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.


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.