to identify this hour-star. Using only two stars per one
hour-star pattern would not have provided any recognizable pattern. Therefore,
the number of stars per each hour-star pattern must have been at least three,
or probably more. This means that the Ancient Egyptians must have observed at
least $24\times 3=72$ bright stars, and in this case the visual brightness of
the 72th brightest star was about $2.^{\mathrm{m}}5$ in 1240 B.C. Thus, it is
certain that the 51st brightest star Algol ($2.^{\mathrm{m}}12$) with an ideal
position in the night sky was included into some hour-star pattern (FAQ 7).
The names of hour-stars in the Ramesside star clocks include for example
various different body parts and equipment of the Giant, the Bird and the
Hippopotamus, suggesting that the stars were members of a known constellation
(i.e. an hour-star pattern). According to Belmonte et al. (2009, p157-194), a
complete set of constellations formed the Egyptian celestial diagram, i.e.
every star belonged to some constellation.

Unlike Astronomy, Egyptology is not an exact science and few questions can be
answered with absolute certainty. For our ten arguments I-X presented in
Sects. 4.1-4.10 it is not important if Algol was an actual hour-star or only a
member of some hour-star pattern or related constellation (FAQ 7). Algol has
not yet been unambiguously identified in any hour-star lists because only the
names of Sirius, Orion and the Plough have reached a widespread consensus
among egyptologists. Algol ($\beta$ Persei) is the second brightest star in
the modern constellation of Perseus. Egyptologists have presented their own
differing identifications of the represented stars with actual stars so it is
difficult to say which decan or group would have included Algol (Belmonte et
al. 2009, p161-162), (Böker 1984) and (Conman 2003). Lull and Belmonte (2009,
p157-158) claim to have uncovered nearly three quarters of the Egyptian
firmament by deciphering the star names of aforementioned tomb of Senenmut,
the clepsydra (water clock) of Amenhotep III and the circular zodiac of the
temple of Hathor at Dendera, which is from Late Period and already
incorporates Mesopotamian and Greek influences. According to them, the ancient
Egyptian constellation of the Bird includes the modern constellations
Triangulum and Perseus but there is no precise identification of the
individual stars of the Bird. Böker (1984) suggested that the correct reading
for the decan Khentu, a group of three stars, is the ``snorting one'' ▲. The
decan is later depicted as a red-haired warrior with fierce attributes
reminiscent of Perseus in Greek mythology. The decan is also known as ``the
lower Khentu'' ▲, and mentioned in the decan lists of the Astronomical ceiling
of the tomb of Senenmut (ca. 1473 B.C.), tomb of Seti I (1313-1292 B.C.) and
the Osireion, a temple in Abydos dated to the time of Seti I (Neugebauer and
Parker 1960, p23-26).

The number seven seems to carry plenty of mythological connotations for the
Egyptians, such as the seven failed attempts of ``Seth'' to lift the foreleg
into the heavens (Leitz 1994, p28). It is reminiscent of the various names of
the Pleiades, a distinct open cluster of bright stars located near Perseus
(i.e. Algol), known for example as Seven Sisters, Starry Seven and Seven
Dovelets (Allen 1899, p391-403). Pleiades may have been connected to the
``Followers'' or the ``Ennead'' which were both connected to
$P_{\mathrm{Algol}}$ in CC (Jetsu and Porceddu 2015, p14-15).

A list of decan deities in papyrus Carlsberg I mentions certain stars that
cause ``sickness'' ▲in fish and birds, while CC speaks of "a star with
bitterness in its face". According to Leitz (1994, p307), ``bitterness'' ▲is
the name of a sickness that plagued the Egyptians. Decans in the inscription
406 of the temple of Esna are portents of death to the "rebel" (von Lieven
2000, p48). Thus, even without knowing which stars exactly they referred to in
these passages, we may conclude that the ancient Egyptians strongly believed
in stars influencing the lives of men.

Argument I: For thousands of years, the ``hour-watchers'' practiced the
tradition of timekeeping by observing hour-stars. If Algol was not an hour-
star, it certainly belonged to some hour-star pattern or related
constellation.

###  4.2 Crucial timing of nightly rituals

Proper timing was considered crucial for the efficacy of nightly religious
rituals.

Astronomy is often considered to be one of the oldest sciences practiced by
mankind despite ancient star observing being carried out for the benefit of
religious practices. Babylonians were probably the first people to make
systematic notes of the Moon and the planets and also to perform calculations
of their celestial motions. As opposed to this, ancient Egyptian records that
span three millennia are almost obsolete in any quantitative approach despite
the culture's special attention to the Sun, the Moon, the planets and the
stars as divine entities (Neugebauer 1951, p71-72). The phases of the Moon and
the heliacal rising of Sirius played a part in determining the date and time
of New Year and several other important festivals.

In ancient Egypt, the scribal professions were the most valued ones, as the
entire functioning of the highly developed culture and state with its complex
bureaucracy was based on written communication (Shaw 2012, p24-38). Many of
the professional scribes had several titles emphasizing their specialized
knowledge (Clagett 1989, p18-24), such as ``physician'' ▲, ``healer'' ▲,
``hour-watcher'' (astronomer who observed stars for timekeeping purposes) and
``mathematician'' ▲.

Beside the religious background, the scribes had plenty of social, political
and personal motivation to perform their job with utmost expertise. Many
scribes received the title of ``king's favourite'' during their lifetime and
such persons were among the most high ranking members of the Egyptian society
(Clagett 1989, p195).

What would have been the ancient Egyptian scribes' interest in the behaviour
of a variable star? Knowing the period and the phase of the Moon was important
for regulating the religious festivities but the scribes would also have paid
attention to any unexpected changes in the observed hour-star patterns. The
hour-watchers' activity required the mapping and measuring of the heliacal
rising and setting, as well as the meridian transit of stars (Magli 2013,
p55). The average width for a suitable ancient Egyptian hour-star pattern
would have been about one hour in modern right ascension, i.e. 15 degrees.
Thus, during every year, Algol's hour-star pattern determined the beginning of
the night during half a month ($\sim 15^{\mathrm{o}}/360^{\mathrm{o}}$), the
epoch of midnight during another half a month, and the end of the night during
yet another half a month. During thousands of years of timing observations,
the discovery of Algol's variability would have been most probable during
these particular time intervals of every year (FAQ 7).

It is essential to note that the professional class of scribes was responsible
for both astronomical observations and religious traditions. To be involved in
the science of Astronomy (e.g. hour-watching) was also to be involved in the
priesthood (e.g. nightly rituals) (Shaw 2012, p11-16). To properly observe the
ritual cycle and recite the magical words at the exactly right time was of
foremost importance to the Egyptian priests, since it was a matter of life and
death (Assmann and Lorton 2001, p64-68). The hour-stars were used for this
exact timing that was crucial for keeping cosmic order (Clagett 1989, p195)
and (Magli 2013, p2). During the night, the Sun was considered to sail across
the underworld where the prayers or incantations of the priests opened the
gates of the underworld and appeased the terrible guardians of the gates
(Wiebach-Koepke 2007, p37-58). If everything went absolutely right, the Sun
was reborn on the 12:th hour of the night. Any failure by the priests in
observing the nightly rituals would mean running a risk of the Sun not rising
the next morning (Assmann and Lorton 2001, p68-73).

The consequential purpose of astronomical observations was ``religious
astronomy'' (von Lieven 2000, p188). Any unpredictable change in the hour-star
patterns observed by the scribes would have been a shock. The ritual activity
of the ancient Egyptian priesthood functioned for the very purpose of
maintaining the known universe in a stable condition in order not to plunge
into chaos. They needed to use all their resources to keep ``Maat'' ▲, the
cosmic order (Magli 2013, p2). If they found out that a star is variable in
brightness, observing its cycle would most certainly have gained their extra
attention and intellectual effort.

Argument II: Proper timing of the nightly religious rituals relied on the
fixed hour-star patterns.

###  4.3 Constellation change

Any unpredictable change in the fixed and known hour-star patterns would have
been alarming. A naked eye observer witnesses a radical hour-star pattern
change during Algol's eclipse.

The naked human eye can detect brightness differences of 0.1 magnitudes in
ideal observing conditions. Hence, naked eye eclipse detection is
theoretically possible for 7 hours when Algol is more than 0.1 magnitudes
dimmer than its brightest suitable comparison star $\gamma$ Andromedae (see
Jetsu et al. 2013, their Figure 5a). For 3 hours, Algol is also dimmer than
its other five suitable comparison stars $\zeta$ Persei, $\eta$ Persei,
$\gamma$ Persei, $\delta$ Persei and $\beta$ Trianguli. The detection of
Algol's eclipse is easy during this 3 hour time interval.

With $P_{\mathrm{Algol}}=2.850=57/20$ days, these opportunities for easy
detection follow the sequence of ``3+3+13=19'' days (Jetsu and Porceddu 2015,
p20-21). A plausible hypothesis is that the ancient Egyptians first discovered
the variability of Algol, like Montanari did in 1669, when they were observing
Algol's hour-star pattern. During primary eclipses, Algol lost its brightness
gradually for five hours until it was outshined by its six dimmer nearby
comparison stars. The dimming was followed by a brightening that lasted for
another five hours. This entire 10 hour eclipse event could be observed during
a single night, but that was rare event, because it occurred only every 19th
night. Algol's hour-star pattern change was so noticeable that the priests on
duty as hour-watchers could hardly have missed this event. For the hour-
watchers, it would have been useful to communicate among themselves the
knowledge about the strange behavior of this hour-star pattern and there is a
good possibility, considering the diligent scribe mentality of Egyptian
officials, that they would have made written notes about the times of the
eclipses.

The modern constellation of Perseus is one of the easiest to perceive. It
occupied a prominent position high in the ancient Egyptian night sky, because
the maximum altitude of Algol was 88 degrees. Algol is the brightest member of
another ancient constellation of four stars which was already recognized by
authors like Vitruvius (Vitruvius and Morgan 1960, p266-267) and Ptolemy
(Ptolemy et al. 1915, p31). The other three stars are $\pi$ Per
($4.^{\mathrm{m}}7$), $\omega$ Per ($4.^{\mathrm{m}}6$) and $\rho$ Per
($3.^{\rm m}4-4.^{\mathrm{m}}0$). During the primary eclipses, the brightness
of Algol falls from $2.^{\rm m}1$ to $3.^{\rm m}4$, i.e. these other three
stars never appear to be brighter than Algol. The shape of this constellation
resembles a diamond and it was therefore called the Head of Gorgon or the Head
of Medusa (Allen 1899, p332) in the Hellenistic culture. The angular
separation between Algol and the other three stars is less than two degrees,
and this constellation is therefore ideal for detecting variability because
atmospheric extinction does not mislead brightness comparisons even at low
altitudes close to the horizon. At its brightest, Algol visually dominates
this diamond shaped constellation, because it is clearly much brighter than
the other three stars. Hence, a naked eye observer can easily notice the
significant constellation pattern change during Algol's eclipse. Wilk (1996)
suggested that from this may have arisen the myth of the Medusa losing its
head.

In principle, other variable stars besides Algol, like the disappearing and
reappearing Mira, also called Omicron Ceti, might also have been discovered by
the ancient Egyptians. However, the eleven month period of Mira is so long
that it can not be rediscovered with statistical methods in CC (Jetsu et al.
2013, p9, see Criterion $C_{2}$).

Argument III: A naked eye can easily discover the significant hour-star
pattern change caused by Algol's eclipse.

Figure 6:  Mid-epochs of Algol's primary eclipses. (a) The yearly mid-epochs
of 56 night-time eclipses occurring $\pm 5$ hours at both sides of midnight
with the period $P_{\mathrm{Algol}}=2.850$ days (blue circles). The red
circles denote the 70 daytime eclipses. Note that only 20 circles are visible,
because the 56 red or 70 blue circles overlap. The eclipse phase angles
$\Theta_{\mathrm{P=19}}$ are computed for the period $P=19$ days. The values
of the Rayleigh test statistic for these eclipse epoch time points with $P=19$
days (Jetsu and Porceddu 2015, $z$ in Eq. 7) are given for the night-time
eclipses $(z_{\mathrm{nights}})$, the daytime eclipses $(z_{\mathrm{days}})$
and all eclipses $(z_{\mathrm{both}})$. (b) The same as in (a), except that
the current period value, $P_{\mathrm{Algol}}=2.867$ days, is used.

###  4.4 ``3+3+13=19'' and ``19+19+19=57'' days eclipse rules

Algol's night-time eclipses followed the regular 3+3+13=19 and 19+19+19=57
days cycles with $P_{\mathrm{Algol}}=2.850=57/20$ days. (Jetsu and Porceddu
2015).

If an eclipse first occurred in the end of the night, then after three days
the next eclipse occurred close to midnight. After another three days the next
eclipse occurred in the beginning of the night. After these three eclipses, it
took thirteen days, before the next eclipse was observed, i.e. the first
eclipse in the end of the night was repeated again (Jetsu and Porceddu 2015).
This is the ``3+3+13=19'' days rule, i.e. a sequence of three night-time
eclipses was repeated every 19 days. The midnight eclipses, when both the
dimming and the brightening of Algol could be observed during the same night,
occurred at 19 days intervals. Coincidence or not, the CC prognosis of the
first day of every month is GGG and the prognosis of the 20th day of every
month is SSS. These two particular days of every month are separated by the
``3+3+13=19'' days eclipse cycle.

With the period of $P_{\mathrm{Algol}}=2.850=57/20$, eclipses separated by 57
days would have occurred at exactly the same time of the night. This is the
``19+19+19=57'' days rule, i.e. this entire 57 days eclipse sequence kept on
repeating itself over and over again. For this reason, there are only 20 small
blue or red circles (night-time or daytime eclipses) in Figure 6a, although
this figure shows all 126 eclipses occurring during 360 days. If the hour-
watchers began to make notes of the eclipse epochs, the ``3+3+13=19'' days
eclipse cycle would have been discovered first, then later the ``19+19+19=57''
days eclipse cycle. The first regularity is easier to discover, because all
Algol's night-time eclipses follow this ``3+3+13=19'' cycle with
$P_{\mathrm{Algol}}=2.850=57/20$ days. The second cycle of ``19+19+19=57''
days is more difficult to discover, because it requires the measurement of the
nightly shifts of these night-time eclipses within the ``3+3+13=19'' cycle.
Although the ancient Egyptians had no way of knowing that Algol's eclipses
also happened during daytime, their long-term eclipse records may have
eventually led to the discovery of this 57/20 days ratio.

Simulations have shown that modern period analysis would rediscover the 2.850
days period in CC, if the scribes had recorded nearly all observed night-time
eclipses (Jetsu et al. 2013). The fact that the 19 days period is also present
in these data (Jetsu et al. 2013) confirms that only the night-time eclipses
were used in the construction of CC. This result can be deduced from Figure
6a. It is highly unlikely that the ancient Egyptians knew that Algol's
eclipses also occurred at daytime, because these events could not be observed.
If an eclipse occurred at the daytime, then no eclipse was observed on the
previous night or the next night. However, this absence of night-time eclipses
could be observed. This absence was observed in 16 nights of every 19 days
cycle. Our Figure 6a shows that there are two plausible alternatives that
exclude each other. Either the presence (Figure 6a:
$z_{\mathrm{nights}}=27.8$) or absence (Figure 6a: $z_{\mathrm{days}}=22.8$)
of night-time eclipses was used as a criterion for selecting the lucky
prognoses connected to Algol's variability. In other words, if the ancient
Egyptians had used both the night-time and the daytime eclipses in the
construction of CC, this would have erased the 19 days signal completely away
from CC (Figure 6a: $z_{\mathrm{both}}=0.0$). The alternative that the absence
of night-time eclipses was used in assigning lucky prognoses is more probable,
because the lucky prognoses mentioning ``Horus'', ``Sakhmet'' and ``Wedjat''
concentrate on the bright phases of Algol (Figure 1). This alternative is
later discussed in greater detail in Sect. 4.8 (principle I).

The scribes seem to have applied a very accurate value for the synodic period
of the Moon, 29.6 days, to correctly predict the lunar phases (Jetsu et al.
2013). If they intended a correct prediction of Algol's eclipses they would
have also needed an accurate period value. Our Figure 6b shows how the
``3+3+16=19'' days cycle breaks down with the currently observed 2.867 days
period of Algol. Modern period analysis detects the 19 days and the 2.850 days
periods in CC, but it does not detect the current 2.867 days period of Algol
(Jetsu et al. 2013). Our Figure 6b confirms that if the period of Algol in
those days had been the same as the current period, 2.867 days, there would be
no signs of the 19 days period in CC ($z_{\mathrm{nights}}=0.4$,
$z_{\mathrm{days}}=0.3$ and $z_{\mathrm{both}}=0.0$).

Argument IV: The scribes could have discovered Algol's $2.850=57/20$ days
period from long-term observations of the regular 19 and 57 days eclipse
cycles.

###  4.5 Ancient Babylonian and Chinese lunar eclipse prediction

The ancient Egyptian discovery of $2.850=57/20$ days ratio in Algol's eclipses
would resemble the ancient Babylonian and Chinese discoveries of the lunar
eclipse cycle. Here, we discuss how these two analogous ancient astronomical
cycle discoveries were made, and how the ancient Egyptians may have utilized a
similar approach to observe Algol's variability.

Firstly, the process that may have lead to the ancient Egyptian discovery of
the periodicity of Algol's eclipses could have been analogous to the process
which led to the ancient Babylonian detection of the cycle in the lunar
eclipses. The unexpected occurrence of solar and lunar eclipses must have
deeply impressed ancient civilizations. The motivation for prognostications
was obvious. There is a description of how the Babylonians determined the
Saros cycle, 18 years and 11.33 days, in the occurrence of lunar eclipses
(Pannekoek 1989, p57-62). Detailed long-term records were kept of these
events. Somewhere between 750 and 650 B.C. there was a more or less complete
record of the observed eclipses and by fairly basic analysis it was found out
that within 223 lunar months there were 38 eclipse possibilities. Also these
phenomena always occurred in a series of four, five or six consecutive lunar
eclipses. A theory of the past and future lunar eclipses, which was based on
the table called ``Saros-Canon'', emerged sometimes after 280 B.C. Brack-
Bernsen and Steele (2005, p181) have argued that the Babylonians may even have
invented sophisticated methods to determine the anomalies (between 6 and 9
hours) of the Saros cycle. Ancient Egyptians were also fluent in arithmetic
and geometric calculations and devised plenty of tabulations to aid in
practical life (Clagett 1999, p24-42) and (Rossi 2004, p57-60). Papyrus
Carlsberg 9 reveals that they used long-term observations to determine the
correct period of the Moon (Clagett 1995, p23-28). It has also been shown that
the $P_{\mathrm{Moon}}$ value in CC was so accurate it must have been
determined from observations made over more than one year (Jetsu et al. 2013,
p7). These two cases (Jetsu et al. 2013; Clagett 1995) confirm that the
ancient Egyptians were capable of using long-term observations to determine
the periods of celestial objects. By the Late Period, the Egyptians were also
familiar with the use of the Saros cycle of 223 synodic months to predict
lunar and solar eclipses. Ptolemy's commentators referred to these periods as
having been used already by ``the ancients'' (Steele 2000, p88).

Secondly, at about 400 B.C. the Chinese were using an astronomical calendar to
predict the dates of lunar eclipses (Steele 2000, p175-178). This calendar
made use of a cycle of 135 months which includes 23 lunar eclipse
possibilities. They approximated that eclipse possibilities occur periodically
every 5 and 20/23 months and counted them arithmetically. The ancient
Egyptians could have used an analogous approach to Algol's eclipses.

The alternating period changes of Algol are so small (Biermann and Hall 1973)
that they do not mislead long-term period determination based on naked eye
observations. However, the synodic period of the Moon varies between 29.3 and
29.8 days in a year (Stephenson and Baolin 1991). Thus, Algol's eclipses are
far easier to predict than lunar eclipses, because Algol's period is constant,
while that of the Moon is not. The 2.850=57/20 days ratio means that 20
eclipses of Algol occur during every 57 days cycle (Figure 6a). If the scribes
performed long-term observations of these cycles, these observations would
have eventually revealed that all nine Algol's night-time eclipses could
always be observed within every 57 days cycle.

There are three alternative methods of how the period of 2.850 days ended up
into CC. We may never find out which one of these three methods was used.
However, in every alternative the scribes would have discovered the eclipses
of Algol, and recorded the period of these events.

  * 1st

method: The scribes might have calculated the numerical value
$P_{\mathrm{Algol}}=2.850$ days from long–term observations.

  * 2nd

method: The scribes may not have had any reason for even trying to calculate
the actual period. If they noticed that Algol's three consecutive night-time
eclipses followed the 19 days cycle, while nine consecutive night-time
eclipses occurred in every 19+19+19=57 days cycle, these cycles would have
appeared extremely stable over very long periods of time. The minor nightly
shifts in the epochs of the eclipses within individual 19 days cycles
cancelled out within every 57 days cycle. The scribes may not even have
noticed these minor nightly shifts within individual 19 days cycles, because
the eclipses at the end of the night, close to the midnight or at the
beginning of the night always returned back exactly to the same moments of
night after every 57 days cycle. If the real period of Algol was 57/20=2.850
days, these 19 and 57 days cycles would have enabled the scribes to foretell
what would be observed in the sky. While the scribes may have recorded into CC
only the 19 or 57 days cycles in the night-time eclipses of Algol, modern
period analysis confirms that these rules worked perfectly only if the real
period of Algol was 57/20=2.850 days in those days. Hence, the scribes have,
although only perhaps unknowingly, also recorded the 57/20=2.850 days period
into CC.

  * 3rd

method: The scribes only used the observed epochs of night-time eclipses as
such in the construction of CC. This could be achieved even without ever
solving a numerical estimate of $P_{\mathrm{Algol}}=2.850$ days, or without
ever discovering the 19 and 57 days cycles. Also in this case, the scribes
have unknowingly recorded the 57/20=2.850 days period into CC.

Argument V: The ancient Egyptian scribes may have calculated the $57/20=2.850$
days period of Algol from long-term observations (1st method). They may not
have calculated this 2.850 days period, because the 19 days and 57 days cycles
already perfectly predicted all night-time eclipses of Algol (2nd method), or
they may have just recorded the observed night-time eclipses into CC (3rd
method).

###  4.6 Indirect references to protect cosmic order

After the scribes had discovered Algol's variability, it would have been
attributed religious significance and described accordingly.

Lack of direct references to Algol's variability in ancient Egyptian records
raises questions. We may draw parallels to solar eclipses, which were
experienced by Egyptians during the New Kingdom and could not have passed
unnoticed. Yet, these events are not mentioned directly in written records
(Smith 2012). The first plainly written Egyptian eclipse records are found in
the demotic papyri Berlin 13147 + 13146 which date to the first century B.C.
(Steele 2000, p85-91). Smith (2012) interpreted references to solar eclipse
events in New Kingdom texts and concluded that astronomical events were
described indirectly by using religious terminology and the reference might
even have been made deliberately obscure. In Late Period demotic writings
eclipses were mentioned and described as fearsome and unlucky portents (von
Lieven 1999, p102).

In general, ancient Egyptian scribes seem to have avoided direct references to
celestial events, because writing was considered in itself to have a magical
power that allows the scribe to communicate with the gods. This could be the
reason why they avoided direct references to the celestial events, i.e. the
observed actions of divine deities in the sky. The scribes would have avoided
making direct textual references to the uncanny behaviour of Algol in order to
preserve cosmic order. Divine names that may have been related to Algol are
various, depending on the context. In particular, the names ``Horus'',
``Wedjat'' and ``Sakhmet'' had similar phase angle $\Theta_{\mathrm{Algol}}$
distributions in Figures 1 and 2. We conclude that references to astronomical
events are indirect but undoubtedly present throughout the whole CC text, not
unusual for Egyptian mythological texts in general (Leitz 1994, p285-286).

Argument VI: To avoid violating cosmic order, the scribes would have referred
to Algol's changes only indirectly.

###  4.7 Two legends

Lists 1 and 2 are given in Appendix C and D, respectively. These lists are
full of extracts from two well known legends: ``the Destruction of mankind''
(hereafter LE1) and ``the Contendings of Horus and Seth'' (hereafter LE2). CC
texts seem to refer to many lesser known legends as well, but most of the
extracts on these lists, chosen by the occurrence of the selected SW, are
clearly connected to these two well known legends (for example "peace between
Horus and Seth", "pacify the Wedjat"). Those without a clear connection to
either of these two such as "Horus hears your words in front of every god" or
"Horus is proceeding while Deshret sees his image" could fit into other
narratives. However, these are of generic or fragmentary nature and the
associations would be too speculative to help to understand the significant
periodicity in the prognoses. It is not our ambition here to explain the
prognosis of all individual dates but to find the logic behind the
periodicity, caused by a larger group of connected dates and prognoses.

####  4.7.1 Destruction of Mankind (LE1):

In this legend, Re sends the Eye of ``Horus'' (i.e. ``Wedjat'') to punish the
rebellious mankind.

The legend of the ``Destruction of Mankind'' is a mythological narrative that
figures repetitively in CC. It concerns the Eye of ``Horus'', also called
``Wedjat'' or ``The Raging One'' ▲(Leitz and Budde 2003, p361), fighting
against the rebels who oppose the Sun god ``Re''. In the beginning of this
legend, Re sends auxiliary gods in the form of fishes to overhear the plots of
the rebellious mankind (Guilhou 2003). The impudence of mankind causes ``Re''
to send the Eye of ``Horus'' to kill all the rebels. As the Eye takes the form
of the lion-goddess ``Sakhmet'', the destruction of the entire mankind is
imminent. The gods deceive the goddess ``Sakhmet'' by colouring beer mash with
hematite to make it look like human blood. ``Sakhmet'' drinks the beer mash,
is pacified and mankind is saved (Lichtheim 1976, p197-199).

####  4.7.2 Contendings of Horus and Seth (LE2):

In this legend, Horus and Seth contend for the kingship of Egypt.

Another explicitly quoted legend in CC is the ``Contendings of Horus and
Seth''. After being murdered, the divine ruler ``Osiris'' is in the underworld
and the contenders for his office are his son ``Horus'' and his brother
``Seth'', who was responsible for the death of ``Osiris''. The dispute is
decided by the council of nine gods called the ``Ennead'', ruled by the Sun
god ``Re''. Various contests are ordered for ``Horus'' and ``Seth'' to
determine who is the able and rightful ruler of Egypt, ``Horus'' being
described as physically weak but clever, ``Seth'' stronger but with limited
intelligence. ``Seth'' is defeated and two parallel judgements conclude the
myth. The first verdict of the gods is a division of the kingdom between the
two, but the second verdict is gods giving ``Horus'' the entire inheritance of
his father ``Osiris'' (Redford 2001, p294-295). Due to parallel judgements,
the legend ends either to the crowning of both ``Horus'' and ``Seth'', or only
of ``Horus''.

Argument VII: Even a quick glance on List 1 ($\Theta_{\mathrm{Algol}}$ order)
and List 2 ($\Theta_{\mathrm{Moon}}$ order) reveals that numerous CC texts are
excerpts from the LE1 and LE2 legends.

###  4.8 Astronomical beliefs behind Lucky and Unlucky Days

The scribes used the LE1 and LE2 legends to describe the phases of Algol and
the Moon. Here, we show how the events of LE1 and LE2 appear in Lists 1 or 2.
This reveals the three principles that they possibly used to describe
celestial variability as activity of deities.

The periods $P_{\mathrm{Algol}}$ and $P_{\mathrm{Moon}}$ were discovered from
over 500 time points of lucky prognoses (Jetsu et al. 2013). These large
samples were used to determine the zero epochs $t_{0}$ of ephemerides Eq. 5
and 6. The phase angles of lucky prognoses concentrated at these epochs
$\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$ at Aa in Figure 1 and
$\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}$ at Ma in Figure 3. This means that
these two particular phase angles were considered to be the luckiest in the
cycle of Algol and the Moon, respectively.

When the CC passages are read in temporal order from one day to the next, the
general sequence of events appears disorganized. However, the contents of the
passages begin to make sense when these passages are rearranged and read in
the order of increasing $\Theta_{\mathrm{Algol}}$ and $\Theta_{\mathrm{Moon}}$
(Lists 1 and 2). This result could have been accomplished even without solving
the zero epochs $t_{0}$ of Eqs. 5 and 6, because the same stories are repeated
in the same phase angle order of Algol and the Moon. In other words, the
$\Theta_{\mathrm{Algol}}$ and $\Theta_{\mathrm{Moon}}$ order of these CC
passages is unambiguous.

The first principle in assigning Lucky and Unlucky Days seems to have been

  * principle I: The middle of the bright phases of Algol and the Moon is lucky for mankind.

For $P_{\mathrm{Algol}}$, point Aa denotes the luckiest phase angle
$\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$ in Figure 1. Out of all 28 SWs
studied by Jetsu and Porceddu (2015), the $n=14$ lucky time points $g(D,M)$ of
``Horus'' have the strongest amplifying impact on the $P_{\mathrm{Algol}}$
signal (Figure 1: closed squares). Twelve of these fourteen values, having
$-90^{\mathrm{o}}<P_{\mathrm{Algol}}<+90^{\mathrm{o}}$, amplify this signal.
The closer the phase angle $\Theta_{\mathrm{Algol}}$ of any $g(D,M)$ is to
$\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$, the more this time point amplifies
the $P_{\mathrm{Algol}}$ signal. Here are short excerpts from the five lucky
``Horus'' passages closest to $\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$

  * $g(14,2)$

$\equiv\Theta_{\mathrm{Algol}}=6^{\mathrm{o}}$, ``the majesty of Horus
receiving the white crown''

  * $g(19,12)$

$\equiv\Theta_{\mathrm{Algol}}=13^{\mathrm{o}}$, ``this eye of Horus has come,
is complete, is uninjured''

  * $g(27,1)$

$\equiv\Theta_{\mathrm{Algol}}=19^{\mathrm{o}}$, ``Peace between Horus and
Seth''

  * $g(24,3)$

$\equiv\Theta_{\mathrm{Algol}}=19^{\mathrm{o}}$, ``Onnophris' happiness in
giving his throne to his son Horus''

  * $g(1,7)$

$\equiv\Theta_{\mathrm{Algol}}=32^{\mathrm{o}}$, `` a feast of entering into
heaven (i.e. appearing into the sky). The two banks of Horus rejoice''

These texts suggest that the lucky prognoses of ``Horus'' were connected to
the bright phases of Algol (Jetsu and Porceddu 2015).

For $P_{\mathrm{Moon}}$, the luckiest phase angle
$\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}$ coincides with point Ma in Figure 3.
Leitz (1994, p285-286,474) had already argued that the New Moon occurred
between the unlucky time points on the first of the following two consecutive
days

  * $s(16,7)$

$\equiv\Theta_{\mathrm{Moon}}=173^{\mathrm{o}}$, ``Opening of the windows and
opening of the court. Seeing the portal of the ``western side of Thebes''
▲where his place is. Do not look at the darkness on this day.''

  * $s(17,7)$

$\equiv\Theta_{\mathrm{Moon}}=185^{\mathrm{o}}$, (Figure 4: ``Seth'') ``Do not
speak the name of Seth on this day. Who in his lack of knowledge pronounces
his name, he will not stop fighting in his house of eternity.''

According to his calculations $s(16,7)$ was the New Moon day when one is
forbidden to go outside and see the darkness. The above two unlucky dates are
indeed at both sides of point Mc
$\equiv\Theta_{\mathrm{Moon}}=180^{\mathrm{o}}$, and exactly half a lunar
cycle away from the luckiest phase at Ma. This result for the phase angle of
the New Moon, Mc $\equiv\Theta_{\mathrm{Moon}}=180^{\mathrm{o}}$, suggests
that the epoch $t_{0}$ of our ephemeris of Eq. 6 is correct. Hence, the Full
Moon is at Ma $\equiv\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}$.

The lucky time points $g(D,M)$ having phase angles $\Theta_{\mathrm{Moon}}$
close to $\Theta_{\mathrm{Moon}}=0^{\mathrm{o}}$ amplify the
$P_{\mathrm{Moon}}$ signal. The lucky points of ``Earth'' and ``Heaven'' have
the strongest impact on the $P_{\mathrm{Moon}}$ signal (Jetsu and Porceddu
2015, p16).

There are only three lucky time points that mention both ``Horus'' and
``Seth'' during the same day (Figure 3: dark squares and open triangles. Note
that the ``Horus'' and ``Seth'' texts at
$\Theta_{\mathrm{Moon}}=73^{\mathrm{o}}$ in List 2 are from two different
days). The excerpts of these three days are

  * $g(27,1)$

$\equiv\Theta_{\mathrm{Algol}}=19^{\mathrm{o}}$,
$\Theta_{\mathrm{Moon}}=278^{\mathrm{o}}$, ``Peace between Horus and Seth''

  * $g(27,3)$

$\equiv\Theta_{\mathrm{Algol}}=38^{\mathrm{o}}$,
$\Theta_{\mathrm{Moon}}=287^{\mathrm{o}}$, ``Judgement between Horus and Seth.
Stopping the fight''

  * $g(29,3)$

$\equiv\Theta_{\mathrm{Algol}}=291^{\mathrm{o}}$,
$\Theta_{\mathrm{Moon}}=312^{\mathrm{o}}$, `` The white crown is given to
Horus and the red one to Seth''

In all these three cases, Algol is also at its brightest and the Moon is
waxing gibbous, since half moon had occurred at phase angle Md
$\equiv\Theta_{\mathrm{Moon}}=270^{\mathrm{o}}$. These texts suggest the
reconciliation of the two gods, ``Horus'' and ``Seth'', thus peace in Egypt
and lucky days, when Algol and Moon were simultaneously bright. The White
Crown (Hedjet) represented the kingship of Upper Egypt and the Red Crown
(Deshret) the rulership of Lower Egypt (Goebs 2001, p321-325).

We know from an unrelated text from Edfu that ``Horus'' would benefit from the
brightening of the Moon: ``When he completes the half month, he assumes
control of the sky rejuvenated'' (Kaper 2001, p480-482). The CC text

  * $s(26,2)$

$\equiv\Theta_{\mathrm{Moon}}=270^{\mathrm{o}}$, ``Do not lay the foundation
of a house. Do not stock a workshop. Do not order any job. Do not do any work
on this day. It is day of opening and closing the court and the windows of
Busiris.''

coincides with Md $\equiv\Theta_{\mathrm{Moon}}=270^{\mathrm{o}}$, the first-
quarter moon.

At the moment of Full Moon, ``Horus'' was declared ``true of voice'' and
``joyful''; related to his victory over ``Seth'' in the divine tribunal (Kaper
2001, p480-482). The unlucky prognoses of ``Seth'' also show a connection to
$P_{\mathrm{Moon}}$, especially to the darker phases of Moon (Figure 4: open
triangles).

The only lucky time point of ``Horus'' overlapping the thick curved line in
Figure 1 (Algol's primary eclipse), mentions

  * $g(28,3)$

$\equiv\Theta_{\mathrm{Moon}}=164^{\mathrm{o}}$, ``The gods are in jubilation
and in joy over the making of will for Horus''

it refers to the will made by ``Osiris'' that raises ``Horus'' to be the ruler
of Egypt. Because of this we can not say that the dark phase of Algol would be
always unlucky. However, unlucky days follow immediately after Algol's eclipse
because the regular distribution of unlucky time points $s(D,M)$ of ``Horus'',
``Wedjat'' and ``Sakhmet'' concentrates at
$\Theta_{\mathrm{Algol}}=270^{\mathrm{o}}$ (Figure 2).

The ``second principle in assigning Lucky and Unlucky Days'' probably was

  * principle II: Use elements from LE1 and LE2 to indirectly describe Algol's changes (List 1).

List 1 contains the translated full CC passages mentioning ``Horus'',
``Wedjat'' and ``Sakhmet''. The short excerpts below give a compact
presentation of these passages. We indicate the cases with a clear connection
to LE1 or LE2. Our notation ``NC'' means that there is no clear connection
specifically to either myth.

  * $g(14,2)$

$\equiv\Theta_{\mathrm{Algol}}=6^{\mathrm{o}}$ (Figure 1: ``Horus'') ``Horus
receiving the white crown'' (LE2)

  * $s(5,8)$

$\equiv\Theta_{\mathrm{Algol}}=6^{\mathrm{o}}$ (Figure 2: ``Horus'') ``Horus
is proceeding while Deshret sees his image'' (NC)

  * $g(1,5)$

$\equiv\Theta_{\mathrm{Algol}}=13^{\mathrm{o}}$ (Figure 1: ``Wedjat'',
``Sakhmet'') ``Re … Sakhmet, … pacify the Wedjat''  (LE1)

  * $g(19,12)$

$\equiv\Theta_{\mathrm{Algol}}=13^{\mathrm{o}}$ (Figure 1: ``Horus'') ``This
eye of Horus has come, is complete, is uninjured'' (LE1)

  * $g(27,1)$

$\equiv\Theta_{\mathrm{Algol}}=19^{\mathrm{o}}$ (Figure 1: ``Horus'') ``Peace
between Horus and Seth'' (LE2)

  * $g(24,3)$

$\equiv\Theta_{\mathrm{Algol}}=19^{\mathrm{o}}$ (Figure 1: ``Horus'') ``giving
his throne to his son Horus'' (LE2)

  * $g(1,7)$

$\equiv\Theta_{\mathrm{Algol}}=32^{\mathrm{o}}$ (Figure 1: ``Horus'')
``entering into heaven. The two banks of Horus rejoice'' (NC)

  * $g(27,3)$

$\equiv\Theta_{\mathrm{Algol}}=38^{\mathrm{o}}$ (Figure 1: ``Horus'')
``Judgement between Horus and Seth. Stopping the fight'' (LE2)

  * $g(15,11)$

$\equiv\Theta_{\mathrm{Algol}}=38^{\mathrm{o}}$ (Figure 1: ``Horus'') ``Horus
hears your words in front of every god''  (NC)

  * $g(1,9)$

$\equiv\Theta_{\mathrm{Algol}}=51^{\mathrm{o}}$ (Figure 1: ``Horus'') ``Feast
of Horus, the son of Isis''  (NC)

  * $g(3,2)$

$\equiv\Theta_{\mathrm{Algol}}=57^{\mathrm{o}}$ (Figure 1: ``Wedjat'') ``Re …
gave the inscription of pacification of Wedjat-eye'' (LE1)

  * $g(7,9)$

$\equiv\Theta_{\mathrm{Algol}}=88^{\mathrm{o}}$ (Figure 1: ``Horus'')
``followers of Horus … in the foreign land'' (LE1 or LE2)

  * $g(28,3)$

$\equiv\Theta_{\mathrm{Algol}}=164^{\mathrm{o}}$ (Figure 1: ``Horus'') ``the
making of will for Horus'' (LE1 or LE2)

  * $g(1,10)$

$\equiv\Theta_{\mathrm{Algol}}=240^{\mathrm{o}}$ (Figure 1: ``Horus'')
``Horus… Osiris… Chentechtai… Land'' (NC)

  * $s(26,1)$

$\equiv\Theta_{\mathrm{Algol}}=253^{\mathrm{o}}$ (Figure 2: ``Horus'') ``day
of fighting between Horus and Seth'' (LE2)

  * $s(11,11)$

$\equiv\Theta_{\mathrm{Algol}}=253^{\mathrm{o}}$ (Figure 2: ``Horus'') ``the
eye of Horus raging in front of Re'' (LE1)

  * $s(10,6)$

$\equiv\Theta_{\mathrm{Algol}}=259^{\mathrm{o}}$ (Figure 2: ``Wedjat'')
``coming forth of Wedjat'' (LE1)

  * $s(27,8)$

$\equiv\Theta_{\mathrm{Algol}}=265^{\mathrm{o}}$ (Figure 2: ``Sakhmet'')
``majesty of Sakhmet violates'' (LE1)

  * $g(16,4)$

$\equiv\Theta_{\mathrm{Algol}}=278^{\mathrm{o}}$ (Figure 1: ``Sakhmet'') ``day
of the feast of Sakhmet and Bastet'' (NC)

  * $s(13,6)$

$\equiv\Theta_{\mathrm{Algol}}=278^{\mathrm{o}}$ (Figure 2: ``Sakhmet'')
``arrival of Sakhmet … ``slaughterer-demons''… loose'' (NC)

  * $s(7,10)$

$\equiv\Theta_{\mathrm{Algol}}=278^{\mathrm{o}}$ (Figure 2: ``Sakhmet'')
````slaughterer-demons'' ▲of Sakhmet'' (NC)

  * $g(23,7)$

$\equiv\Theta_{\mathrm{Algol}}=291^{\mathrm{o}}$ (Figure 1: ``Horus'') ``Feast
of Horus in Athribis'' (NC)

  * $g(29,3)$

$\equiv\Theta_{\mathrm{Algol}}=291^{\mathrm{o}}$ (Figure 1: ``Horus'') ``white
crown is given to Horus … red one to Seth'' (LE2)

  * $s(20,9)$

$\equiv\Theta_{\mathrm{Algol}}=291^{\mathrm{o}}$ (Figure 2: ``Horus'')
``angered on the island … inspected by … Horus.'' (NC)

  * $g(9,5)$

$\equiv\Theta_{\mathrm{Algol}}=303^{\mathrm{o}}$ (Figure 1: ``Sakhmet'')
``gods are joyful over the matter of Sakhmet'' (LE1)

  * $g(30,10)$

$\equiv\Theta_{\mathrm{Algol}}=303^{\mathrm{o}}$ (Figure 1: ``Wedjat'')
``coming forth of Shu to bring back Wedjat'' (LE1)

  * $g(29,5)$

$\equiv\Theta_{\mathrm{Algol}}=309^{\mathrm{o}}$ (Figure 1: ``Sakhmet'')
``sole mistress Sakhmet the great … gods are pleased'' (NC)

  * $g(18,1)$

$\equiv\Theta_{\mathrm{Algol}}=322^{\mathrm{o}}$ (Figure 1: ``Horus'')
``magnifying the majesty of Horus over his brother'' (LE2)

  * $g(6,9)$

$\equiv\Theta_{\mathrm{Algol}}=322^{\mathrm{o}}$ (Figure 1: ``Wedjat'') ``they
catch Wedjat together with their followers'' (LE1)

The lucky prognoses of the days related to the main protagonists of LE1,
``Horus'', ``Wedjat'' and ``Sakhmet'', had a strong impact on the
$P_{\mathrm{Algol}}$ signal (Jetsu and Porceddu 2015, p8). All time points of
these three deities have extremely similar $\Theta_{\mathrm{Algol}}$
distributions (Figures 1 and 2). The lucky prognoses are centered at Aa at
$\Theta_{\mathrm{Algol}}=0^{\mathrm{o}}$ in the middle of the brightest phase
of Algol (Figure 1). The unlucky prognoses of these three SWs concentrate
close to Ad at $\Theta_{\mathrm{Algol}}=270^{\mathrm{o}}$, immediately after
Algol's eclipse (Figure 2).

The passages between $\Theta_{\mathrm{Algol}}=6^{\mathrm{o}}$ and
$51^{\mathrm{o}}$ describe feasts or peaceful actions by ``Horus'', ``Wedjat''
and ``Sakhmet''. Those regarding the crowning or judgement of ``Horus'' are
related to LE2. The bringing back of ``Wedjat'' is related to LE1, as well as
the pacifications of ``Sakhmet'' or ``Wedjat''. The only exception is ``Horus
is proceeding while Deshret sees his image'' at
$\Theta_{\mathrm{Algol}}=6^{\mathrm{o}}$ which might not be directly related
to either LE1 or LE2. Notably, this unlucky date close to point Aa is the only
unlucky time point that deviates from the other seven unlucky time points
concentrated close to Ad in Figure 2.

The text at $\Theta_{\mathrm{Algol}}=57^{\mathrm{o}}$ refers to the order that
``Re'' gave in LE1 to save mankind from the wrath of ``Wedjat'' (Eye of
``Horus''), synonymous with ``Sakhmet''. As Algol's eclipse is approaching at
$\Theta_{\mathrm{Algol}}=88^{\mathrm{o}}$, ``Horus'' enters ``the foreign
land'' (LE1 or LE2). His ``will is written'' at
$\Theta_{\mathrm{Algol}}=164^{\mathrm{o}}$ (LE1 or LE2). This particular time
point $g(28,3)$ coincides with the moment when Algol's eclipse can be observed
with naked eye (Figure 1: thick curved line at Ac).

The interval between $\Theta_{\mathrm{Algol}}=253^{\mathrm{o}}$ and
$\Theta_{\mathrm{Algol}}=291^{\mathrm{o}}$ is filled with descriptions of
anger and aggression by ``Horus'', ``Wedjat'' and ``Sakhmet'', or contains
hostile elements such as the ``slaughterer-demons''. These specific demons
were believed to punish mankind on behalf of ``Sakhmet'' and were considered
the cause of diseases and symptoms whose pathology was not well understood
(Lucarelli 2010). Some texts are clearly related to the transformation of
``Wedjat'' into the raging ``Sakhmet'' (LE1). Two out of the three prognoses
at $\Theta_{\mathrm{Algol}}=291^{\mathrm{o}}$, are lucky. After these three,
all remaining texts are lucky, i.e. there are only good prognoses in the end
of Algol's cycle. This suggests the pacification of ``Sakhmet'', as in LE1. ``
The coming forth of Shu to bring back Wedjat'' is mentioned at
$\Theta_{\mathrm{Algol}}=303^{\mathrm{o}}$ (LE1). ``Shu'' succeeds in his
intention, `` they catch Wedjat together with their followers'', at
$\Theta_{\mathrm{Algol}}=322^{\mathrm{o}}$ (LE1). Then, the same stories begin
anew at $\Theta_{\mathrm{Algol}}=6^{\mathrm{o}}$.

In short, all the texts of ``Wedjat'' and ``Sakhmet'' that were not discussed
earlier (Jetsu and Porceddu 2015), also support the idea that principle II was
used to assign the prognoses of CC. The order of List 1 texts connected to LE1
more or less follows the plot of LE1, but the order of texts connected to LE2
does not follow the plot of LE2. This is not unexpected, because several
events are related to ``Seth'', who is not connected to the
$P_{\mathrm{Algol}}$ signal (Jetsu and Porceddu 2015).

The ``third principle in assigning Lucky and Unlucky Days'' could have been

  * principle III: Use LE2 for indirect description of the lunar phases (List 2).

The unlucky prognosis texts of ``Seth'' and ``Osiris'' support this idea.
These prognoses are very clearly concentrated to the dark phases of the Moon
(Figure 4: open triangles and closed circles). The descriptions from List 2
are

  * $s(12,2)$

$\equiv\Theta_{\mathrm{Moon}}=100^{\mathrm{o}}$ (Figure 4: ``Seth'') ``His
head, who did rebel against his lord, is cut off''

  * $s(13,3)$

$\equiv\Theta_{\mathrm{Moon}}=117^{\mathrm{o}}$ (Figure 4: ``Seth'' and
``Osiris'') ``day of severing''

  * $s(14,3)$

$\equiv\Theta_{\mathrm{Moon}}=129^{\mathrm{o}}$ (Figure 4: ``Osiris'') ``gods
are sad over the action against Osiris' place''

  * $s(11,12)$

$\equiv\Theta_{\mathrm{Moon}}=137^{\mathrm{o}}$ (Figure 4: ``Seth'')
``repelled the confederacy of Seth to the eastern desert''

  * $s(14,5)$

$\equiv\Theta_{\mathrm{Moon}}=139^{\mathrm{o}}$ (Figure 4: ``Osiris'')
``Weeping of Isis and Nephthys''

  * $s(18,3)$

$\equiv\Theta_{\mathrm{Moon}}=178^{\mathrm{o}}$ (Figure 4: ``Seth'') ``tumult
by the children of Geb: Seth and his sister Nephthys'''

  * $s(17,7)$

$\equiv\Theta_{\mathrm{Moon}}=185^{\mathrm{o}}$ (Figure 4: ``Seth'') ``Do not
speak the name of Seth on this day''

  * $s(17,8)$

$\equiv\Theta_{\mathrm{Moon}}=190^{\mathrm{o}}$ (Figure 4: ``Seth'') ``The
going of Seth, … they repelled his followers''

  * $s(19,4)$

$\equiv\Theta_{\mathrm{Moon}}=195^{\mathrm{o}}$ (Figure 4: ``Osiris'')
``Making of ointment for Osiris''

  * $s(20,2)$

$\equiv\Theta_{\mathrm{Moon}}=197^{\mathrm{o}}$ (Figure 4: ``Seth'') ``The
rebels against their lord were overthrown''

  * $s(26,1)$

$\equiv\Theta_{\mathrm{Moon}}=266^{\mathrm{o}}$ (Figure 4: ``Horus'' and
``Seth'') ``day of fighting between Horus and Seth''

  * $s(24,8)$

$\equiv\Theta_{\mathrm{Moon}}=275^{\mathrm{o}}$ (Figure 4: ``Seth'') ``Do not
pronounce the name of Seth''

Seth's rebellion is first referred to at
$\Theta_{\mathrm{Moon}}=100^{\mathrm{o}}$. It is followed by a long story of
``the day of severing'' on $s(13,3)$. On the next day $s(14,3)$, ``gods are
sad over the action against Osiris's place''. ``Seth'' is repelled in a tumult
($\Theta_{\mathrm{Moon}}=137^{\mathrm{o}}$). The goddesses Isis and Nephthys
are mourning the death of ``Osiris''
($\Theta_{\mathrm{Moon}}=139^{\mathrm{o}}$). ``Seth'' causes another tumult
($\Theta_{\mathrm{Moon}}=178^{\mathrm{o}}$). CC advises not to pronounce his
name at New Moon ($\Theta_{\mathrm{Moon}}=185^{\mathrm{o}}$). His influence
begins to wane ($\Theta_{\mathrm{Moon}}=190^{\mathrm{o}}$). ``Making of
ointment for Osiris'' follows at $\Theta_{\mathrm{Moon}}=195^{\mathrm{o}}$.
``Seth'' is overthrown and judged ($\Theta_{\mathrm{Moon}}=197^{\mathrm{o}}$).
Yet, ``Seth'' and ``Horus'' continue their fight
($\Theta_{\mathrm{Moon}}=266^{\mathrm{o}}$). In the end, CC advises not to
pronounce the name of ``Seth'' ($\Theta_{\mathrm{Moon}}=275^{\mathrm{o}}$).
All these texts are connected only to LE2. They mostly follow the plot of LE2
in the order of $\Theta_{\mathrm{Moon}}$, but there are contradictions. For
example, the beginning of the cycle would be the expected place for the fight
at $\Theta_{\mathrm{Moon}}=266^{\mathrm{o}}$.

``Osiris'' is also connected to LE2 in the CC. Therefore, it is logical that
the unlucky texts of ``Osiris'' were connected to $P_{\mathrm{Moon}}$ (Jetsu
and Porceddu 2015, p18).

The three previously discussed lucky prognoses on $g(27,1),g(27,3)$ and
$g(29,3)$ mentioning both ``Seth'' and ``Horus'' show a reconciliation (LE2)
when Algol was at its brightest and the Moon was waxing gibbous after Md
$(\Theta_{\mathrm{Moon}}=270^{\mathrm{o}})$.

It would not be logical to study the texts of ``Seth'' in connection with LE1,
since he is not one of the protagonists in the myth. ``Horus'' personally
plays a part only in LE2, but his Eye is a protagonist in LE1. Neither the
lucky, nor unlucky, prognoses of ``Horus'' were connected to
$P_{\mathrm{Moon}}$ (Jetsu and Porceddu 2015, p7), i.e. they were randomly
distributed as a function of $\Theta_{\mathrm{Moon}}$ (Figures 3 and 4: closed
squares).

Because ``Horus'' is present as a name in both LE1 and LE2, the same texts
mentioning ``Horus'' are repeated when arranged in the order of increasing
$\Theta_{\mathrm{Algol}}$ (List 1) and when arranged in the order of
increasing $\Theta_{\mathrm{Moon}}$ (List 2). This means that we have (or the
scribes had) a choice to interpret those texts either in relation to the cycle
of Algol or the cycle of the Moon. We do not know if the scribes faithfully
repeated always the same mythological texts in relation to the cycle of Algol,
the cycle of the Moon or both of these cycles. However, if they did, these
``Horus'' texts may have a double relation.

The 1st example of a double relation to both Algol and Moon, or to LE1 and
LE2, is the text

  * $g(28,3)$

``The gods are in jubilation and in joy over the making of will for Horus, son
of Osiris, to pacify Onnophris in the underworld.''

At $\Theta_{\mathrm{Algol}}=164^{\mathrm{o}}$, it can describe the return of
the lost Eye of Horus during the eclipse of Algol (LE1), but more likely it
refers to the will made by ``Osiris'' that raises ``Horus'' to be the ruler of
Egypt at $\Theta_{\mathrm{Moon}}=300^{\mathrm{o}}$ (LE2). It is also related
to the time of gestation and infancy of ``Horus'', when he was hidden from
enemies' sight by his mother Isis. All of these three alternative
interpretations symbolize the cyclic rejuvenation of royal power over Egypt.

The 2nd example is the text

  * $g(7,9)$

``The crew and followers of Horus have assembled in the foreign land, to make
known that Horus smites him who rebels against his lord.''

On this day, the phase angles of Algol and the Moon are nearly equal
$\Theta_{\mathrm{Algol}}=88^{\mathrm{o}}$ and
$\Theta_{\mathrm{Moon}}=73^{\mathrm{o}}$. The smiting of the rebels is equally
applicable to punishing the mankind for its wicked ways (LE1) and to the
battle against ``Seth'' with his followers (LE2).

However, only the unlucky texts of ``Seth'' and ``Osiris'' clearly follow the
principle III, and perhaps the three lucky texts mentioning both ``Horus'' and
``Seth''. Many of the remaining ``Horus'', ``Seth'' or ``Osiris'' texts may
have more complicated connections to LE1 and LE2, or to other myths.

We can confirm principle I which connects the lucky prognoses to the moments
in the middle of the bright phases of Algol and the Moon. More tentatively, we
present principles II and III, i.e. a specific connection of the LE1 and LE2
myths to the phases of Algol and the Moon. The hemerological tradition had
already existed for centuries, if not millennia, and accordingly had
accumulated into itself cultural layers from different historical eras.
However, this uncertainty in principles does not alter our main result that
the scribes connected the texts of the two legends LE1 and LE2 to the phases
of Algol and the Moon.

Argument VIII: The texts of List 1 ($\Theta_{\mathrm{Algol}}$ order) and List
2 ($\Theta_{\mathrm{Moon}}$ order) show that the LE1 and LE2 legends could
have been used to describe indirectly the regular changes of Algol and the
Moon.

###  4.9 Rejuvenation and kingship

Rejuvenation, the power to disappear and reappear, was associated with
``Horus''.

Everything in the ancient Egyptian worldview was repeating in a cyclic manner:
the sky, the celestial phenomena, the Nile, the winds, the clouds, the
migrating birds and fish, all life in its individual or holistic sense (Leitz
1994, p452-459). Even the highest power in Egypt, the divine king, was subject
to a continuous cycle, first representing ``Horus'' on Earth and then
``Osiris'' in afterlife. The ancient Egyptian deities were subject to the same
eternal rules of recurrence.

The discovery of Algol's variability might have first astonished the scribes.
Had the cause of variability been non-repeating, such as supernova, they would
have been baffled by it, but it must have been reaffirming to them to notice
the regularity of Algol's variability, well suited to their worldview: the
idea of the continuous struggle between the forces of chaos and order (Magli
2013, p2). It must have taken some effort to incorporate this new phenomenon
into their religion and mythology, but evidently they were able to do that.

It was probably of utmost importance to the scribes to determine the period of
this phenomenon, because this would have allowed them to interpret correctly
the divine events relating to it and also to incorporate these events into
their explanation of cosmos.

We must remember that the ancient Egyptians did not practice natural sciences
in the modern sense, but expressed their worldview and all observable
phenomena in the context of religion and myths (von Lieven 2000, p188). Many
of their discoveries were known only by the experts of religion and magic, the
scribes, who could interpret the indirect mythological descriptions of the
observed phenomena. The intention of the writers of CC was not necessarily to
prevent outsiders from understanding the connections to Algol and the Moon,
but for three millennia their indirect mythological references have hidden
from sight the basic periodic principles of assigning Lucky and Unlucky Days
(i.e. principles I, II and III).

The idea of rejuvenation was important in ancient Egyptian mythology. The Moon
was used as a symbol of rejuvenation, called ``the one that repeats its form''
(Kaper 2001, p480-482). Likewise, a vanishing and reappearing star would have
been suggestive of the restoration of the eye of ``Horus'' alongside his
kingship (Allen 2005, p6), (Edwards 1995, p233-240) and (Troy 1989, p132-133),
and reaffirming of the Egyptians' cyclic worldview. The eyes of ``Horus'' were
associated with the crowns of the kings, as symbols of sovereignty (Griffiths
2001, p476-480). Hence, a regularly variable star would naturally be linked
with the divine cycles ( ``Horus'' and his eye), much like the Moon was seen
to have rejuvenative power (Kaper 2001, p480-482). Algol's eclipses could have
been considered the blinding of the eye of ``Horus'' by ``Seth'', but on the
other hand the return of the Eye of ``Horus'' would have reaffirmed the
restorative, life-giving powers of the gods and kings of Egypt. ``Horus'' or
his eye are supposed to have been linked with varying celestial objects
depending on the context. Krauss (2016, p137-139) interpreted the Eye of Horus
in CC to be Venus because the narration regarding it can also be taken to mean
the yearly absence of Venus from the night sky as it is transformed from a
morning star into an evening star. The dramatic changes in the brilliancy of
Venus would have been indicative of divine fighting, injuring and rivalry.
Additionally, Krauss suggests that the stars called sehed in the Pyramid Texts
such as Horus are planets because the text attributes to them the ability to
move freely but could this also include the ability to vanish from the sky? It
is logical that Algol would have been represented in mythological texts as
``Horus'' or his eye, as were most of the planets in the known Egyptian
astronomical texts (Clagett 1995, p114). Considering the significance of the
name ``Horus'' to the periodic signal, the ancient Egyptians could have
believed in some connection between Algol, royal power and the cosmic order.

Argument IX: Algol could have been naturally associated with ``Horus'' and
called as such, because Algol can disappear and reappear.

###  4.10 Astrophysical evidence

The period of Algol must have been shorter three millennia ago (Kwee 1958;
Biermann and Hall 1973; Soderhjelm 1975).

It is a fascinating idea that the ancient Egyptians would have discovered
Algol's variability over three thousand years ago, noticed its regularity,
determined its period and incorporated this phenomenon into their mythology.
Our modern interpretations of their concepts are mostly circumstantial. The
interpretation of ancient Egyptian texts is complex when the phenomena are not
everyday concrete matters such as agriculture, climate, weather, time keeping,
geography or geometry. Although the CC deals with astronomy only indirectly,
it contains evidence that the scribes made recordings of a concrete phenomenon
later discovered by modern natural science: the regular changes of the
eclipsing binary Algol. Several astronomical and astrophysical considerations
(Jetsu et al. 2013; Jetsu and Porceddu 2015) support the idea that their
prolonged naked eye observations revealed the same discoveries of Algol that
Montanari (variability) and Goodricke (regular variability) made about three
millennia later.

Naked eye can discover regular variability in the Sun, the Moon, the planets
and the variable stars. Jetsu et al. (2013) formulated eight astronomical
criteria which showed that only the periods of the Moon and Algol could be
discovered from CC, and it was exactly these two periods that they
rediscovered. Their period analysis also showed that 2.850 days is the
strongest real periodicity governing the assignment of the lucky prognoses in
CC, after the lucky prognoses connected to the synodic lunar month are
removed. The mass transfer from Algol B to Algol A is a well established
phenomenon (Soderhjelm 1975; Sarna 1993). This mass transfer should cause a
period increase (Kwee 1958; Biermann and Hall 1973). Yet, no-one had confirmed
the presence of this phenomenon in over 230 years, since Goodricke determined
the period of Algol in 1783. The period of 2.850 days in CC is 0.017 days
shorter than the current orbital period of Algol, 2.867 days. The mass
transfer between Algol B and A could have caused this period increase during
the past three millennia. The required mass transfer rate (Jetsu et al. 2013,
p8) was in excellent agreement with the predictions of the best evolutionary
model of Algol (Sarna 1993, p540). Furthermore, it was shown that Algol's
inclination has remained stable (Zavala et al. 2010; Baron et al. 2012), i.e.
eclipses occurred also in that historical era and the ancient Egyptians would
have been able to record these events (Jetsu et al. 2013).

Argument X: Astrophysical considerations support the idea that the 2.850 days
period in CC can be the period of Algol.

###  4.11 Cultural evidence and lack of it

Our arguments prove that the ancient Egyptians could have recorded Algol's
period into CC, but the very same arguments do not definitively prove that
they did so. However, such cultural aspects are not important for all of our
arguments. We will first discuss the arguments that pose no problems, and then
those that do.

There is cultural evidence about the ``hour-watchers'', as well as about the
connection between their observations and the religious rituals (Arguments I
and II). Algol's variability could have been easily observed in any ancient
culture, unless the geographical location of this culture prevented
observations (Argument III). Many excerpts in List 1 and List 2 are
definitively connected to the LE1 and LE2 legends (Argument VII). The order of
these excerpts in our Lists 1 and 2 is based on statistical analysis, not on
cultural aspects. It is therefore not accidental that this order makes sense
(Argument VIII). Statistical, astronomical and astrophysical evidence
(Porceddu et al. 2008; Jetsu et al. 2013; Jetsu and Porceddu 2015) supports
our last argument (Argument X).

If the ``hour-watchers'' noticed Algol's variability and just recorded the
observed eclipses into CC, then modern period analysis would detect the 2.850
days periodicity (Arguments IV and V). If the references to Algol and the Moon
in CC are indeed indirect, it is, and it will be, very difficult to find any
definitive cultural proof about Argument VI. Statistical analysis by Jetsu and
Porceddu (2015) has revealed a connection between the 2.850 days period and
``Horus'', but our cultural interpretation can be questioned (Argument IX).

We do admit that a specific identification of Algol is missing (FAQ 7) but
this is a general problem regarding all but a few stars and planets. The
Pyramid Texts make it obvious that Horus is a star, but the identification of
the star is a matter of debate. Late Period texts identify Horus-son-of-Isis
as god of the morning (mentioned as a star also in Coffin Texts from a much
older period, the First Intermediate Period), from which Krauss (2016,
p137-141) concluded that Horus-son-of-Isis and Haroeris (the elder Horus) are
Venus as morning star and evening star. This does not exclude other
interpretations, particularly since we know that many other celestial objects
received the title or association to Horus. Descriptions such as ‘Horus who
ascends as gold from upon the lips of the akhet’ (Coffin Texts 255), are
applicable to all but circumpolar stars. Krauss analyzed other passages as
well from Coffin Texts and the Book of the Dead which prove the association of
the Eye of Horus to something else than the Sun or the Moon. It would be
worthwhile to also study the celestial diagrams of the Late Period which
include Greek and Mesopotamian influences, to discover in greater detail the
connection of the Greek constellation of Perseus to the ancient Egyptian star
names.

##  5 Conclusions

We have presented ten arguments which show that the ancient Egyptian scribes,
the ``hour-watchers'', had the possible means and the motives for recording
the period of Algol in CC. Those arguments are combined here.

  * Argument I: For thousands of years, the ``hour-watchers'' practiced the tradition of timekeeping by observing hour-stars. If Algol was not an hour-star, it certainly belonged to some hour-star pattern or related constellation.

The scribes observed regularly about 70 bright stars, or most probably a lot
more, in a region where there are about 300 clear nights every year. They
practiced this tradition for the timing of religious rituals.

  * Argument II: Proper timing of the nightly religious rituals relied on the fixed hour-star patterns.

The ``hour-watchers'' probably discovered Algol's variability from the changes
that its eclipses caused in its hour-star pattern.

  * Argument III: A naked eye can easily discover the significant hour-star pattern change caused by Algol's eclipse.

These changes followed the regular ``3+3+16=19'' and ``19+19+19=57'' days
cycles.

  * Argument IV: The scribes could have discovered Algol's $2.850=57/20$ days period from long-term observations of the regular 19 and 57 days eclipse cycles.

Three alternative methods could have been used in recording the 2.850 days
period.

  * Argument V: The ancient Egyptian scribes may have calculated the $57/20=2.850$ days period of Algol from long-term observations (1st method). They may not have calculated this 2.850 days period, because the 19 days and 57 days cycles already perfectly predicted all night-time eclipses of Algol (2nd method), or they may have just recorded the observed night-time eclipses into CC (3rd method).

The scribes did not describe Algol's regular changes directly.

  * Argument VI: To avoid violating cosmic order, the scribes would have referred to Algol's changes only indirectly.

Two legends, ``the Destruction of Mankind'' and ``the Contendings of Horus and
Seth'', could have been used to indirectly describe the changes of Algol and
Moon.

  * Argument VII: Even a quick glance on List 1 ($\Theta_{\mathrm{Algol}}$ order) and List 2 ($\Theta_{\mathrm{Moon}}$ order) reveals that numerous CC texts are excerpts from the LE1 and LE2 legends.

These two legends were probably used to describe celestial phenomena as
activity of gods.

  * Argument VIII: The texts of List 1 ($\Theta_{\mathrm{Algol}}$ order) and List 2 ($\Theta_{\mathrm{Moon}}$ order) show that the LE1 and LE2 legends could have been used to describe indirectly the regular changes of Algol and the Moon.

Algol would have been considered as a manifestation of ``Horus''.

  * Argument IX: Algol could have been naturally associated with ``Horus'' and called as such, because Algol can disappear and reappear.

We have presented evidence that the period of Algol in CC was 0.017 days
shorter than today (Porceddu et al. 2008; Jetsu et al. 2013; Jetsu and
Porceddu 2015).

  * Argument X: Astrophysical considerations support the idea that the 2.850 days period in CC can be the period of Algol.

It is not even necessary to present this many arguments to convince the reader
that it would be more complicated to explain the statistically significant and
accurate $2.850\pm 0.002$ days period with something else than Algol. No-one
disputes that Algol's variability is easy to observe. The lack of other
reasonable alternatives is also a part of the evidence. It is not only that
Algol is the simplest explanation but so far no-one has been able to think of
any other reasonable alternative explanation for the 2.850 days periodicity in
CC. If no-one can answer our last Frequently Asked Question (Appendix B: FAQ
11), then the following famous aphorism by the Italian Nobel Prize winner
Luigi Pirandello must be true: a thing ``is so if it seems so'' (Bentley
1986).

We dedicate this paper to the memory of our two co-authors and friends, PhD
Jaana Toivari-Viitala (May 16th, 1964 - May 12th, 2017) and prof. Tapio
Markkanen (Jan 27th, 1942 - Aug 28th, 2017). We acknowledge PhD Patricia Berg,
Dr. Robert J. Demaree, PhD Heidi Jauhiainen, prof. Karri Muinonen and prof.
Heikki Oja for their comments of the text of this article.

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##  Appendix A List of translations to Ancient Egyptian language

##  Appendix B Frequently asked questions (FAQ)

We have rewritten innumerable versions of this manuscript. The list below
contains some of the most frequently asked questions (FAQ) about our research,
as well as our short answers. Some of these questions have arisen from
misunderstandings of our research. However, we want to answer also those
questions, because it was impossible to foresee them coming, or to prepare
answers to them beforehand. The sections and the paragraphs (par), where our
more detailed answers can be found, are given in parenthesis. These answers
are aimed to ensure that our research better stands the test of time and
sceptics. The last question 11 is our own. So far, we have not received a
reasonable alternative answer to this simple question.

  1. 1.

Several CC passages mention different deities, but not all of them are
included in your analysis. How do you know which one of them determines the
calendar date?
Answer: The deities do not determine the dates. The dates are known (Eqs. 1
and 2, Sect. 2.1: 4th and 6th par, Sect. 2.2: 4th last par).

  2. 2.

Your statistical analysis is flawed, because the exact historical dating of CC
is uncertain.
Answer: The results of our statistical analysis do not depend on the zero
epoch in time. The time points within CC are unambiguously defined and
computed (Eqs. 1 and 2). Hence, the results of our statistical analysis would
be the same if CC was dated to ice age, stone age or a million years ahead
into the future (Sect. 2.2: 4th last par).

  3. 3.

Your samples are too small to allow reliable statistical conclusions.
Answer: The ephemerides of Eqs. 5 and 6 are based on large samples of over 500
time points. We use these ephemerides to rearrange the CC passages into the
unambiguous order of Lists 1 and 2. These ephemerides work like an accurate
modern clock when studying Lists 1 and 2. Furthermore, the binomial
significance estimates (Jetsu and Porceddu 2015) $Q_{\mathrm{B}}$ were
reliable even for small samples (Sect. 2.2: 3rd last par).

  4. 4.

Something must be seriously amiss in your method if a term like "Earth" has a
strong correlation with Algol's period.
Answer: The ephemeris of Eq. 5 reveals what the authors of CC wrote when they
were observing Algol at its different phases. Nothing is ``amiss'' in our
method, unless those CC texts were not written on Earth (Sect. 2.2: 3rd last
par). This is natural because feast dates where often described as feasts in
``Earth'' and in ``Heaven''.

  5. 5.

Your CC translations are incomplete.
Answer: We have used the hieroglyphic transcription of Leitz (1994) and his