Precession and the Pyramid
Astronomical Knowledge in Ancient Egypt
Jim Fournier
May 5, 1996
PAR #999
David Ulansey
Note: the conversion to html lost the extensive footnotes and references,
and I haven't had time to deal with converting them back into html.
Introduction
There is one point about ancient Egypt which stands out above all others,
an insight critical not only to our understanding of Egypt, but also to
our overall understanding of history. The ancient Egyptians observed, and
to an important degree understood, the precession of the equinoxes. This
point is really a subsidiary correlate to the realization that at least
circa 2500 BC the Egyptians knew the size of the earth very precisely. Precise
geodetic knowledge is contingent upon precise astronomical observations,
and both taken together imply an advanced understanding of geometry, as
well as precession. It follows that the ancient Greeks should be taken at
their word when they claim that their knowledge is of great antiquity and
was derived from Egyptian sources. Indeed it is nothing if not bizarre that
modern scholars of the Greek world should go to great lengths to dismiss
such claims on the part of the authors of the primary texts themselves,
to instead rely on the advice of modern Egyptologists that the ancient Egyptians
had no such knowledge.
I believe this situation is the result of a number of factors which have
together conspired to distort our understanding of the very ancient world.
These factors are to a large degree artifacts of our own cultural history.
First, there was the Judeao-Christian creation myth which placed the moment
of creation at some fixed date in relatively recent pre-history. As late
as last century an exact date for such an event was still staunchly defended
by eminent scholars as having occurred in 4004 BC. When this line of thinking
finally gave way to the scientific theory of evolution it was supplanted
by an implicit belief in a doctrine of progress. It was assumed that time
was linear, and that human knowledge and civilization have grown steadily
from a state of primitive chaos to a state of enlightened order. There were
a few set backs, but the only really glaring one was the dark ages in Europe,
and for the most part this did not detract from our overall belief in this
trend. Finally, ever since the renaissance there has been an assumption
that ancient Greece was the birth place of civilization and philosophy;
in spite of the troubling tendency of the Greeks themselves to understand
the trend of time in terms of a descent from a Golden Age. This was essentially
the opposite of our modern understanding, but because we saw that we were
obviously right about progress in our own time, it was easy to conclude
that the Greeks were wrong about their view of their own time, and to write
the whole thing off as a myth which the early Greek mind had concocted to
explain its own origins. We, as moderns, still have a sort of schizophrenic
attitude toward the ancient Greek thinkers, on the one hand we want to revere
them above all others, but on the other hand we dismiss their own beliefs
whenever it suits us.
As a result of investigating ancient Egyptian material I have come to suspect
that the ancient Greeks were not the first to understand the geometry and
mathematics attributed to them, but rather they were the first to write
about these insights, publicly, in texts which have survived and in a language
we could understand. There may very well have been a cross fertilization
from a northern shamanic culture with ancient Greece, as well as an invigoration
of their own culture brought about through the ongoing use of entheogens
at the Eleusinian Mysteries, but when accounts tell us that Pythagoras attributed
his knowledge to having been initiated in Egypt, we should take these seriously.
It was widely believed that in the ancient Egyptian tradition sacred knowledge
was only transmitted to the properly initiated. It would follow that if
Pythagoras was initiated into the Egyptian tradition he would have been
strongly admonished to maintain secrecy. Thus the most obvious explanation
for Pythagoras' emphasis on secrecy would be that the Greek initiate simply
adopted the practice of his Egyptian teachers. Pythagoras might have learned
much from a shamanic initiation in the lands north of Greece, but his emphasis
on secrecy was not likely to be part of it. Secrecy is unnecessary where
the content of the ecstatic experience is itself unspeakable, but where
specific information is passed on to an initiate it becomes far more critical.
There was a doctrine of secrecy involving initiation rites at Eleusis. But
there is apparently no evidence Pythagoras was an initiate at Eleusis, and
the fact that so many were initiated there, over such a long period of time,
without disclosing any 'secret' suggests that these rites involved an essentially
unspeakable experience rather than the transmission of any specific information
which could be divulged. The 'secret' at Eleusis may very well have involved
the experience of a brilliant flash of light, which like the impact of a
plot twist in a good suspense story, would have lost much of its impact
from being anticipated in advance. Thus it could only be fully experienced
once, and might have been kept secret largely for the benefit of future
initiates.
Just the opposite seems to be true of Pythagoras. Here we have attributed
to one individual the transmission of a huge body of very concrete and coherent
information involving harmony, proportion and geometry, as well as doctrines
of the transmigration of the soul and reincarnation. It seems unlikely that
Pythagoras could have developed such a comprehensive harmonic geometric
doctrines solely as a result of initiation into a shamanic culture. Although,
it does seem highly plausible that being instructed in a mathematically
advanced esoteric tradition in ancient Egypt, combined with personal experience
of shamanic initiation could have caused an entirely new level of insight
to arise in an individual such as Pythagoras. The insight he articulated
then apparently diffused out into Greek thought, through Plato and others,
but it was always attributed to sources of greatest antiquity. It is relevant
to bear in mind that at the time of the ancient Greeks, the roots of Egypt
were already of greater antiquity to the Greeks than the ancient Greeks
are to us today.
In many cases the Greeks themselves were honest enough to admit that they
did not invent these ideas. They were simply the first ones to write down
what had previously been disclosed only to the initiated in the form of
closely guarded secret teachings. Once the Greeks did start to openly discuss
these ideas, many Greek thinkers did have genuinely original insights as
a result of this new information flooding their culture. But it should be
obvious from the fact that it was Alexandria which became the focal point
of the Hellenistic world that there was a reason why that particular city
was so important. Its library contained the Egyptian knowledge, or what
was left of it.
It does seem likely that by Ptolemaic times some knowledge had either been
lost by the Egyptians, or more likely was simply withheld from the loud
mouthed Greeks by an Egyptian initiate priesthood who had long experience
with being invaded and ruled by less advanced civilizations over the prior
three thousand years. It is not even entirely clear that the initiated priests
always shared all of their knowledge with the presiding pharaoh, if he did
not warrant their trust. In any case, it is clear that some of the knowledge
which the Egyptians appear to have possessed at earlier times, and very
likely even in Ptolemaic times, was not passed on to the Greeks directly,
but was instead rediscovered by Greeks either entirely on their own, or
through less than perfect interpretation of Egyptian texts, or from partial
or corrupted knowledge of Egyptian sources.
In many cases what we believe we know about ancient Egypt is based largely
on Greek accounts. On the one hand it often seems clear that the Greek authors
of these accounts possessed a limited understanding of what they were writing
about, or were basing their descriptions on fragmentary evidence. Yet at
the same time, the mathematics and astronomy exhibited in the late Ptolemaic
Egyptian temples and texts are often discounted as evidence of Egyptian
achievement because it is assumed that any astronomy or mathematics exhibited
can be attributed to Greek influences. This is entirely unfounded, especially
when it seems clear from the first sympathetic translations of Greek astrological
texts that they are based on an exceedingly ancient and well developed Egyptian
tradition. While the new Greek knowledge of the movements of the planets
was undoubtedly based largely on Babylonian sources, attributions as to
the natures of individual 'fixed' stars and decans is almost certainly of
Egyptian origin. This is not difficult to understand given the well accepted
Egyptian fascination with the sequence of stellar decans documented in a
wealth of Egyptian hieroglyphic texts depicting stars, and maps of sequences
of stars, spanning three thousand years. What is still hotly disputed is
what we may infer from translations of these Egyptian texts.
The Great Pyramid:
Primary Evidence Written in Stone
The most compelling evidence of Egyptian achievement is not written
in texts but in stone. The most awe-inspiring example of this is the Great
Pyramid at Giza. Perhaps no other object, structure or human artifact has
inspired so many theories, speculations and certainly, in at least some
cases, fantasies. Many theories attempt to claim to explain the Great Pyramid
by themselves, as was the fashion during the period in which they were formulated.
A few might succeed in doing so, and yet enough of them are so equally compelling
that it becomes virtually impossible to choose one alone at the expense
of the others. My own suspicion is that like a design of nature, the Great
Pyramid is not mono-dimensional, but rather it simultaneously solves many
problems and expresses many truths at once. Such an explanation is perhaps
uniquely dissatisfying to a reductionist scholar, so in the interest of
making the best possible argument as to its transcendence, as both an expression
of human spirit, and human reason, I will attempt to document the most inarguable
facts first, and then progressively move toward more speculative realms.
The Great Pyramid is located at the extreme northern edge of a limestone
plateau at the edge of the Nile flood plain where the river meets the Delta.
Napoleon's savants noticed, when they arrived in Egypt in 1798, that the
Great Pyramid is situated at the exact apex of the Nile Delta such that
an arc centered on the Pyramid defines the extent of the Delta, perfectly
enclosing its outer perimeter. The northern promontory of the Delta is due
north of the Pyramid, and the extended North-West and North-East edges of
the Pyramid define the Delta within a perfect sector, or quadrant of ninety
degrees, centered on the Pyramid. In 1882, Robert T. Ballard pointed out
that this placement of the Great Pyramid would have allowed the residents
of the Nile Delta to easily resurvey their fields every year after the annual
flood using only a plumb line, by sighting on the apex of the Great Pyramid.
He further demonstrated that the combination of the three Giza Pyramids
would have improved this operation and provided more information than a
single pyramid by itself could have.
The meridian defined by the Pyramid in relation to the Nile Delta was, according
to Livio Catullo Stecchini, the central meridian of ancient Egypt. The establishment
of this meridian bisecting the Nile Delta (at 31* 14' east) apparently predated
the building of the Pyramid, and was seen as the central axis of Egypt from
the earliest antiquity. Incidentally, if this were true, and it seems inescapable
that it was true for at least some significant period prior to the construction
of the Great Pyramid, then the natural outcrop which formed the head of
the Sphinx (carved or un-carved) would have previously performed that same
function. The deeper importance of this particular location will become
apparent in a moment. Stecchini also claims that a number of other locations
throughout the ancient world were located in exact geodetic relation to
the longitude meridian of the Great Pyramid. Of these the Persian capital
Persepolis, whose location otherwise appears senseless to scholars, is perhaps
the most straight forward to explain. Persepolis was located at 30* 00'
north latitude, and three units of exactly 7* 12' east of the meridian of
the Great Pyramid. The reason for this 7* 12' unit was that the Persian
Empire of King Darius the Great was idealized as three geodetic squares
of 6 of latitude, stretching from 30* to 36* north. At 33* north, the midpoint
of this distance, 6* of latitude is equal to 7' 12" of longitude, thus
making these regions true squares. Among the other ancient sites exhibiting
similar geodetic precision, according to Stecchini, are: Nimrod, Sardi,
Susa, Mycenae, Dodona and Delphi, as well as the Kaaba at Mecca, and Mt.
Gerizim, the original Jewish holy center, before it was moved to Jerusalem
in 980 BC.
The apparent placement of these of these other sites in relationship to
the meridian of the Great Pyramid becomes even more understandable when
we recognize that the Great Pyramid was located at 30* north latitude (currently
29* 58' 51"). At first glance it appears that the builders made an
error of 1' 9" in its location. However, without a correction for atmospheric
refraction, 29* 58' 22" north latitude appears to be exactly 30*, based
on purely astronomical observation. Thus there could instead be an error
of 29" in the other direction. Or, there could be an error of only
20" if, as Piazzi Smyth suggests, they had intended to split the difference
and try for the intermediate value of 29* 59' 11". This idea becomes
more plausible when one realizes that the atmospheric error is in the opposite
direction for an alignment based on solar observations, and thus it would
make sense that they might have used an intermediate value between the solar
and stellar calculations. It is equally likely that they simply could not
place the Great Pyramid any farther north, and still remain on their prime
meridian bisecting Egypt, because the Giza Plateau ends. As it stands, the
Great Pyramid is closer to the cliff at the northern edge of the Giza Plateau
than many engineers would have thought feasible. It is even remotely possible
that the earth's crust has shifted slightly over the intervening 4500 years
and the Pyramid was originally placed at a minutely different latitude.
In any case, the precision with which it is placed is astounding, certainly
more than accurate enough to prove both their intention and their ability.
The most important aspect of the Great Pyramid is the precision with which
its overall dimensions encode the measurement of the earth. In 1925, J.H
Cole, a professional surveyor, was commissioned by Ludwig Borchardt to make
a truly accurate survey of the Great Pyramid. This remains the most precise
data available with respect to its overall dimensions and orientation. Prior
to that time there had been a series of survey attempts, each successively
better than the previous one in terms of accuracy, although, as it turned
out, the most astute theory proved to be one of the first. Unfortunately,
that theory was first conceived based on some accidentally fortuitous, but
technically incorrect data. The result, as those measurements were soon
ruled to be incorrect, was to discredit that entire line of thinking for
over a century; confirming the prejudice in the eyes of mainstream Egyptologists
that the ancient Egyptians could not have had anything more than
primitive astronomy and mathematics. This position has been built on a foundation
which presupposes a priori that one must dismiss any line of thinking
which asserts that the ancient Egyptians might have possessed accurate geodetic
knowledge. The following assertion made by the preeminent Egyptologist,
Ludwig Borchardt is typical. He is commenting here on an Egyptian inscription
stating that the distance between Behdet (at the northern tip of the Nile
Delta) and Syene (at the first cataract near Aswan in the south) was 106
atur, "one must absolutely exclude the possibility that the ancients
may have measured in degrees." Borchardt gives absolutely no grounds
for this assertion. It is instead invoked as an article of faith. It is
ironic that it was Cole's survey of the Great Pyramid, commissioned by Borchardt
himself, which provided Stecchini with his best evidence to refute this
long standing prejudice. It should be pointed out, however, that Stecchini
derived his knowledge of Egyptian geodetic measurement from his reading
and interpretation of hundreds, if not thousands, of hieroglyphic texts.
In the case of Borchardt's quote cited above, if one simply checks the distance,
it does in fact measure 106 geodetic atur. An atur was 15,000
royal cubits, which was also equal to 17,000 of the older geodetic cubits.
The figure 106 atur is significant because it is 1/12 of the length
of the meridian from the equator to the pole.
A brief chronological overview of the exploration of the Great Pyramid
follows. This section has been taken almost entirely from Peter Tompkins
book Secrets of the Great Pyramid, as this offers the best way to
efficiently provide the necessary background information from his excellent
book.
In 24 BC, the Pontine geographer Strabo visited Egypt and wrote an extensive
history. It is now lost, but in a surviving geographical appendix he tells
us that there was a perfectly concealed swivel door in the north side of
the Great Pyramid. This door has been lost, but a similar one was found
at the south pyramid of Dashur. The door in the Great Pyramid gave access
to a less than four foot square passage which descended 374 feet to a rough,
damp vermin infested pit carved out of the solid rock beneath the apex of
the Pyramid. This account is confirmed by the existence of the initials
of Greek and Roman tourists scrawled with torches on the ceiling of the
pit and still visible in modern times.
In 820 AD a well educated Arab prince, Abdullah El Mamun, seeking accurate
knowledge of the length of a degree of latitude, as well as gold and treasure,
forced his way into the Great Pyramid. It was no easy undertaking. At that
time the Pyramid was still fully clad in polished limestone casing stones
whose outside face had hardened from centuries of exposure to the air. They
could not find the hinged door, and his men, armed with iron tools could
not chip the surface, so they were forced to build fires and then douse
the red hot stone with cold vinegar to break in. His men then tunneled straight
in for over a hundred feet but found nothing. They were on the verge of
giving up when, according to the legend, they heard a muffled thud and tunneled
to the east where they broke into the Descending Passage. The sound had
apparently been made by a block being dislodged from the ceiling of the
Descending Passage by the vibration of their tunneling. This block, which
had concealed the bottom of the Ascending Passage, exposed the first of
a series of three huge granite plugs in the Ascending Passage. Again, they
could make no progress in chipping these granite blocks, so they tunneled
alongside them through the limestone core blocks of the Pyramid. Where the
granite plugs ended, limestone ones began. These could, however, be broken
up and eventually gave way to the Ascending Passage. Once inside, they found
both the Queen's Chamber and the King's Chamber absolutely empty and swept
clean. Both names were based solely on an Arab tradition of making women's
burial vaults with pitched ceilings and men's with flat ceilings. The King's
chamber was also larger and higher in the structure. The only object they
found anywhere inside the Pyramid was a lidless 'sarcophagus' in the King's
Chamber. There appears to be wide agreement among scholars that this Arab
account is largely accurate. What is most important to understand, based
on this account, is that when the Pyramid was first opened, no body, or
burial, or any evidence of any burial was found, nor has any actual evidence
of an intended burial ever been found since.
Except for El Mamun's hole, the casing stones remained intact when the Great
Pyramid was visited by an Arab historian in the early thirteenth century.
But, over the course of the fourteenth century, apparently following an
earthquake which dislodged some casing stones (and destroyed Cairo), the
rest were systematically stripped off to rebuild the mosques and palaces
of Cairo. A century later, during the Italian renaissance, Girolamo Cardano,
a Milanese physician and mathematician, and close friend of Leonardo Da
Vinci, maintained that an advanced science must have predated the Greeks.
He suspected that knowledge of a far more exact degree of latitude must
have existed hundreds, if not thousands, of years before the Alexandrian
Greeks, and he believed that the place to look for it would be in Egypt.
However, it was not until the enlightenment that the first European investigation
of the Great Pyramid was made.
In 1638, an English mathematician named John Greaves joined by an Italian,
Tito Livio Burattini, made the first European survey of the Great Pyramid.
Greaves estimated the height at 499 feet (within 12 feet of correct) and
the base at 693 feet (70 feet too short), but the base was still totally
covered by debris at that time. Upon his return to England, Greaves discussed
his findings in Egypt with many, including a Dr. William Harvey who had
discovered the circulation of the blood. Dr. Harvey was surprised to learn
that Greaves had not discovered any means of ventilation which would allow
fresh air into the interior of the Pyramid. He insisted that some form of
ventilation shafts must exist. Greaves and Burattini did, however, measure
the King's Chamber very accurately and it was on the basis of these figures
that Sir Isaac Newton deduced his 'profane' cubit of 20.63 inches. A cubit
of this dimension was implied by the 1:2 proportions of the King's Chamber
which suggested to Newton that it must measure 10 X 20 cubits. Newton also
postulated a longer 'sacred' cubit of between 24.80 and 25.02 British inches,
based on the Jewish historian Josephus's description of the circumference
of the pillars of the Temple of Jerusalem. Newton was interested in the
exact length of a cubit because he too was convinced that accurate geodetic
information was encoded in the dimensions of the Great Pyramid, and he needed
to know the size of the earth in order to test, and thus to prove, his theory
of gravitation before he would publish it.
In 1798, Edme-Francois Jomard visited the Great Pyramid as a young savant
on Napoleon's expedition. The French had the debris cleared away from the
two northern corners of the Pyramid and discovered the corner sockets where
the corner casing stones had apparently originally been placed. These were
ten by twelve foot mortises, perfectly level, and perfectly level with each
other, cut twenty inches into the limestone bedrock. Although, there were
still piles of rubble between them, Jomard was able to measure the north
side of the base to be 230.902 meters (757.5 feet). For the height, he measured
each step. They added up to a total of 144 meters (481 feet). By means of
trigonometry Jomard calculated a slope of 51* 19' 14", and an apothem
of 184.722 meters. Because the casing stones were missing, these figures
were both estimates, but the length of the apothem looked virtually perfect
in light of various ancient classical texts which Jomard was familiar with.
Diodorus Siculus and Strabo both claimed that the apothem of the Great Pyramid
was one stadium long. The Olympic stadium was 600 Greek feet, and was supposed
to be related to the size of the earth. Jomard found the stadium of Eratosthenes
and Hipparchus to be 185.5 meters, and thus within one meter of his figure
for the apothem. He also found that distances quoted by the ancients in
stadia matched the distances found by Napoleon's surveyors, if a stadium
was taken to be 185 meters. The ancient stadium was also reported to have
been 1/600 of a degree. When Jomard took the length of a degree at what
he believed to be the mean latitude of Egypt, 110,827.68 meters, and divided
it by 600, he arrived at a stadium of 184.712 meters, which was within ten
centimeters of his figure for the length of the apothem! In addition, several
Greek authors had reported that the perimeter of the base was equal to half
a minute of a degree. This would mean that a degree of latitude divided
by 480 should equal the length of one side of the base. Again Jomard used
the length of a degree at his mean latitude of Egypt, 110,827 meters, and
dividing by 480 arrived at 230.8 meters, again within 10 centimeters of
his measured base. According to Herodotus there were 400 cubits in a stadium.
So, Jomard divided his apothem length, by 400 to arrive at a cubit of .4618
meters. This turned out to be the common cubit still in use in Egypt in
Jomard's time. Other Greek sources stated that the length of one side of
the base was 500 cubits, which when multiplied by his cubit length yielded
a side of 230.9 meters, which was exactly what he had found the base to
measure. The theory looked perfect! Until two other Frenchmen re-measured
the base and found it two meters longer. They also measured the height with
a special instrument and found that Jomard's apothem was too short. The
apparent defeat of Jomard's theory led to a long period of confusion and
dispute regarding the design of the Great Pyramid which still continues,
and may never end. Yet at this point, following Stecchini's interpretation
of Cole's data, there is, in my opinion, little excuse for it.
The next major investigations of the Pyramid were hardly surveys, but did
reveal some important features. In the 1830's Captain G. B. Caviglia cleared
the descending passage of debris, exposing the 'pit' for the first time
since the Pyramid had first been opened by Al Mamun. Caviglia also discovered
and opened the 'well,' an enigmatic irregular shaft which descends almost
vertically from the base of the Grand Gallery. Its upper opening was concealed
at the point where the horizontal passage to the Queens Chamber branches
off, just above the highest point to which the Ascending Passage could have
been filled with limestone plug stones. From there, the Well leads almost
straight down to an odd chamber called the 'grotto' and then angles down
to near the bottom of the Descending Passage. Clearing the debris from the
Well did improve the otherwise stifling air quality in the Pit and Descending
Passage, but the crude nature of this shaft is in such sharp contrast to
the rest of the structure that it seems inconceivable that it was part of
the original design and construction. Theories and speculation as to its
origin and purpose could, and do, fill volumes. I will not go into them
here.
Caviglia was superseded by Colonel Howard-Vyse who spent a small fortune
and several years of his life exploring the Pyramid. Howard-Vyse blasted
his way up, above Davison's Chamber, above the King's Chamber to discover
four more small 'relieving' chambers above the King's Chamber, each only
a few feet high and totally sealed, and each, except the last, roofed with
huge granite beams polished on their ceiling face, but forming rough uneven
floors. It has been claimed by at least one engineer that these chambers
served the purpose of relieving the tremendous load of the Pyramid from
the King's chamber. They might serve this purpose, though this has been
disputed by other engineers, but whatever their purpose the structural explanation
cannot answer the question of why only the bottom face of each was so carefully
smoothed. More interesting to Egyptologists were some hieroglyphic markings
found on some of the limestone core blocks forming the walls of these chambers.
Among the markings, many of them upside down, were the cartouche 'Khufu.'
It is almost solely on the basis of these markings that the building of
the Great Pyramid was attributed to a pharaoh named Khufu. Mark Lehner has
reportedly had pigment from one of these 'quarry marks' (taken from a remote
corner of one these chambers) carbon dated, and it came out c2500 BC.
Howard-Vyse also discovered the 'air shafts' in the north and south walls
of the Kings Chamber, thus fulfilling Dr. Harvey's prediction. It was not
hard to assume they were simply meant to be air shafts because when they
were cleared of sand there was a rush of cool air into the previously stifling
chamber, whereupon its temperature dropped to a steady 68 Fahrenheit. This
could be seen as evidence in favor of the idea, first put forward by Jomard,
that the Sarcophagus in the King's Chamber served as a standard of weights
and measures. Howard-Vyse also cleared away the ruble from the middle of
the north side and discovered the first two intact casing stones on the
Pyramid. They were each about twelve feet long, by five feet high, by eight
feet deep. The angle of the face could finally be accurately measured and
was found to be 51* 51'. Howard-Vyse described the blocks as being, "in
a sloping plane as correct and true almost as modern work by optical instrument
makers. The joints were scarcely perceptible, not wider than the thickness
of silver paper." On the north side, he also uncovered some of the
paving stones. They extended out from under the casing stones of the Pyramid,
which were placed on top of them. With this base level, and precise angle,
combined with the length of the side measured by the French, it was now
possible, in 1840, to compute the height of the Pyramid to its apex as 485.5
feet or 147.9 meters.
John Taylor was the first of many modern mathematicians to become fascinated
with the Great Pyramid. A gifted mathematician and amateur astronomer, working
from the new data, as well as Herodotus' reports attributed to Egyptian
priests, Taylor concluded that this particular pyramid constituted a unique
mathematical solution, in which the surface area of each face is equal to
the square of its height. When he divided his figure for the perimeter by
twice his figure for its height he arrived at 3.144, a very good approximation
of pi = 3.14159 Therefore, the height of the Pyramid is in the same
proportion to its perimeter as the radius of a circle is to its circumference,
or more precisely, it is in the same relationship as the radius of a sphere
is to the circumference of its great circle. Thus the pyramid not only squares
the circle, it cubes the sphere. With this relationship in mind it is obvious
why Taylor would have expected the height of the Pyramid to relate to the
radius of the earth, while the perimeter would be expected to relate to
the circumference of the earth (as Jomard had asserted). So, Taylor looked
for an integer ratio which would express the relationship in likely units
of measure. When he expressed the approximation of pi in terms of
the ratio 366:116.5, the larger number, representing the perimeter of the
Pyramid and thus the circumference of the earth, matched the number of sidereal
days in a solar year. If he converted this perimeter into inches, it came
out very close to 100 x 366. When he divided one side of the base by 366,
he obtained a 25 inch+ cubit, virtually identical to Newton's 'sacred' cubit.
At the same time, at the beginning of the nineteenth century, the astronomer
Sir John Herschel had proposed a unit of measure half a hair longer than
a British inch, as the only reasonable earth based unit of measure. Herschel
argued that the length of every meridian on the earth was different due
to surface fluctuations, and thus the French meter was a flawed approach.
(In addition the French had calculated their meridian about 2000 meters
too short, and thus the meter was, and is, .0002 short of the theoretically
correct length.) Herschel argued for a unit based on the length of the earth's
polar axis. A recent British Ordinance Survey had just concluded, based
on an average of all known meridians, that this length was 7898.78 miles,
or 500,500,000 inches. Herschel proposed that this be treated as an even
five hundred million inches, and that the inch be lengthened by half the
thickness of a human hair. Fifty of these inches would make a yard, which
would be the ten millionth part of the axis. By the same argument twenty
million of the Newton-Taylor 'sacred' pyramid cubits would be/were equally
perfect measures. The International Geophysical Year in 1957-58 confirmed
the precise value with orbiting satellites as 25.02614284 British Inches,
the same as the Newton-Taylor sacred pyramid cubit to three decimal places.
Taylor didn't live long enough to see this final corroboration of his cubit,
but he did undertake a massive study of virtually every ancient unit of
measure ever known, in an effort to find the common roots underlying the
inch and all units of measures. In 1859, close to death, he recorded his
theories in a volume titled The Great Pyramid: Why Was it Built &
Who Built It?
Unfortunately for the fate of his theories, his religious fervor was at
least as intense as his mathematical talent. He argued fervently that the
perfection of these measures found in the Pyramid was due to divine intervention
and guidance. This not only insured that his ideas failed to find much favor
in Victorian society, but served to thoroughly discredit his whole line
of mathematical and metrological thinking in academic circles. This was
especially true in the emergent field of Egyptology, which was based firmly
and solely on the philological territory opened up by Champollion's deciphering
of the Rosetta stone, and the subsequent translation of ancient Egyptian
hieroglyphic, hieratic and demotic texts.
Piazzi Smyth, Astronomer Royal for Scotland, became convinced that Taylor's
mathematical reasoning was sound with respect to a cubit, first hypothesized
by Newton, of 25.025 British Inches. Smyth believed that the modern inch
had been derived from this cubit, and that it was the same cubit which had
been used by Moses to construct the tabernacle and by Noah to build the
Ark. Smyth presented a paper in support of Taylor's work to the Royal Society
of Edinburgh, which had included him as a member because of his work on
spectroscopy. In spite of his reputation, he received no better reception
than Taylor had in academic circles. Smyth and Taylor corresponded intensely
during the last weeks of Taylor's life, and when Taylor died in 1864, Smyth
decided to go to Egypt to measure the Pyramid. Smyth undertook the first
truly scientific survey of the Pyramid using instruments he had specially
made for the project by expert opticians. The slope he measured for the
descending passage, 26* 27', was by far the most precise reading to date.
He measured every interior detail he could, including especially the sarcophagus
in the King's Chamber, which he concurred was a perfect standard of linear
and volumetric measures, with its precisely polished interior dimensions
carefully maintained at an unchanging temperature, and humidity within the
King's Chamber.
With his long experience in astronomy Smyth was well prepared to make the
precisely accurate astronomical observations necessary to determine the
location and alignment of the Pyramid. To determine its exact latitude he
took sightings from its peak to avoid the distortion its mass might otherwise
have exerted on his plumb line. He arrived at a figure of 29* 58' 51"
north, but he also pointed out that due to atmospheric refraction a latitude
of 29* 58' 22" north would instead appear as if it were at exactly
30* 00' 00" north. He concluded that the precision of its alignment,
which was superior to that of the famous observatory of Tycho Brahe, must
have been achieved by observing a polar star from the Descending Passage.
The great astronomer Sir John Herschel had previously suggested that Alpha
Draconis would have been the Pole star about 4000 years ago. By subtracting
the 26* 27' angle of the Descending Passage from the 30* north latitude
of the Pyramid, he arrived at an angle of 3* 43' north. According to Smyth's
calculations Alpha Draconis made a lower culmination at that angle in 3440
BC, and again 2123 BC. He concluded that either date might have been when
the Pyramid was laid out, but because the Pleiades had also been crossing
the meridian on the autumn equinox in 2170 BC, he preferred this date.
Smyth remained primarily concerned with proving Taylor's hypothesis regarding
the encoding of pi in the proportions of the Pyramid. The sharp edges
of the casing stones discovered by Howard-Vyse had been vandalized by resentful
Arabs, and then by souvenir hunters. Smyth dug up more and they all measured
52*, confirming the theory of a pi based relationship between the
height and the perimeter of the base. But, he sought a more exact measure.
By observing the silhouette of the entire Pyramid he obtained an angle of
51* 49'. Sir John Herschel had calculated a figure of 51* 52' 15.5"
from the figures reported by Howard-Vyse. Smyth used the mean of these two,
51* 51' 14.3" along with the mean of the 763.62 foot baseline measured
by the French with Howard-Vyse's 764 foot baseline, to arrive at 763.81
feet. These were arbitrary compromises, but produced a very exact value
for pi of 3.14159+, perfect to five decimal places!
By this time there had been four different actual measurements of the base:
757.5 feet = 9,090 inches measured by Jomard, but soon discredited
763.63 feet = 9,163.56 inches measured by the French c1800
764 feet = 9,168 inches measured by Howard-Vyse
759.17 feet = 9,110 inches measured by some visiting Scotsmen for Smyth
Versus: 9,140.18 inches The figure Smyth needed to prove Taylor
Smyth could convince himself that it might be correct, by averaging actual
measurements, but proof of the solar year theory rested on a precise measurement
to within a fraction of an inch over a distance of hundreds of feet. Literally
the argument came down to a precision of one part in ten thousand. Most
were unwilling, or unable, to understand the subtleties of his argument,
and Smyth's recourse to Biblical divine instruction as the explanation of
its origin did not help his case. Smyth's case was further compromised,
if that were possible, by his association with another Scotsman, Robert
Menzies. Menzies propounded a theory that the passageways in the Great Pyramid
represented a chronological map of time, and that a biblically based system
of prophesy could be read from them at the scale of one Pyramid inch to
one year. In the end, Smyth, like Taylor was for the most part dismissed
by the academic world as a crank, and the mathematical basis of his arguments
was not so much refuted as ignored.
A mechanical engineer named William Petrie became interested in the theories
of Taylor and Smyth and set out to design instruments which would settle
the matter once and for all. It wasn't easy to improve on Smyth's instruments,
which were quite good to begin with. William Petrie spent over twenty years
building the instruments without ever mounting the expedition. In 1880 his
twenty six year old son, by then a professional surveyor named William Flinders
Petrie, set off ahead of his father with the instruments. Though he could
not remove the rubble, Petrie had his father's exceedingly accurate theodolite
capable of reading a single second of a degree. A second of a degree is
described as the angle subtended by a dime at the distance of a mile. Thus,
by means of vast numbers of triangulation readings taken over the whole
of the Giza plateau Petrie was able to establish a set of measurements of
the dimensions of the exterior of the Great Pyramid always accurate to within
a quarter of an inch, and often to within a tenth of an inch. He described
the Pyramid as, "a triumph of skill. Its errors, both in length and
in angles, could be covered by placing one's thumb on them."
Inside, Petrie's equipment allowed him to measure within 1/100th inch and
when required, to within 1/1000th inch. He used sightings on the elongation
of Polaris to measure the Descending Passage and found that it deviated
from perfectly straight by only 1/50th inch in 150 feet, and by only 1/4
inch in 350 feet. Petrie found that the proportions of the King's Chamber
are governed by both phi, and by the 2 - sq root5 -3 and 3 - 4 -
5 Pythagorean triangles. The floor plan as defined by the walls is 1 : 2,
expressed as 10 x 20 cubits. The east and west end walls are 2 : sq root5,
with a diagonal of 3, expressed as 10 x 11.18 cubits with a diagonal of
15 cubits. The diagonal of the room's volume is 25 cubits. Thus a triangle
composed of the diagonal of the end wall, the long edge of the room, and
the volumetric diagonal, has lengths 15, 20 and 25 cubits, or proportions
3 : 4 : 5.
Petrie dug holes looking for more casing stones and eventually found some
intact. They were equally as impressive as those found by Howard-Vyse, and
Petrie quantified the measure of their perfection. The mortar filling the
gap between them was 1/50th inch, and on their face the mean variation from
a straight line and true square was 1/100th inch over 75 inches. His greatest
discovery, however, was that the corner sockets did not actually hold the
corner stones of the Pyramid, but rather base paving stones upon which the
casing stones rested. Therefore, the base of the pyramid should be measured
at an elevation some twenty inches higher than previously thought, and the
dimensions of its base were therefore smaller than had been thought by Smyth
and Taylor (and by the French who had dismissed Jomard). Petrie came up
with a figure for the base which he interpreted as being 440 of the smaller
20.63 inch cubits, which were also used for the dimensions of the Kings
Chamber. The height Petrie figured at 280 cubits. While this spelled the
end of Smyth's theory about the length of the perimeter being connected
to the number of days in a year, it did confirm the connection to p, because
22/7 = 3.14286, which is a very good working approximation for pi.
Following the completion of Petrie's survey in 1883, the scholarly community
was only too happy to forget Smyth and Taylor, as they had never really
paid any attention to them in the first place. Petrie became Sir Flinders
Petrie, and it was to be his work alone which would be quoted with respect
to the measurement of the Great Pyramid, as he rapidly became the most respected
authority on the matter.
One of the great mysteries about the Great Pyramid remained the apparently
incomprehensible design of the Grand Gallery, an elaborate corbelled vault
forming the upper half of the Ascending Passage leading to the King's Chamber.
One of the most elegantly simple, coherent, and widely ignored theories
about the Great Pyramid was the astronomer Richard Anthony Proctor's explanation
of this aspect of its design. Proctor was inspired by a passage in the neo-Platonic
philosopher Proculus's commentary on Plato's Timaes, which mentioned that
before the Great Pyramid was completed it was used as an observatory. Based
on his reading of the account he surmised that when the Pyramid was completed
to its fiftieth course, i.e. to the level of the top of the Grand Gallery,
which was also the floor of the Kings Chamber, it would have made an excellent
observatory. He documented his theory in a book published in the late nineteenth
century titled The Great Pyramid, Observatory, Tomb and Temple.
In Search of a Plausible Model of Stellar Alignment
Based on the calculated alignment of the southern air shaft of the Queens
Chamber with Sirius, and the southern air shafts of the King's Chamber with
Zeta Orionis, c2450 BC, Robert Bauval dates the construction (of at least
the upper portion) of the Great Pyramid to that period. He also claims that
the northern air shaft of the King's Chamber aligned with Alpha Draconis
during that same period, while the northern shaft of the Queen's Chamber
is supposed to have aligned with Kochab in Ursa Minor. Bauval's date is
also in agreement with Mark Lehner's carbon dating of quarry marks found
above the King's Chamber. Thus the c2450 BC date looks very good. It also
works very well with Manetho's king list, which for a long time was virtually
the sole means by which a valid chronology for assigning dates was claimed
by Egyptology.
There is however one question about this which still bothers me. For many
years prior to Bauval's work there was a theory, first promulgated by Sir
John Herschel and later repeated and advocated by Sir Flinders Petrie among
others, that the descending passage had been aligned by means of a circumpolar
star. William Proctor's very convincing explanation of the design of the
Ascending Passage was also contingent upon the use of such a star, as it
would have provided not only the alignment of the Descending Passage, but
also insured the perfect alignment of the Ascending Passage and Grand Gallery
with the meridian. Taken together with the textural evidence for the 'stretching
of the cord' ceremony, everything suggests that the descending passage must
have been aligned on a circumpolar star. The astronomers immediately picked
out Alpha Draconis because it was in about the right place in about the
right time, and it was third magnitude, which although not bright, would
be easily visible. The dates given by Herschel for when Alpha Draconis made
a lower culmination at the appropriate angle were 3440 BC and 2123 BC. Assuming
those calculations are still valid, we would be forced to select the earlier
date, if it was in fact Alpha Draconis which was used. This appears more
than a little troubling as it leaves the Pyramid in an unfinished condition
for a thousand years, a little long even for Proctor's taste. Thus it would
appear that we must find a different star, of less than third magnitude,
which was at the right location c2500 BC. It occurs to me that we might
also look for evidence of a super nova in that location about 4,500 years
ago, but that seems an extremely remote possible explanation.
Two other thoughts on the question come to mind which I have not seen explicitly
discussed in print. First, it has been pointed out repeatedly that the angle
of the Descending Passage is very close to the diagonal of a double square,
26* 33' 54" . The most recently measured angle for the Descending Passage
is 26* 30' 53", while the Ascending Passage measured 26* 02' 30".
The coincidence between the angle of alignment of these passages and the
angle of the diagonal of a double square have led some to claim that the
passages were aligned solely on the basis of this geometry. However, to
me the perfection of their north-south alignment strongly suggests that
a faint circumpolar star was chosen which made a lower culmination at this
point, thus allowing the angle of the shafts to also define a double square.
This is, I believe, an example of one of those instances where the design
can be demonstrated to simultaneously integrate and reconcile at least two
issues. Indeed, I strongly suspect that one of the chief objectives in the
design of all sacred architecture is to express the harmonious integration
of all aspects of creation. If one believes that this is in fact the underlying
nature of reality, then one will undertake the design process with the belief
that it will be possible to arrive at such solutions, and unlikely as it
might appear to the modern mind, mired in a reductionist belief system,
one finds that such solutions do exist. I know this from my own experience
as an architect who spent a great deal of time studying the nature of pure
geometry, and solar astronomy, at the drafting board. It is on the basis
of this experience that I make my second observation, which is about the
angle of inclination, versus the ceiling height, of the Grand Gallery.
It occurred to me that the height of the ceiling might have been chosen
such that when the Pyramid was completed to the 50th course, as in Proctor's
theory, the Sun just reached the northern end of the floor of the Grand
Gallery on the Winter Solstice. At noon, on the Winter Solstice, at 30*
north latitude, in our era, the Sun is about 36* 33' above the horizon.
If we subtract the angle of the Grand Gallery from this figure we would
arrive at a Sun angle of about 10* 30' greater than the slope angle. By
taking the tangent of this angle, multiplied by the effective length of
the Grand Gallery, we find that, given the height of the ceiling, the Sun
would not penetrate to the back wall. It would strike the floor about 90%
of the distance to the back wall. But, the angle of obliquity of the ecliptic
has slowly shifted over time. Stecchini tells us that Egyptian texts say
it was 23* 51' when their geodetic system was established. If we rerun these
admittedly rough calculations with this figure, we arrive at a slightly
smaller sun angle of 10* 09'. By taking the tangent of this angle, times
the effective length of the Grand Gallery, we find that the Sun appears
to reach to about 96% of the distance to the back wall. Close enough that
it would appear to be worth checking the geometry accurately to determine
at what angle of obliquity the sun would have reached all the way to the
back. Unfortunately, the rate of change of the obliquity of the ecliptic
appears to be uneven and therefore unpredictable by any model yet developed.
Returning to the question of which star might have been used for the alignment
of the Descending Passage, there is another piece of evidence which I noticed
at Saqqara, which I have not seen mentioned in the context of discussions
of the alignment of the Great Pyramid, or anywhere else for that matter.
On the north side of the Step Pyramid, attributed to Zoser, there is an
odd little box or room with a statue inside, seated looking up and out to
the north through two eye holes. The figure is clearly sighting on the lower
culmination of a circumpolar star to layout the meridian which defines true
north for the pyramid, and for the entire complex. The slab forming the
roof of the box is slanted at the same angle at which the figure is sighting,
just under 15*. The Step Pyramid at Saqqara is attributed to the 3rd dynasty,
less than two hundred years before construction of the Great Pyramid at
the Giza Plateau. There is apparently no evidence of monumental stone construction
in Egypt prior to the Step Pyramid at Saqqara and the complex appears to
me to be a goedetic laboratory. The design is attributed to Imhotep, who
was later canonized as a demigod in Egyptian mythology, while it was not
even recorded in any surviving text or inscription which pharaohs built
the pyramids at Giza. All of this, suggest that Imhotep may very well have
been responsible not just for the design the Step Pyramid, but possibly
for the design of the entire pyramid complex at Giza as well. In any case,
there is the question of why the figure at Saqqara is sighting on a star
at just under 15* when, apparently less than two hundred years later, the
Descending Passages is aligned at an angle of 26* 30'. It seems obvious
that the answer is that they were using two different stars, and that the
most easily observed circumpolar star was the one they used first, at Saqqara,
observable then at just under 15*. This would also appear to further corroborate
my contention that by the time the Great Pyramid was built they had selected
a much fainter star, but at the angle which would simultaneously give them
the diagonal of a double square.
If an appropriate star could be found with a sighting tube, but were too
faint to be used for sighting in construction, then an extension like the
one proposed by E. M. Antoniadi might be used at first. In this arrangement,
a candle might be mounted on trestle aligned in front of the star to sight
on at first. Once the passage got deeper, the star would work no matter
how dim it was, because the screening effect would amplify the star as if
magnified. The Grand Gallery would have formed a vertical swath with transiting
stars each appearing at one edge and moving steadily westward, in say one
second of time. Would the width of the Grand Gallery not likely encode the
cubit distances used to construct the larger whole? If the time it takes
a star to transit the gap were proportional to, or identical to, the time
it took the sector of earth described by the Pyramid to pass through the
same period of movement then the width of the Grand Gallery might be the
most logical place to look for the derivation of cubits.
Cole's Survey
It should be remembered that the only reason we can even measure these subtle
variations or 'errors' in the alignment of the Great Pyramid is because
it was intentionally and deliberately constructed to such exact tolerances
that these subtitles are discernible. This, combined with the fact that
the casing stones were finished and placed with such precision that we can
measure the angle of their faces to two significant figures in arc seconds,
and their mortar joints in hundredths of an inch, illustrate that the degree
of precision attained in both its design and execution should not be underestimated.
These are Stecchini's conclusions based on his analysis of Cole's data:
The circumference of the base is equal to one half of one minute of a degree
of latitude at the equator. The length of one side is also equal to the
distance swept by the rotation of the earth at the latitude of the Pyramid
in one second of time. The length of the apothem, which was equal to one
tenth of a minute of a degree of latitude at the Pyramid, without its pyramidion,
also gave the length of one tenth of a degree of latitude at the north pole
by including the pyramidion. Values for intermediate latitudes might thus
have been inscribed as marks ascending the pyramidion. The common consensus
ever since Petrie has been that the proportions of the Great Pyramid were
as follows: 280 cubits high, with a 440 cubit base, giving a median triangle
of 220 cubits, and an apothem of 356 cubits. These lengths give very nice
approximations for both pi and the golden mean phi, because
22 / 28 approximates pi / 4 with pi = 3 1/7, while 356 / 220
= 89 / 55 which is a very good Fibonacci approximation of the golden mean.
Stecchini, however, claims that these numbers were only the first approximation
which was then adjusted slightly. He points out that this was necessary
because, in the first place, it is impossible to make a right triangle with
the edge lengths listed; 356 squared = 126,736 while the sum of the squares
of the other two sides equals 126,800. Thus the length of average side of
the base had to be reduced slightly, he believes to 439 1/2 cubits while
the height was adjusted to 279 15/28 cubits. All Egyptian fractions are
made up of sums of unit fractions. This one would be 1/2 + 1/28.
Analysis of Cole's figures for the alignment of the sides with the true
cardinal directions shows that the north and west sides are within an error
of 0* 00' 02" less than perfectly perpendicular to each other. (Roughly
the angle subtended by a quarter at a distance of a mile). The north side
faces 0* 02' 28" west of true north, while west side faces exactly
0* 02' 30" south of due west. The east side faces exactly 0* 05' 30"
north of due east, and the south side faces 0* 02' 03" east of true
south with an apparent error of 0* 00' 03" Stecchini points out that
if a consistent (intentional) rotation of 0* 02' 30" counter clockwise
is assumed, then the east side faces exactly 0* 03' 00" north of virtual
east, while the south side actually faces 0* 00' 30" west of virtual
south with a 0* 00' 03" error. A consistent intentional deviation of
0* 02' 30" seems plausible when seen against the precise incremental
values of angular variation found in all four faces. It would seem reasonable
to believe that this is not due to an error, or a shift in the actual alignment
of the earth's crust because the discrepancy is so precisely two and one
half arc minutes. It is more plausible that it was due to the intentional
introduction of an astronomically symbolic rotation. Stecchini points out
that it represents the relationship between time and space expressed as
the amount precessional motion changes the angular alignment of the earth
in space, in exactly three years time. An alternative explanation might
be that this represents a consistent error in their timing or their method
of astronomical observation. However, if this rotation were due to an error,
it would seem to contradict the equally precise adjustments in the alignment
of the east and south faces, which deviate from the cardinal axes by equally
precise amounts in other directions. All of this taken together suggests
that the ancients were engaged in articulating a remarkably subtle geometric
geodetic language.
Stecchini claims that the best evidence of the intentional manipulation
of the lengths of the edges of the Great Pyramid may be the location of
a 'midpoint' mark found near the center of the base of the north face. This
mark was located by Cole at a point 115.090 meters from the north-western
corner, but 115.161 meters from the north-eastern corner, indicating to
Stecchini that it was in fact the point due north from the apex, allowing
for the shortening of one end of the north face, due to the angled orientation
of the eastern face. (See "critical notes"
attached to this paper.)
The deviation of the Second Pyramid from square, as measured by Petrie,
appears to bear this out, as it too demonstrates an apparently intentional
shortening of the north side with respect to the south. But here the north
and south sides are parallel, with the west side again perpendicular to
the north side, and thus to the south side as well, while the east side
is again angled to face north of east. Unfortunately, at the time of Petrie's
survey true north in Egypt had not yet been adequately (re)calculated so,
as Petrie himself advises, his measurements can only be used to establish
relative angular relationships, not absolute compass directions.
Once one understands the alignment in this way, it appears that the east
and south faces of the Great Pyramid are intentionally tweaked out of square
and it raises the question, why? Stecchini doesn't really address this except
in the context of variation of height to base ratios in the north and west
side sections. He hypothesizes that the west side represented p exactly
while the north side represented j, the golden mean. His argument is very
precisely worked out and I am inclined to suspect he is correct. However,
this says nothing about the other two faces, except that their lengths were
adjusted to maintain the overall average in the correct ratio to the height
(with and without the pyramidion). What would justify this rotation? It
would have to be a compelling reason. Stecchini suggests that the north
sides of both major pyramids were shortened. He connects this to a similar
feature in the design of the Parthenon and apparently other temples as well.
He goes on to say he can not understand why, although I immediately suspect
this may have involved the representation of the shortening of a degree
of longitude as one moves north, perhaps even the shortening encountered
over that very distance? This would require some fairly extensive research
to substantiate if it were true.
However, another thought about the implications of the angled alignment
of the east and south faces occurred to me. It has been pointed out that
the faces of the Pyramid were so highly polished that they would have cast
reflections, something like inverse shadows. These reflections and shadows
moved over the ground surrounding the pyramids, and across the face of each
other under some conditions. When the sun was exactly due east, the reflection
would point due east, while the shadow would point due west, if the faces
were aligned perfectly with the points of the compass. The fact that the
two most significant faces for this phenomenon were not, suggests to me
that something significant was intended, or was being corrected for. It
immediately occurred to me that the east face is associated with the rising
sun, and what in modern astrology has come down to us (most likely from
the ancient Egyptians) as the Ascendant in an astrological chart, while
the south face represents the midday sun, or the Midheaven of a chart. To
intentionally capriciously tweak the very faces of the Pyramid which would
measure and represent these two most important points seemed not only unthinkable,
but reversed. If it was a question of which faces to tweak in the service
of two others, the west would be subjugated to the east and the north would
most likely be adjusted to preserve the south. So, it seemed more likely
to me that if the east and south were adjusted it was with some immediate
purpose related to their relationship to the sun. I have not come to any
deeper hypothesis as to the details of this manipulation, although the obvious
suspicion is that it has to do with correcting for the effect of atmospheric
refraction. It might adjust the timing of a solar event to be better reconciled
with the time scale of stellar decans. This might also explain why the correction
to the east, at sunrise, must be greater than to the south, at midday, when
the sun penetrates less atmosphere. This is far from a conclusive theory,
but seems worth pursuing as a working hypothesis.
Lockyer at Karnak
In the course of searching for a copy of Cole's survey, I found bound with
it a copy of a survey of Karnak published in 1920. This was the first and
most accurate survey of the alignment of the central axis of that temple
made after the rubble had been entirely cleared from its axis. The author,
F. S. Richards, first quoted Lockyer's claim, and then gleefully explained
that according to the new definitive survey, the correct alignment of the
axis was too far north for the setting sun to have been used for its alignment.
He elaborated a then standard formula for the rate of change in the angle
of obliquity of the ecliptic as proof that the date at which the sun would
have aligned was so far back as to be absurd. More recent sources have stated
that extrapolation of any formula from modern observations of the rate of
change of the ecliptic is unreliable, as nobody has been able to devise
a reliably predictive mathematical model of this (perhaps chaotic) movement.
However, I suspect that the formula in question was the same one which Lockyer
himself had used, and I doubt that the degree of uncertainty is large enough
to salvage Lockyer's claim of an alignment based on the angle of the sun
at sunset on the winter solstice. It occurs to me that one way to check
the formula, in fact the only plausible corroboration I can imagine, would
be to use the Great Pyramid and Egyptian astronomical texts as a calibration
point. Stecchini claims, on the basis of his reading of hieroglyphic texts,
that the ancient Egyptians explicitly say that the angle of obliquity of
the ecliptic was 23* 51' when they established geodetic measurement of Egypt
in pre-dynastic times. Most mainstream scholars appear no more ready to
believe Stecchini than Lockyer, and Stecchini seems to feel that the establishment
of geodetic measure substantially predated the building of the Pyramid.
But, if we assume that the Pyramid was in fact built c2500BC, and that Stecchini's
reading of the texts is correct then the angle of the obliquity of the ecliptic,
which is 23* 27' now, would have had to have been 23* 51' or less, c2500
BC. The value plotted from the formula in the survey paper was 23* 58' 44"
c2500 BC. Thus, if Stecchini's reading of the texts is correct, then the
formula is wrong, but in the opposite direction from that which would be
required to redeem Lockyer's solar alignment hypothesis. If the formula
were to be correct, and Stecchini's reading was also correct, than the geodetic
system would not have been established until roughly 1500BC. This is at
odds with both Buaval's astronomical data on the alignment of the Great
Pyramid and Lehner's carbon dating evidence. Thus it appears likely that
if Stecchini is right about the texts, then the formula overestimates the
rate of change in the obliquity of the ecliptic. There are still fluctuations
due to nutation to take into account, but that becomes even more arcane.
I suspect that the 1920 survey paper on Karnak was the key piece of evidence
used by Egyptologists to successfully refute and dismiss Lockyer early in
this century. Most were already hostile to his entire method of investigation.
It frequently led him to claim dates for temple alignments in the neighborhood
of 4,500BC. While his methods were heretical, such dates were not necessarily
out of line with those of mainstream Egyptologists. Lockyer would not, I
believe, have argued that the temples in question (or even their foundations?)
were older than the Giza Pyramids, he and many others scholars of his time
simply had a more expanded chronology than the one around which the modern
consensus has formed. The more recent dating of the Great Pyramid c2450BC
looks very good, so it seems safe to conclude he was wrong, at least about
the specifics of some of his claims. His error may be more a question of
which astronomical objects made the alignments, rather than whether temples
were in fact aligned with heavenly bodies.
There are a great many Egyptian (and Mesoamerican) temples with their principal
axes diverging from true north, but not by enough to align with the sun
at sunrise or sunset on the summer solstice, which is the day the sun rises
and sets farthest north. This troubled me a great deal when I was there,
and it almost as troubling that in most cases, accurate surveys showing
the alignment of these temples, measured in terms of their angular deviation
from true north, are not easily available, if they exist at all. Karnak
was apparently only surveyed so accurately in 1914 to refute Lockyer.
The Axis of the Universe is a Pregnant Hippopotamus
In all of the ancient Egyptian astronomical diagrams there is one figure
which is always larger than all the rest, and most frequently found at the
center of what appears to be a horizontal parade of figures. This figure
is Taweret "the Great one", a goddess depicted as a pregnant hippopotamus
standing upright. It is no mystery that this figure represents a northern
constellation associated, at least in part, with our modern constellation
of Draco the dragon.
What is more of a mystery is why this particular constellation should be
so important. In the Dendera zodiac it is found near the center of the circle,
but it does not contain the north star, which even in Ptolemaic times was
close to Polaris which is at the center of the entire circular diagram.
What the Taweret figure in the Dendera zodiac does contain, literally as
if it were the heart inside her chest, is the center point of a circle which
defines the constellations of the zodiac and thus the ecliptic. The center
of this circle is thus the exact location of the pole or axis of the ecliptic.
This is not an immediately obvious or visible point, but it is the point
about which the polar axis of the earth's equator gyrates. In other words,
it is the axis about which the precession of the equinoxes revolves. It
would make sense that the icon representing this point would be depicted
larger than all other features of the sky, she is, after all, the mother
of all other cycles - one might even say Mithras' grandmother.
With this in mind, it occurred to me that this point could be understood
as a direction in an angular relationship to true north, where the angle
would change over the course of the precessional cycle. This angle might
best be depicted in terms of two different directions in the plane of the
prime meridian, one line pointing toward equatorial axis, what we think
of as the north star, and a second, divergent by the angle of obliquity
of the ecliptic, pointing toward the pole of the ecliptic. It is possible
that these two directions may coincide with orientation of the northern
air shafts of the King's and Queen's chambers of the Great Pyramid. But,
in addition, one might also portray this angle as a projection onto the
surface of the earth. In this configuration depending upon day chosen, say
the summer solstice, the angular deviation from true north would be a function
of that point in time in the precessional cycle. Thus it is possible to
imagine a system in which the alignment of different temples axes, each
deviating slightly from a true north axis, actually portrayed different
points in the long term precessional cycle by their respective deviations
from true north. I do not claim that this is actually the case, only that
it is in theory plausible and therefore worth investigating further. It
is also not clear to me whether this line of investigation might have already
been pursued by Lockyer himself, but never published. It was Lockyer who
first pointed out that the Babylonians distinguished the pole of the equator
which they called Bil, from the pole of the ecliptic which was called
Anu.
Neugebauer on Stellar Calendars
One of the chief bodies of evidence cited in support of the claim that the
ancient Egyptians did not know about the precession of the equinoxes is
the work of Otto Neugebauer and Richard Parker on ancient Egyptian astronomy.
The earliest sources used for the basis of this work are diagonal star clocks,
or calendars, taken from coffins dating from the ninth and tenth dynasties.
The Great Pyramid in contrast is attributed to the fourth dynasty. If one
assumes that knowledge grows and improves over time, then it would seem
reasonable to expect that examining later evidence would only indicate a
maximum against which earlier periods could be compared. However, in the
case of Egypt this modern intuitive assumption may not be valid. The basis
of Neugebauer's own argument is that the information found in the coffin
texts themselves declines over time. This is well documented in his work,
where it easy to see that the earliest star clock is virtually perfect,
but each of the others are then progressively more corrupt. He takes this
as evidence that the Egyptians did not understand precession and therefore
the system of clock/calendars became progressively more corrupt over time.
What is implicit in this explanation, however, is that at some point in
time someone understood the system well enough to make a clock that worked.
The first ones are almost perfect. It is assumed that subsequently it was
mindlessly copied without any insight, and those that did tinker with it
attempting to correct the growing errors could do nothing to correct the
underlying problem because they did not understand precession which was
giving rise to the errors. While this may demonstrate that during the period
in question those attempting to patch up the now traditional ceremonial
calendars could not totally rework them, it does not necessarily prove that
no one understood what was causing the problem. It is even more questionable
to claim that as a result of this several hundred year long process no one
figured out what was going on with precession. These calendars also clearly
do not show that several hundred years earlier, at the time the Great Pyramid
was built, those who designed the Pyramid had suffered from the same ignorance.
Indeed the more interesting question to me seems to be one of how that knowledge
was apparently maintained over the intervening several hundred years, such
that the first star clock is accurate, but subsequent ones remain static
and thus progressively become less accurate.
Indeed, one could even devise a thought experiment in which it is assumed
that the first rule among those who receive initiation into the highest
knowledge is that the knowledge may not be written down explicitly. This
knowledge might be maintained by an initiate priesthood who would continuously
evaluate the trustworthiness of initiates, including pharaohs. The high
priests might then initiate each individual only to the degree to which
they demonstrated their capacity and dedication to maintaining the integrity
of the secret. At first glance this may seem an absurd proposition, but
it may fit the observed phenomena better than any other explanation. Particularly
where we do already know that someone maintained a great deal of knowledge
over the span of long intermediate periods between dynasties, which only
flower during certain phases. Such a model might explain how, or why, the
evidence we do find in texts is frequently inferior to the sophistication
of the knowledge implicitly embodied in the geometry of the architecture.
I am not arguing that this can actually be proven, or even that it is rigorously
correct in all cases, but I do suspect that the depth and sophistication
of all dynasties was not equal, and that this may be due in part to variations
in the degree to which different pharaohs, or lines of pharaohs, inspired
the confidence of the perennial priesthood who may well have been the seat
of real knowledge. It is also entirely possible that the accuracy and depth
of understanding waxed and waned over the three thousand year span of dynastic
rule. It seems clear that at the time of construction of the Great Pyramid
astronomical and geodetic knowledge, including the precession of the equinoxes
must have been exceedingly complete. It is possible that this knowledge
might have declined to the point seven hundred years later, when the coffin
text calendars were written, that Egyptians knew something had been known,
but not exactly how it all worked. It might then have taken several hundred
years of cumulative errors to arise due to precession, before the correct
understanding was once again reestablished, at least in some circles. This
would still be over a thousand years before Pythagoras. However, it seems
highly unlikely that a civilization which lasted for over three thousand
years, intensely watching and recording their observations of the heavens
over that period would not become aware of the phenomenon of precession.
This becomes even more striking when we see a dramatic shift from an iconography
portraying bulls, to one portraying rams, at precisely the time when the
vernal equinox precessed from Taurus into Aries.
Tentative Conclusions
The more I study the situation, the more I find that it is difficult, if
not impossible, to come to any absolute conclusions about ancient Egypt.
There are so many contradictory opinions and theories based on so much contradictory
evidence. However, this is not to say that I believe we must abandon the
field to the minimalists. The confusion no more proves them right than the
physical evidence of the Pyramid proves the biblical prophecy mystics right.
After reviewing the opinions and work of the best of the Egyptian astronomical
tradition: Sir Issac Newton, Sir John Hershel, and Sir Norman Lockyer, Neugebauer
& Parker, Livio Catullo Stecchini, Robert Bauval, and even Schwaller
de Lubicz, and finally, visiting the key sites myself, I believe the situation
we are faced with is one in which it can be demonstrated that c2500 BC someone
designed and oversaw the construction of an object, the Great Pyramid, which
encoded exceedingly accurate geodetic information along with profound geometric
insight and subtly. While it is disorienting to recognize that so early
in the chronology of human civilization there stands such a discontinuous
alpha point, it exists, and attempting to dismiss its implications is no
substitute for grappling honestly with them.
If Egypt did spring from primitives to pyramids in a scant few hundred years,
it would raise serious questions about the nature of the evolution of consciousness.
It would imply an episode of punctuated equilibrium which might only be
matched by the tempo of our own times. The acceptance of such an event at
the root of history might ultimately force one to an archetypal view which
could threaten to make the ideas of biblical creationists look almost trivial
in comparison. I am not yet prepared to reject such an eventuality out of
hand, but it would appear to be so at odds with everything else we have
been able to observe in more recent history that I am inclined to first
look for a more reasonable rational explanation.
The most obvious explanation would be that the Great Pyramid, and perhaps
the entire Giza Plateau, was the culmination of a lineage of observational
astronomy and mathematics which had developed over a long period of time.
Since the archeologists have found little or no evidence of this in Egypt,
it would appear that it would have to have been brought to Egypt from elsewhere,
perhaps by the same people who designed the Great Pyramid. Such a hypothesis
is appealing from a number of perspectives. First, it addresses not only
the question of the time needed to evolve the knowledge, but it also leaves
open the possibility that those responsible might subsequently have departed
Egypt, perhaps taking with them key parts of their knowledge. A residue
of their more complete knowledge might then have deteriorated in later dynasties
in Egypt, or have been maintained by only a select cadre of initiates.
Something like this scenario would appear to be necessary to reconcile the
existence of the Great Pyramid with Neugebauer and Parker's interpretation
of the diagonal star clocks taken from Middle Kingdom coffins. These appear
to show the steady deterioration of a calendar over time based on cumulative
errors due to precession of the equinoxes. Neugebauer and Parker take this
as proof that the authors of these diagrams, and their respective pharaohs,
did not recognize or understand precession.
This may be true, but it does not disprove the possibility that the keepers
of the highest information may not have divulged it, even to the presiding
pharaoh. It is more likely that the star clocks themselves became tradition
bound iconographic symbols which were patched up and made to work as long
as possible because it was culturally too difficult to abandon them, as
long as they could be adjusted. This is not unprecedented in human cultural
behavior. There is, however, a deeper critique of Neugebauer and Parker.
How is it that we can be sure that after watching these star calendars continuously
slide out of alignment over several hundred years, the later Egyptians did
not figured out precession based on those very errors. Can we really conclude
that the ancient Egyptians were really that ignorant and stupid for that
long? Especially in light of their fascination with recording the movements
of the stars, and their reputation for keeping knowledge secret.