Pyramid Quest (42 page)
Authors: Robert M. Schoch
Tags: #History, #Ancient Civilizations, #Egypt, #World, #Religious, #New Age; Mythology & Occult, #Literature & Fiction, #Mythology & Folk Tales, #Fairy Tales, #Religion & Spirituality, #Occult, #Spirituality
Tastmona (1954) and Temple (2000; see also 2001) have both suggested that the ancients, and the ancient Egyptians in particular, had optical lenses that could be used not only for magnification of small, close objects but also to construct telescopes and what Temple refers to as “proto-theodolites” for precise surveying. Such ancient optical instruments could conceivably have been used to help accurately align various structures, including the Great Pyramid.
Concerning the orientation of the Great Pyramid, Proctor (1883, pp. 148, 149 n.) noted:
We have seen that the Great Pyramid is so perfectly oriented as to show that astronomical observations of great accuracy were made by its architects. No astronomer can doubt this, for the simple reason that every astronomer knows the exceeding difficulty of the task which the architects solved so satisfactorily, and that nothing short of the most careful observations would have enabled the builders to secure anything like the accuracy which, as a matter of fact, they did secure. Many, not acquainted with the nature of the problem, imagine that all the builders had to do was to use some of those methods of taking shadows, as, for instance, at solar noon (which has to be first determined, be it noticed), or before and after noon, noting when shadows are equal (which is not an exact method, and requires considerable care even to give what it
can
give—imperfect orientation), and so forth. But to give the accuracy which the builders obtained, not only in the orientation, but in getting the pyramid very close to latitude 30° (which was evidently what they wanted), only very exact observations would serve. . . . In the first place, many seem quite unaware of the difficulty of orienting a building like the Great Pyramid with the degree of accuracy with which that building actually has been oriented. One gravely asks whether (as Narrien long since suggested) a plumb-line, so hung as to be brought into line with the pole-star, would not have served as well as the great descending passage. Observe how all the real difficulties of the problem are over looked in this solution. We want to get a long line—a line at least 200 yards long—in a north and south position. We must fix its two ends; and as the pole-star is not available as a point along the line, we set our plumb-line at the northern end of the line, and our observing tube or hole, or whatever it may be (only it is not a telescope, for we are Egyptians of the time of Cheops, and have none), at the other. The pole-star being an altitude of 26½ degrees, the plumb-line should be nearly 100 yards long, to be seen (near the top), coincident with the pole-star, from a station 200 yards away . . .” Then its upper part (thus to be seen
without telescopic aid at night
) would be 260 yards away. The observer’s eyesight would have to be tolerably keen. (italics in the original)
can
give—imperfect orientation), and so forth. But to give the accuracy which the builders obtained, not only in the orientation, but in getting the pyramid very close to latitude 30° (which was evidently what they wanted), only very exact observations would serve. . . . In the first place, many seem quite unaware of the difficulty of orienting a building like the Great Pyramid with the degree of accuracy with which that building actually has been oriented. One gravely asks whether (as Narrien long since suggested) a plumb-line, so hung as to be brought into line with the pole-star, would not have served as well as the great descending passage. Observe how all the real difficulties of the problem are over looked in this solution. We want to get a long line—a line at least 200 yards long—in a north and south position. We must fix its two ends; and as the pole-star is not available as a point along the line, we set our plumb-line at the northern end of the line, and our observing tube or hole, or whatever it may be (only it is not a telescope, for we are Egyptians of the time of Cheops, and have none), at the other. The pole-star being an altitude of 26½ degrees, the plumb-line should be nearly 100 yards long, to be seen (near the top), coincident with the pole-star, from a station 200 yards away . . .” Then its upper part (thus to be seen
without telescopic aid at night
) would be 260 yards away. The observer’s eyesight would have to be tolerably keen. (italics in the original)
As already pointed out, Petrie (1883, 1885) and Cole (1925) sighted on the stars using modern instruments in order to determine true north and therefore determine the accuracy of the orientation of the Great Pyramid.
Hawass and other Egyptologists have suggested that the unusual layout of the chambers and passages in the Great Pyramid is simply the result of the ancient builders changing their minds a couple of times during construction. According to Hawass (
Update to Petrie,
1990, p. 99):
Update to Petrie,
1990, p. 99):
the first burial chamber of Khufu is located underneath the pyramid and was left unfinished. Next, the Overseer Of All The Works of Khufu moved the king’s burial chamber up further within the pyramid to what is known as the queen’s chamber. But for reasons I believe are connected to the cult change made by Khufu, the burial chamber, which contains the sarcophagus, was moved even higher up into the pyramid.
Hawass explains the cult change he refers to as follows.
At all times, the kings was thought to be the reincarnation on earth of Horus, the Son of Ra. When the king died, he was believed to become Ra, the sun god and Osiris, king of the dead. I believe that Khufu changed this cult and became identified with Ra during his lifetime. Stadelmann suggested this idea because the name of Khufu’s pyramid “Horizon of Khufu,” as noted above, indicates that Khufu is placed with Ra, whose natural location is on the horizon. Furthermore, he notes that Djedefra and Khafra, the sons and immediate successors of Khufu, were the first kings to bear the title “Sons of Ra” suggesting that their father, Khufu, was Ra. (Hawass, 1990, p. 99)
Pi, π: Ratio of the circumference of a circle to its diameter, namely
C
/
d
= π or 2π
r
= C (where
r
is the radius of the circle and thus 2
r
is the diameter); 3.14159 . . . is the best six-digit approximation of π. Pi is an incommensurable number.
C
/
d
= π or 2π
r
= C (where
r
is the radius of the circle and thus 2
r
is the diameter); 3.14159 . . . is the best six-digit approximation of π. Pi is an incommensurable number.
The pi theory of the Great Pyramid is that the shape of the pyramid was determined by setting the height of the pyramid equal to the radius of a hypothetical circle, and then making the perimeter of the base of the pyramid equal to the circumference of the same hypothetical circle; thus each of the sides of the pyramid (assuming all four sides are of equal length) will be one-quarter of the circumference of the circle defined by the height = radius of the pyramid; or let
L
be the length of one of these sides and let
h
be the height of the pyramid, then 2
h
π = 4
L,
or π = 2
L
/
h.
Now let
a
be the distance from the middle of one side of the Great Pyramid along the horizontal to the point directly below the apex; then 2
a
=
L
. Now substituting into the last equation, π = 4a / h. The tangent of the slope of the Great Pyramid, if built according to the pi theory, would be
h
/
a
= 4 / π (by rearranging the last equation), so we can calculate the theoretical slope and compare it to the estimated “genuine” slope of the Great Pyramid. The theoretical pi-slope is 51.854° (Herz-Fischler, 2000, p. 67). Of course, this is based on a modern approximation of pi. If the ancient Egyptians applied the pi theory but used a different approximation of pi, such as 22 / 7 = 3.1428571, then a different slope will ensue. Using 22 / 7 as pi, the slope will be approximately 51.843°.
L
be the length of one of these sides and let
h
be the height of the pyramid, then 2
h
π = 4
L,
or π = 2
L
/
h.
Now let
a
be the distance from the middle of one side of the Great Pyramid along the horizontal to the point directly below the apex; then 2
a
=
L
. Now substituting into the last equation, π = 4a / h. The tangent of the slope of the Great Pyramid, if built according to the pi theory, would be
h
/
a
= 4 / π (by rearranging the last equation), so we can calculate the theoretical slope and compare it to the estimated “genuine” slope of the Great Pyramid. The theoretical pi-slope is 51.854° (Herz-Fischler, 2000, p. 67). Of course, this is based on a modern approximation of pi. If the ancient Egyptians applied the pi theory but used a different approximation of pi, such as 22 / 7 = 3.1428571, then a different slope will ensue. Using 22 / 7 as pi, the slope will be approximately 51.843°.
In the equation that summarizes the pi-theory for the Great Pyramid,
h
/
a
= 4 / π, this is basically the rise over the run. So if the rise is 4 for every run of π, the theory will be corroborated. Pi, however, is a pretty inconvenient number to cut or measure blocks to; it is much cleaner to have the rise and run in nice whole numbers for building purposes (and any units can be used, since in
h
/
a
units cancel out). If 22 / 7 is used as an approximation for π, then
h
/
a
= 4 / π = 4 / (22 / 7) = 28 / 22 = 14 / 11, so use a rise of 14 for every horizontal run of 11, and one will be building a pi-theory pyramid. Petrie (1883, 1885) believed just this relationship holds in the Great Pyramid. 14 / 11 = 1.272727 . . . which is the tangent of the angle 51.842767°, or approximately 51.843°, is remarkably close to the angle of 51.844 ± .0180546° given by Petrie (see hereafter) for the north face mean of the Great Pyramid, or even his 51.866 ± .0333° mean approximation.
h
/
a
= 4 / π, this is basically the rise over the run. So if the rise is 4 for every run of π, the theory will be corroborated. Pi, however, is a pretty inconvenient number to cut or measure blocks to; it is much cleaner to have the rise and run in nice whole numbers for building purposes (and any units can be used, since in
h
/
a
units cancel out). If 22 / 7 is used as an approximation for π, then
h
/
a
= 4 / π = 4 / (22 / 7) = 28 / 22 = 14 / 11, so use a rise of 14 for every horizontal run of 11, and one will be building a pi-theory pyramid. Petrie (1883, 1885) believed just this relationship holds in the Great Pyramid. 14 / 11 = 1.272727 . . . which is the tangent of the angle 51.842767°, or approximately 51.843°, is remarkably close to the angle of 51.844 ± .0180546° given by Petrie (see hereafter) for the north face mean of the Great Pyramid, or even his 51.866 ± .0333° mean approximation.
The angle of inclination of the original sides, covered with their casing stones, of the Great Pyramid is a matter of uncertainty. Petrie (1885, p. 12) gives values that he measured on the few remaining in-place casing stones on the north face, as well as fragments found around the north face, and he gives one value for a casing stone on the south face. For the north face, Petrie’s measurements range from 51° 44’ 11” ± 23” (51.736° when converted to decimal form) to 51° 53’ 20” ± 1’ (51.889°), and for the south face he gives a value of 51° 57’ 30” ± 20” (51.958°). As the north face mean, Petrie gives the value 51° 50’ 40” ± 1’ 5” (51.844°). Petrie (1885, p. 13) concludes: “On the whole, we probably cannot do better than take 51° 52’ ± 02’ [51.866°] as the nearest approximation to the mean angle of the Pyramid, allowing some weight to the South side.” Assuming that the Great Pyramid originally came to a point at the top, Petrie (1885, p. 13) continues: “The mean base being 9068.8 ± .5 inches [230.3475 meters] this yields a height of 5776.0 ± 7.0 inches [146.7104 meters].”
Unfortunately, in a sense, the pi-theory is almost identical in practical terms to the so-called seked theory (Herz-Fischler, 2000, p. 30-45), and thus it is virtually impossible to distinguish which one is correct just on the basis of the data of the external dimensions of the Great Pyramid, and in fact it may not be that one is mutually exclusive of the other. The modern seked theory for the shape of the Great Pyramid is based on information and example problems in the Rhind Papyrus, an ancient Egyptian mathematical text. The actual manuscript probably dates from the Fifteenth Dynasty, a thousand or so years after the building of the Great Pyramid, but is apparently a copy of a Twelfth Dynasty text (which itself was still written about 700 years after the construction of the Great Pyramid). On the basis of the Rhind Papyrus, a seked can be considered the run relative to a rise of one cubit. A royal cubit, or simply a cubit, consisted of 7 palms (hands) consisting of 4 fingers (digits) each; thus a cubit was 28 fingers. According to the modern seked theory, the Great Pyramid was built with a seked of 5 palms and 2 fingers (equals 22 fingers), or, in modern terms, the rise was 1 cubit and the run was 5 palms and 2 fingers; this can be expressed as a rise of 28 fingers and a run of 22 fingers, which equals 28 / 22 = 14 / 11, which is the same rise and run that would be used for the pi theory if the approximation of pi used is 22 / 7.
One should note the comment of Herz-Fischler (2000, p. 34): “Despite having made an extensive search of the literature, I was unable to find unequivocal proof that the
seked
method was used [by the ancient Egyptians as an actual architectural and construction technique].” Still, “the assumption that the Egyptians used the
seked
method to determine the inclination of the faces of pyramids is widely accepted among Egyptologists” (Herz-Fischler, 2000, 43, italics in the original).
seked
method was used [by the ancient Egyptians as an actual architectural and construction technique].” Still, “the assumption that the Egyptians used the
seked
method to determine the inclination of the faces of pyramids is widely accepted among Egyptologists” (Herz-Fischler, 2000, 43, italics in the original).
According to Herz-Fischler (2000, p. 70), the pi theory originated in 1838 with the publication by a certain H. Agnew of
Letter from Alexandria on the Evidence of the Practical Application of the Quadrature of the Circle in the Configuration of the Great Pyramids of Egypt.
Letter from Alexandria on the Evidence of the Practical Application of the Quadrature of the Circle in the Configuration of the Great Pyramids of Egypt.
According to Herz-Fischler, Agnew applied the pi theory not to the Great Pyramid but to the Third (Menkaure) Pyramid, and it was later applied by Taylor (1859) to the Great Pyramid, although Taylor never acknowledged any debt to and never cited Agnew. Howard Vyse (1840) included a summary of Agnew’s book. Herz-Fischler (2000, p. 72) says:
Taylor does not mention Agnew in his book, but we can be fairly certain that he was familiar with Agnew’s work, because Taylor mentions Vyse’s book several times and the latter has an explicit statement of Agnew’s pi-theory. Another link to Agnew may have been provided by the mathematician De Morgan who had published a very short notice about Agnew’s book and with whom Taylor was acquainted.
The pi theory was widely popularized by Smyth with the publication of
Our Inheritance in the Great Pyramid
(1864), which went through five editions (1864, 1874, 1877, 1880, 1890) and continues to be reprinted. Smyth was quite famous in his lifetime and immediately after his death for his Great Pyramid work and fantasies, and he is still well known today. There was even a fictionalized account of his life and work by Max Eyth, published in Germany:
Der Kampf um die Cheopspyramide
(The Struggle over the Pyramid of Cheops).
Our Inheritance in the Great Pyramid
(1864), which went through five editions (1864, 1874, 1877, 1880, 1890) and continues to be reprinted. Smyth was quite famous in his lifetime and immediately after his death for his Great Pyramid work and fantasies, and he is still well known today. There was even a fictionalized account of his life and work by Max Eyth, published in Germany:
Der Kampf um die Cheopspyramide
(The Struggle over the Pyramid of Cheops).
Mendelssohn (1974, p. 73) acknowledges the apparent pi relation found in the Great Pyramid but argues that it was an outgrowth of the practical way in which the Egyptians laid out the Great Pyramid, following an idea that was suggested to Mendelssohn “by an electronics engineer, T. E. Connolly” (Mendelssohn, 1974, p. 64). Mendelssohn says, relative to the theory he espouses:
The explanation is based on the assumption that the ancient Egyptians had not yet formed the concept of isotropic three-dimensional space. In other words, whereas to us measures of height and vertical distance [apparently he meant “horizontal” rather than “vertical”] are the same thing, namely a length for which we use the same unit, this may not have been regarded as natural by the pyramid builders. (1974, p. 73)
He then suggests that the ancient Egyptians used cubits for height but what he calls a “rolled cubit” for horizontal distances, where a rolled cubit is one revolution on the ground of a drum that is 1 cubit in diameter. In terms of the Great Pyramid, if we apply Mendelssohn’s hypothesis, we can suggest that it was designed to have a height of 280 cubits and a length on each side of 140 rolled cubits. The side length of 140 rolled cubits would be equal to 140 × π = 439.8 cubits. More important, using the foregoing terminology,
a
= 70π cubits and
h
= 280 cubits, so
h
/
a
= 280 / (70π) = 4 / π, as is suggested by the pi theory. However, the downside of Mendelssohn’s hypothesis is that there is no evidence “that the ancient Egyptians had not yet formed the concept of isotropic three-dimensional space” (I’m not sure how they would have achieved all of their amazing architectural and sculptural achievements if this were true), and it postulates a unit and way of measuring horizontal distances by the ancient Egyptians for which there is no independent evidence. As Herz-Fischler (2000, p. 226, n. 9) points out, it is not clear with Mendelssohn’s theory how the ancient Egyptians either measured the vertical height up to 280 cubits so accurately, or rolled a drum so many times along the ground to measure the horizontal distance without any perceptible slippage. It is also not inconceivable that the ancient Egyptians used a variation on the scenario suggested by Mendelssohn for the very purpose of incorporating the pi relation into the Great Pyramid rather than the apparent pi relationship simply being an unexpected outcome of the way the Great Pyramid was laid out.
a
= 70π cubits and
h
= 280 cubits, so
h
/
a
= 280 / (70π) = 4 / π, as is suggested by the pi theory. However, the downside of Mendelssohn’s hypothesis is that there is no evidence “that the ancient Egyptians had not yet formed the concept of isotropic three-dimensional space” (I’m not sure how they would have achieved all of their amazing architectural and sculptural achievements if this were true), and it postulates a unit and way of measuring horizontal distances by the ancient Egyptians for which there is no independent evidence. As Herz-Fischler (2000, p. 226, n. 9) points out, it is not clear with Mendelssohn’s theory how the ancient Egyptians either measured the vertical height up to 280 cubits so accurately, or rolled a drum so many times along the ground to measure the horizontal distance without any perceptible slippage. It is also not inconceivable that the ancient Egyptians used a variation on the scenario suggested by Mendelssohn for the very purpose of incorporating the pi relation into the Great Pyramid rather than the apparent pi relationship simply being an unexpected outcome of the way the Great Pyramid was laid out.
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