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Copyright 2000-2002 Andrew Bourmistroff.
All rights reserved.  Reprinted with Permission.

Parameter A and the Egyptian Decans

by Andrew Bourmistroff
Editorial correction by Gilles & Margaret Nullens ( UK )



  • Why ancient Egyptians built up Pyramids ?
  • Was Pyramid symbol of the Sun ?
  • Why ancient Egyptians divided the ecliptic on the 36 Decans only ?


I. Parameter A

In Part I " Hermetic geometry " from my work " Numbers of Thoth " ,  I described how some of my results refer to new properties of Pyramid considered as geometrical figure.

The most important new property of any Pyramid is the maximum difference ( β - α ) :

α - it is the angle of the lateral edge of the  pyramid in relation to its  base ;
β - it is the angle of  the lateral side of the  pyramid in relation to its base ;

The maximum difference ( β - α ) is equal to 9.879 degrees in any Pyramid.
Later I named  the maximum difference ( β - α )
  the universal  Parameter A.   


2. The Egyptian Year and the Decans

It is known that ancient Egyptians used their own  calendar. 

" The ancient Egyptians originally employed a calendar based upon the Moon , and , like many peoples throughout the world , they regulated their lunar calendar by means of the guidance of a sidereal calendar. They used the seasonal appearance of the star Sirius ( Sothis ); this corresponded closely to the true solar year , being only 12 minutes shorter. Certain difficulties arose , however , because of the inherent incompatibility of lunar and solar years. To solve this problem the Egyptians invented a schematized civil year of 365 days divided into three seasons , each of which consisted  of four months of 30 days each. To complete the year , five intercalary days were added at its end , so that the 12 months were equal to 360 days plus five extra days. This civil calendar was derived from the lunar calendar ( using months ) and the agricultural , or Nile , fluctuations ( using seasons ); it was , however , no longer directly connected to either and thus was not controlled by them. The civil calendar served government and administration , while the lunar calendar continued to regulate religious affairs and everyday life. 

In time , the discrepancy between the civil calendar and the older lunar structure became obvious. Because the lunar calendar was controlled by the rising of Sirius , its months would correspond to the same season each year , while the civil calendar would move through the seasons because the civil year was about one-fourth day shorter than the solar year. Hence , every four years it would fall behind the solar year by one day , and after 1,460 years it would again agree with the lunisolar calendar. Such a period of time is called a Sothic cycle." 
[ 1 ]

The 3 seasons are :

  • Aakhet , which correspond to the Nile over flow
  • Pert , which correspond to seed-time
  • Shemu , which correspond to harvest

Aakhet's  season consist of next 4 months :

  • Thoth
  • Paophi
  • Hat-hor
  • Khoiak

Pert's  season consist of next 4 months :

  • Tybi
  • Mekhir
  • Phamenoth
  • Pharmuthi

Shemu's  season consist of next 4 months :

  • Pakhons
  • Pauni
  • Epiphi
  • Mesore

The Epagomenes consist of 5 special days :

  • the first day – Osiris’  day
  • the second day – Horus’  day
  • the third day – Seth’s  day
  • the fourth day – Isis’  day
  • the fifth day – Nephthys'  day


The ancient Egyptians also used an unusual system of time intervals known as the Decans.
The Decans were 36 special bright groups of stars from the sky along the ecliptic. These Decans were used as a special calendar --- each decan would rise above the dawn horizon for ten days every year.

" HERMETICA : EXCERPT VI : From the Discourses of Hermes to Tat.

Tat. In your former General Discourses you promised to explain about the thirty-six Decans; I therefore ask you to tell me about them now, and to explain their working.

Hermes. I am quite willing, Tat; and of all my teachings, this will be of supreme importance, and will stand highest among them. I bid you mark it well.
I have told you before about the zodiacal circle, which is also called the animal-bearing circle, and about the five planet-stars and the sun and the moon, and the several circles of these seven bodies.

Tat. You have, thrice-greatest one.

Hermes. I desire you then, in your thoughts about the thirty-six Decans also, to bear in mind what I have told you, that so my teaching about the Decans also maybe intelligible to you.

Tat. I bear in mind what you have told me, father.

Hermes. I told you, my son, that there is a body which encloses all things. You must conceive the shape of that body as circular; for such is the shape of the universe.

Tat. I conceive its shape as circular, even as you bid me, father.

Hermes. And you must understand that below the circle of this body are placed the thirty-six Decans, between the circle of the universe and that of the zodiac, separating the one circle from the other; they bear up, as it were, the circle of the universe, and look down on the circle of the zodiac.
They retard the all-enclosing body,—for that body would move with extreme velocity if it were left to itself,—but they urge on the seven other circles, because these circles move with a slower movement than the circle of the universe.
And subject to the Decans is the constellation called the Bear, which is centrally situated with regard to the zodiac. The Bear is composed of seven stars, and has overhead another Bear to match it. The function of the Bear resembles that of the axle of a wheel; it never sets nor rises, but abides in one place, revolving about a fixed point, and making the zodiacal circle revolve, transmitting the world from night to day, and from day to night.
Let us understand then that both the ... of the seven planets and all... ;or rather, that the Decans stand round about all things in the Cosmos as guardians, holding all things together, and watching over the good order of all things. 

Tat. Even so I conceive them, father, according to your words.

Hermes. And further, my son, you must understand that the Decans are exempt from the things that befall the other stars. They are not checked in their course and brought to a standstill, nor hindered and made to move backwards, as the planets are; nor yet are they as are the other stars. They are free, and exalted above all things; and as careful guardians and overseers of the universe, they go round it in the space of a night and a day.

Tat. Tell me then, father, do the Decans act on us men also?

Hermes. Yes, my son, they act on us most potently. If they act on the heavenly bodies, how could it be that they should not act on us also, both on individual men and on communities The force which works in all events that befall men collectively comes from the Decans; for instance, overthrows of kingdoms, revolts of cities, famines, pestilences, overflowings of the sea, earthquakes,— none of these things, my son, take place without the working of the Decans.For if the Decans rule over the seven planets, and we are subject to the planets, do you not see that the force set in action by the Decans reaches us also, whether it is worked by the Decans themselves or by means of the planets?
And besides this, my son, you must know that there is yet another sort of work which the Decans do; they sow upon the earth the seed of certain forces, some salutary and others most pernicious, which the many call daemons.

Tat. And what is the bodily form of these beings, father?

Hermes. They do not possess bodies made of some special kind of matter, nor are they moved by soul, as we are; for there is no such thing as a race of daemons distinct from other beings; but they are forces put in action by these six and thirty gods. "  [ 2 ]


3.  36 Pyramids in the Sky 

Taking into consideration that a circle is divided in 360 degrees and that the Egyptian year has 365 days , we can say that the Earth rotates , in average :

360 / 365 = 0.9863 degrees each day.

Or , in other words , the Sun has an average shift in the sky along the ecliptic as 0.9863 degrees per day.

By the way , the movement of the Earth in its orbit around the Sun, does not proceed at a constant rate because of the eccentricity of the orbit of the Earth. 
The maximum rate is 1.0020 degrees per  day ( on 4 January )  while the minimum rate is 0.9701 degrees per day (  on 4 July)  ---- with modern value of eccentricity ( 0.0167 ) .

From my previous data we saw that the Sun has average shift in the sky of 9.863 degrees per ten days or per one Egyptian Decan.

Look again on this data :

The Decans

Parameter A  of  any  Pyramid  =  9. 879 degrees
One Egyptian Decan ( in angular meaning )  =  9. 863 degrees



On this base I conclude that :

  • Pyramids were sacred symbols of the Sun ( Re ) for ancient Egyptians
  • Pyramids were sacred symbols of the Egyptian Decan as general time interval 
    in Egyptian year
  • Ancient Egyptians believed that the ecliptic ( The Path of the Sun ) consisted of 36 Decans (Pyramids) plus 5 additional days 
  • Ancient Egyptians knew the meaning of the Parameter A of  Pyramids and that a circle was divided in 360 degrees

Enclosure # 1 : Definition of Year [ 3 ]

  •  A Year is time required for the Earth to travel once around the Sun, about 365 1/4 days. This fractional number makes necessary the periodic intercalation of days in any calendar that is to be kept in step with the seasons. In the Gregorian calendar a common year contains 365 days, and every fourth year (with a few exceptions) is a leap year of 366 days.
  • The Solar Year is year (365 days 5 hours 48 minutes 46 seconds), also called Tropical Year, or year of the seasons, is the time between two successive occurrences of the vernal equinox (the moment when the Sun apparently crosses the celestial equator moving north). Because of the precession of the equinoxes (an effect of a slow wobble in the Earth's rotation), the solar year is shorter than the Sidereal Year (365 days 6 hours 9 minutes 10 seconds), which is the time taken by the Sun to return to the same place in its annual apparent journey against the background of the stars. 
  • The Anomalistic Year (365 days 6 hours 13 minutes 53 seconds) is the time between two passages of the Earth through  perihelion, the point in its orbit nearest the Sun. 
  • A Lunar Year (used in some calendars) of 12 synodic months (12 cycles of lunar phases) is about 354 days long.
  • A Cosmic Year is the time (about 225 million years) needed for the solar system to revolve once around the centre of the Milky Way Galaxy. 

Enclosure # 2 : Definition of the Decans [ 4 ]

  • The Decans are 36 star configurations circling the sky somewhat to the south of the ecliptic. They make their appearance in drawings and texts inside coffin lids of the 10th dynasty (around 2100 BC) and are shown on the tomb ceilings of Seti I (1318-04 BC) and of some of the Rameses in Thebes. The decans appear to have provided the basis for the division of the day into 24 hours.

    Besides representing star configurations as decans, the Egyptians marked out about 25 constellations, such as crocodile, hippopotamus, lion, and a falcon-headed god. Their constellations can be divided into northern and southern  groups, but the various representations are so discordant that only three constellations have been identified with certainty: Orion (depicted as Osiris), Sirius (a recumbent cow), and Ursa Major (foreleg or front part of a bull). The most famous Egyptian star map is a 1st-century-BC stone chart found in the temple at Dandarah and now in the Louvre. 



[ 1 ]  Encyclopaedia Britannica , 1996.

[ 2 ]  Hermetica ,  p.158 - 160 , Solos Press , 1997.

[ 3 ]  Encyclopaedia Britannica , 1996.

[ 4 ]  Ibid.

Copyright 2003 by Andrew Bourmistroff

All rights reserved.  Reprinted with Permission.


Has the Great Pyramid shifted by 9.85 degrees ? 

By  Andrew  Bourmistroff

Editorial correction by Graham Russell (UK)

The important fact to any researcher is the availability of  precise & true geodesic information which refers to  interesting object.
Doubtlessly such information is in the works by W.M.F.Petrie [1] & J.H.Cole [2] it refers  to the Complex Giza and the Great Pyramid.
For instance, we have next Cole's data :

Table I The Great Pyramid :     The length of the sides
Side North East South West Average
meters 230,253 230,391 230,454 230,357 230,364


Table II The Great Pyramid :     The errors of the 90 ° corners


North - West South - West South - East North - East


- 0 ' 2 " + 0 ' 33 " - 3 ' 33 " + 3 ' 2 "


Table III The Great Pyramid :     The direction of the sides
Side North East South West
direction 0°2'30"  W. of true N. 0°1'57"  N. of true E. 0°5'30"  W. of true N. 0°2'28"  N. of true E.

Concerning to geometry of the Great Pyramid my attention was attracted by fresh work "The Pyramid Paper"  by Terrans Nevin (USA).
He used his own mathematical software for the analysis of the geometry of the Great Pyramid on basis of Cole's data.
In May 1995  he found the so-called special "Four Circles".
Then in March 1997  a mysterious line was  found by Dave Seymour and Terrans Nevin. No satisfactory explaination has yet be found for its line.

Four corners of the base and the apex of the Great Pyramid have a geometrical and constructive shift from each other.
The Great Pyramid has a hidden azimuth of 350.15 degrees (or 9.85 degrees West of true North).

At once I saw that the value 9.85 degrees is very close to my Parameter A of  9.879 degrees.

As you remember, the maximum difference (β - α) or Parameter A is equal to 9.879 degrees of  any Pyramid, where :

α - it is an angle of an inclination of a lateral edge of the  pyramid and it's  base ;

β - it is an angle of an inclination of a lateral side of the  pyramid and it's base ;

But is it possible to aggregate together these two meanings ?

Because the speech goes about only idealized meaning of  9.879 degrees and about the meaning of  9.85 degrees obtained after due consideration of the practical geometry of  Cole's measurements.

It is known from plane analytic geometry that the equation of a straight line is called a linear equation with variable parameters ( x ) and ( y ) to which the coordinates at any point of this straight line may be found .
The general equation of this type is defined as ax + by + c = 0 ( 1 )  and is called as a general equation of a straight line .

The equation of straight line, allowing for a variable ( y ) , i.e. equation of type y = kx + b ( 2 ) is called as an equation with an angular factor . Parameter ( k ) is called an angular factor and it is proportional  to the tangent of the angle of the slope of the straight line to the axis ( ox ) .

k = tan φ ( 3 )

Parameter ( b ) is the value of a section cut by a straight line ( 2 ) . It is considered from the origin of  the coordinates on the axis ( oy ) .

The equation of  type x / a + y / b = 1 ( 4 )  where ( a ) and ( b ) are values of sections cut by a straight line on axes of the coordinates ( Figure 1 ) is known  as a form of the equation of a straight line in sections .

     Figure 1

Figure 1

The angle ( Φ )  between two straight lines  y = kx + b ( 5 ) and  y1 = k1x + b1 ( 6 )  is the angle on which it is necessary to turn the first straight line (with an angular factor ( k ) ) to coincide it with the second straight line (with an angular factor ( k1 ) ) counter clockwise ( Figure 2 ) .


Figure 2

Figure 2

This angle is calculated under using  the next formula :

tan Φ  = (k1 - k) / (1 + k1 k )  ( 7 )

Now we shall consider the formula as a function of an angular difference in trigonometry :

tan (β - α) = (tan β - tan α) / (1 + tan β  tan α)    ( 8 )

The conformity between two formulas is certain :

Φ =  (β - α) = Parameter A

k1 = tan β 

k =  tan α

Hence we can confirm the connection between the hidden  azimuth 9.85 degrees (West of true North) and the universal Parameter A  of the Pyramid equal 9.879 degrees  (Figure 3).

Figure 2


Figure 3



1. W.M.F. Petrie , The Pyramids and Temples of Gizeh , London , Field & Tuer , 1883.

2. J. H. Cole , Determination of the Exact Size and Orientation of the Great Pyramid of Giza , Cairo , Government Press , 1925.

Copyright 2003 by Andrew Bourmistroff

All rights reserved.  Reprinted with Permission.


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