The Keys of Atlantis

A Study of Ancient Unified Numerical and Metrological Systems

by Peter Wakefield Sault

Copyright © Peter Wakefield Sault 1973-2008
All rights reserved worldwide

Sections

1.		Abstract of Prior Knowledge
2.		Earth And Moon

Illustrations

1.		The Pyramids of Giza
2.		Proportions of The Great Pyramid
3.		Squaring The Circle
4.		Theoretical Proportions of The King's Chamber
5.		Theoretical Proportions of The Second Pyramid

Animations

1.

Earth, Moon And Great Pyramid

Tables

1.

Theoretical Dimensions of The King's Chamber

References

3-1. Abstract of Prior Knowledge

Figure 3-1. The Pyramids of Giza

The Great Pyramid, also known as the Pyramid of Khufu, or Kheops, stands on the Giza Plateau in Egypt at 29°58'34" N. × 31°07'50" E.. A plan of the location showing the three main pyramids can be viewed in Appendix C.

Peter Tompkins summarizes the origin of the Great Pyramid thus[1]

“Though all agree that the Great Pyramid is at least four thousand years old, none can say for certain just when it was built, by whom, or why.”

The first thing to note about this observation is that Tompkins refrains from attempting to assign a maximum age to the Great Pyramid.

Tompkins' book ‘Secrets of The Great Pyramid’ (1971) is without doubt the most comprehensive compendium of myths, legends and knowledge about the Great Pyramid ever published and is, as noted on his website*, “not yet faulted in its original content”.

Tompkins goes on to say[2]

“It would be satisfactory to be able to describe the method by which the Great Pyramid was put together, by whom, and when.

“But the builders, whoever they may have been, left no description of their method. No one has even found a later Egyptian report of how the first pyramids were built.”

So, before proceeding any further, the reader must set aside whatever opinions of the Egyptological establishment he or she has encountered, no matter how knowledgeable those opinions might seem to be, in the clear realization that, in the complete absence of actual knowledge, nobody's guess is any better than anyone else's. That includes any guess that the three principal Pyramids of Giza were built by the people we call ‘Ancient Egyptians’, meaning the occupants of the Nile Valley from about 4000 BC onwards, with a recorded history beginning around 3000 BC. There is simply no evidence that would place such a guess above any other and as the present director of the Egyptian Department of Antiquities, Zahi Hawass, has been quoted as saying in respect of the ‘Exodus’ of the Hebrews “Without historical evidence we are forced to say that some things never happened”.

The earliest extant reference to the Great Pyramid comes from Herodotus, who says[7]

“To build the pyramid itself took twenty years; it is square at the base, its height (800 feet) equal to the length of each side; it is of polished stone blocks beautifully fitted, none of the blocks being less than thirty feet long.”

It would be another 2,000 years before the pyramids were accurately surveyed. With the missing parts of the relationship inserted:- [the circumference of a circle whose radius is] the height of the Pyramid is equal to [the perimeter of] the base, using the value 3 ¹/7 for p (pi).

Figure 3-2. Proportions of the Great Pyramid

Some of the earlier surveys contained errors and gave rise to some preposterous theories, none of which will be discussed here, about which Prof. W.M.F.Petrie says[8]

“It is useless to state the real truth of the matter, as it has no effect on those who are subject to this type of hallucination. They can but be left with the flat earth believers and other such people to whom a theory is dearer than a fact.”

The results of the two most reputable surveys of the Pyramids, those of W.M.F.Petrie (1881) and J.H.Cole (1925), are shown in Appendix C. Since these surveys were conducted, certain peculiarities in the form of the Great Pyramid have come to light, not the least being a very slight indentation of the sides which results in the distance across the middle of the base (BC in Figure 3-2), between opposite apothems (AB and AC in Figure 3-2), being shorter than the lengths of the sides parallel to it, by about 1.2 metres. The indentations went unnoticed until discovered by aerial photography and are visible only when the Sun is shining across a face, throwing half of that face into shadow. Their presence testifies to the high degree of precision with which the Great Pyramid was built.

The first person to comment on the apparent p (pi) relationship between the height of the Pyramid and the perimeter of its base was John Taylor, working from measurements made by Howard Vyse in the 1830s. Petrie later opined that it formed the basis of one of the best theories regarding the intended dimensions; in so-called ‘Egyptian’ cubits (each equal to 20.620 English inches, or 523.75mm), 280 for the height and 440 for the width of the base, expressing a ratio between the two of 7:11 and thus embodying the well-known approximation of ¹/7 for p. This means that the height is equal to the radius of a circle whose circumference very closely approximates the perimeter of the base.

Figure 3-3. Squaring The Circle

A Reexamination of The Data: Separating Fact from Fiction

It might seem to the reader that were the Great Pyramid laid out and constructed using an Egyptian unit of measure then that would be evidence of it being an Egyptian monument. However, the cubit to which Petrie refers was contrived by Sir Isaac Newton (1642-1727) from measurements of the G.P. cavity known as the ‘King's Chamber’ made separately by John Greaves (1638) and Tito Livio Burattini (1642). The naming of Newton's cubit as ‘Egyptian’ is based on the prior, unfounded assumption that the Great Pyramid must, of necessity in Newton's view, accord with both the chronology and the metrology of ‘The Holy Scriptures’ and is, therefore, an Egyptian monument. Newton was obsessed with Bible codes and had, moreover, acquired from an unknown source the belief that the builders had encoded the dimensions of the Earth in those of the Great Pyramid. As Tompkins records[3]

“...it was from Greaves's data that Sir Isaac Newton deduced that the Great Pyramid had been built on the basis of two different cubits, one of which he called ‘profane’ and the other which called ‘sacred’. From Greaves's and Burattini's measurements of the King's Chamber, Newton computed that a cubit of 20.63 British inches produced a room with an even length of cubits: 20 x 10. This cubit Newton called the ‘profane’, or Memphis, cubit; whereas a longer, more arcane cubit appeared to measure about 25 British inches.

“This longer, or ‘sacred’, cubit Newton derived from the Jewish historian Josephus's description of the circumference of the pillars of the Temple at Jerusalem. Newton estimated this cubit to be between 24.80 and 25.02 English inches, but believed the figure could be refined through further measurements of the Great Pyramid and other ancient buildings.

“All of this Newton wrote up in a small and now hard-to-find paper called A Dissertation upon the Sacred Cubit of the Jews and the Cubits of several Nations; in which, as taken by Mr. John Greaves, the ancient Cubit of Memphis is determined.

“Newton's preoccupation with establishing the cubit of the ancient Egyptians was no idle curiosity, nor just a desire to find a universal standard of measure; his general theory of gravitation, which he had not yet announced, was dependent on an accurate knowledge of the circumference of the earth. All he had to go on were the old figures of Eratosthenes and his followers, and on their figures his theory did not work out accurately.

“By establishing the cubit of the ancient Egyptians, Newton hoped to find the exact length of their stadium, reputed by classical authors to bear a relation to a geographical degree, and this he believed to be somehow enshrined in the proportions of the Great Pyramid.”

Hence it is plain to see that Newton was working backwards from a purely hypothetical Israelite cubit allegedly described by a now largely discredited historian and of which no physical example existed, via the ratio 6:7 (that of a hypothetical ‘profane’, or ‘common’, cubit to a hypothetical ‘sacred’, or ‘royal’, cubit and an approximation to Ö3:2, that being the exact ratio of the diameters of 30th Parallel and Equator on a sphere), to a division by 10 of the width of the floor of the King's Chamber of the Great Pyramid thence to postulate a ‘Memphis’ cubit that had existed since the foundation of Egypt. Clearly this line of argument is utterly phantasmagorical and no rational person could ever take it seriously. Nonetheless, subsequent authorities bowed to that of Newton and we find, for example, Dr. Charles Piazzi Smyth lauding Newton thus[4]

“How thankful should we be that it pleased God to raise up the spirit of Newton amongst us; and enable him to make one of the most important discoveries of his riper years - though the opposition of the Church of England has caused it to remain unread almost to the present day - that while there undoubtedly was in ancient times a cubit of 20.7 inches nearly... ...and which Newton calls ‘the profane cubit’ there was another which he equally unhesitatingly speaks of as the sacred cubit, decidedly longer.”

We also have Petrie accepting without question that Newton's cubit is a genuine, ancient Egyptian unit of measure[9]

“Thus the total length of the plug-blocks would be about 203 inches, or very probably 206 inches, or 10 [Newton's] cubits, like so many lengths marked out in that passage.”

And then Prof. Livio Stecchini adds his stamp of approval[6]

“It was Newton who, on the basis of the survey conducted by Greaves, realized that the King's Chamber measures 10 by 20 [Newton's] cubits. Having established this fact,...”

The only clear facts that we possess are that the floor-plan of the King's Chamber is in almost exactly 1:2 ratio and that both dimensions of the floor appear to have been intended to be precise, whole-number divisions of both the base-width and the height of the pyramid, making these 44 and 28 chamber-widths respectively. Hence this unit, the width of the chamber, which - unlike Newton's cubit - can be derived without making any arbitrary divisions, is in length 206.3, ±0.2, English inches (5.24m). This is indeed close to Petrie's reported length of the plug-blocks, as noted above. The width of the chamber is to its height as 9:10 and taking this into account the longest common length that can be derived from the proportions is a cubit that is one tenth of the height of the chamber, again unlike Newton's cubit, at ~22.9 English inches (582.2mm). In terms of this Great Pyramid cubit the width of the base and the height of the Great Pyramid come out at 396 and 252 respectively. The figure for the width of the base is remarkably close to Jomard's figure of 400 pyk belady[5].

“In vain Jomard argued that he had found an even more surprising coincidence in that the four-hundredth part of his base of the [Great] Pyramid gave a figure of .5773 meter, which was exactly the length of a longer modern Egyptian cubit called the pyk belady.”

Jomard's measure of the width of the base was, however, slightly excessive since 1/400th of the actual width is 0.5759 metre (22.67 inches), taking J.H.Cole's mean value of 230.364 metres, where 1/396th gives us 0.5817 metre (22.9 inches).

Modern authors seem generally to have made the mistake of thinking that metrological standards in ancient times followed the same principle as those of today and that a given unit of measure would have had exactly the same length no matter who was using it or where. However, as we have seen in the case of the Parthenon stadion, the unit used for the temple, although approximating it, was not the same as that of the general tradesman. It would seem that units were consistent only within the context of their use and that the architect of a given building or complex was at liberty to invent and use his own version of any given unit.

The averaged, corrected measures of the King's Chamber are shown in Table 3-1 below. Petrie's words concerning the condition of the chamber as he found it must be borne in mind.[10]

“The King's Chamber was more completely measured than any other part of the Pyramid; the distances of the walls apart, their verticality in each corner, the course heights, and the levels, were completely observed. On every side the joints of the stones have separated, and the whole chamber is shaken larger. By examining the joints all round the 2nd course, the sum of the estimated openings is, 3 joints opened on N. side, total = .19 [English inches]; 1 joint on E. = .14; 5 joints on S. = .41; 2 joints on W. = .38. And these quantities must be deducted from the measures, in order to get the true original lengths of the chamber. I also observed, in measuring the top near the W., that the width from N. to S. is lengthened .3 by a crack at the S. side.

“These openings or cracks are but the milder signs of the great injury that that the whole chamber has sustained, probably by an earthquake, when every roof beam was broken across near the South side; and since which the whole of the granite ceiling (weighing some 400 tons), is upheld solely by sticking and thrusting. Not only has this wreck overtaken the chamber itself, but in every one of the spaces above it are the massive roof-beams either cracked or torn out of the wall, more or less, at the South side; and the great Eastern and Western walls of limestone, between, and independent of which, the whole of these construction chambers are built, have sunk bodily. All these motions are yet but small - only a matter of an inch or two - but enough to wreck the theoretical strength and stability of these chambers, and to make their downfall a mere question of time and earthquakes.”

It is worth noting Petrie's comment about the levels of floor and ceiling[11].

“The floor of the chamber, as is well known, is quite disconnected from the walls, and stands somewhat above the base of the lowest course. It is very irregular in its level, not only absolutely, but even in relation to the courses; its depth below the first course joint varying 2.29 [English inches], from 42.94 to 40.65.”

What we are dealing with is the art of marrying certain irrational and mutually incommensurable numerical constants, such as p (pi) and Ö2 (square root of 2), using a system of whole numbers and their ratios. In order to do this, some rough-and-ready approximations must be incorporated. The primary approximation expressed by the gross form of the Great Pyramid is that of 6 ²/7, or the perimeter of the base divided by the height, as a representation of the value 2p. Hence there can be no exact expression of any of the constants referenced in any of the physical approximations embodied in any of the structures. In fact, to find a single exact value anywhere would be to exclude all possibility of finding comparable references. The architect has given us a compendium of equally variant approximations.

Figure 3-4. Theoretical Proportions of The King's Chamber

In terms of the proportion 9:10:18, employing the derived cubit of ~22.9 English inches:- The theoretical surface area of the King's Chamber is 864 square cubits. This is equal to the surface area of a cube of edge 12 cubits. The volumes of the chamber and its cube of equal surface area are in the ratio of a naturally tempered musical semitone, 15:16. This is also the ratio of both the widths of the bases (variance 0.34%) and the capacities of the coffers of the Second and Great Pyramids. The sum of the lengths of all the edges of a 12-cubit cube is to that of the chamber as 36:37. The theoretical volume of the Great Pyramid is 13,172,544 cubic cubits.

13,172,544 = 2⁶ × 3⁵ × 7¹ × 11² = 7 × 33² × 12³

That is exactly equal to 7,623 12-cubit cubes; cubes, that is, of surface area equal to that of the theoretical King's Chamber. Now, these 7,623 cubes can be arranged as a stack of 7 equal square tiers of width 33. With the width of the tiers equal to the width of the base of the pyramid, the height of the resulting rectangular block is exactly 1/3rd of the height of the pyramid. Altogether, the figure clearly embodies in whole numbers the arithmetical method for calculating the volume of any pyramid - one third of the height times the area of the base. Such an example cannot be constructed with Newton's cubit for the reasons firstly that it does not give a whole number for the volume of the pyramid, let alone a multiple of the cube of twelve, and secondly that neither dimension, let alone the height, expressed in it is divisible by three a whole number of times.

The width of the base of the Second Pyramid, the one known as Khephren or Khafre, is almost exactly 370 G.P. cubits, making the perimeter 1,480 cubits. Since this pyramid was apparently intended to conform to the proportions of a ‘Pythagorean’ {3-4-5} triangle, this gives theoretical height and apothem of 246 ²/3 and 308 ¹/3 cubits respectively, summing to 555 cubits. In order to express each side as a whole number, it is necessary to derive a unit which is equal to a third part of a cubit. This we shall call a double-palm, since a cubit is normally divided into 6 palms. The theoretical volume of Khephren is then exactly 303,918,000 cubic double-palms.

303,918,000 = 2⁴ × 3¹ × 5³ × 37³ = 6 × 370³

Figure 3-5. Theoretical Proportions of The Second Pyramid

As it happens, the proportions of the Great Pyramid do indeed relate to the size and shape of the Earth as the legends tell us although not, however, in the same manner as some have imagined; the proportions of the Great Pyramid express the ratio of the diameters of Earth - at the very latitude upon which the pyramid is located - and Moon, as explained in Section 3-2 below.

* Peter Tompkins' website was removed from the World-Wide Web in February 2008, a year after his death in January 2007. This author has a copy saved in his archives.

3-2. Earth and Moon