Diagram 16 may look familiar. It is the same construction as Diagram 11 in the Pentagon Section, and shows the manner in which the length of a side of an equal-sided inscribed pentagon may be produced via the finding of the golden ratio division along a circle's radius.

As can be seen above, the right triangle COB has side OC equal to 1 (it is a radius), and the side OB equal to .618034. With the aid of a trigonometric table it can be learned that angle OCB equals 3143', and angle OBC is therefore 5817'. Although the above diagram is of prime significance, it leads to another construction, which may have been recognized by the Egyptian priests as being even more remarkable.

In Diagram 17, one of the right triangle's sides is still equal to .618034 of the radius, but in this instance it is the hypotenuse (and not the adjacent side) that is equal to 1, the circle's radius. With a trigonometric table it can then be learned that the angle opposite the .618034 side equals 3810', and so the remaining angle must be 90 minus this, or 5150'. Using the Pythagorean Theorem, the length of side OD can be found, and it equals .78615 of the radius.

There are two aspects of this construction that make it of particular interest. The first is that the lengths of side ED (.618034) divided by that of side OD (.78615) will, oddly enough, equal the length of this same side OD (.78615). In terms of trigonometric relationships, this means that the tangent (opposite÷adjacent) of 3810' is the same as the cosine (adjacent÷hypotenuse) of 3810' (and, conversely, that the cotangent of 5150' equals the sine of 5150').

The second, and perhaps more important factor, is that the length of side OD (.78615), when multiplied by 4 yields an amount (3.1446) that is almost exactly equal to Pi (3.1416). This finding means that the 3810' right triangle offers a unique and most interesting point of intersection between the Pi ratio and the golden ratio phenomenon.

Were the Egyptian priests of the Old Kingdom period aware of the properties of this triangle? Diagram 18 is a sketch outline of the Great Pyramid. This structure, intentionally or not, was built incorporating the 3810' right triangle in such a way that the sides slope upward at the precise angle of 5150'. If we compare the pyramid cross-section in Diagram 18 to the construction in Diagram 17, we can see that side BC corresponds to .618034 of the radius, side AB corresponds to .78615, and side AC corresponds to 1. (As nearly as can now be determined, in actual Great Pyramid lengths, AB was equal to 481 feet, BC to 377.9 feet, and AC to 611.5 feet).

From here things begin to get really interesting (and hopefully not confusing).

As can be seen, BC above is equal to one half the length of the pyramid's side. Therefore, the perimeter of the base equals BC x 8, and in relative terms this equals .618034 x 8 = 4.9443. The relative height of the pyramid is .78615, and, if one uses this length as the radius of a circle, then the circumference (perimeter) of that circle will also be 4.9443.

How this unexpected agreement comes to be is that: 1) As we saw in the 3810' right triangle, .618034 ÷ .78615 = .78615. This means, that .618034 = .78615 x .78615. Therefore, 8 x .618034 is the same as 8 x .78615 x .78615; and 2). As we also saw, 4 x .78615 is a very close approximation for Pi (). Therefore, 2can be said to equal 8 x .78615. For the circumference of the circle using .78615 as its radius, we then have C =2R = (8 x .78615) x .78615 .As a result, the Great Pyramid can be seen as having the same perimeter length when measured in a horizontal plane, as a square, and in a vertical plane, as a circle.

To this we now add the finding that the actual length of a side of the Great Pyramid, as measured at its base, is 755.73 feet.³⁷ As noted earlier, 8 times this length (i.e. twice the perimeter) is exactly the length of one minute of latitude as measured at the equator, (or only about 12 feet less than the average length of a minute of latitude as measured between 24 and 30 north. See Measuring the Earth discussion). The conceptual unity between this latter "8 times" relationship and the "8 times" relationship in regard to Pi and the 3810' right triangle is startling. Here, then, we may possibly have yet another layer of coincidence embedded in the design details of this building.

Why 360°? Why Base 60?

Throughout these sections we have talked about angles both in terms of degrees and as 1/5th or 1/20th, etc., of a full rotation. We know that the Egyptians had a concept for slope which they called "seked" (equivalent to our cotangent), but beyond this there is no surviving documents explaining how (or even whether) they measured angular separations. Thus far, I have tried to make clear only that the capability existed to make such determinations, regardless of the exact manner in which they may have been done.

However, if the length of a side of the Great Pyramid was intentionally designed to equal 1/8th of a minute of latitude, then the supposition must be that the Egyptian architect was employing a system of 360 with 60 minutes per degree. Is it conceivable that such a system was in use over 2,000 years prior to its recorded emergence?

The idea of a circle being divided into 360 parts (degrees) first appears in the written historical record as an innovation of the Babylonian culture a few hundred years prior to the birth of Christ.³⁸ The division of each degree into 60 'minutes', and each minute into 60 'seconds', etc., is of Babylonian (via Sumerian) provenance as well.

Although there is no surviving written evidence that the ancient Egyptians had previously developed these methods, it is not beyond the realm of possibility that they had. The association of the number 360 during the Old Kingdom with a complete cycle, or circular context, could have come about in a variety of different ways.

As previously mentioned, the Egyptians introduced a 365 day calendar shortly after the unification of Upper and Lower Egypt in about 3,000 B.C.³⁹ With this change, the year was divided into three seasons, each containing four 30 day months. Each of these months was next further divided into three ten-day weeks. As a result, a year contained 36 ten-day weeks for a total of 360 days, with the calendar year's five remaining days being added somewhat ceremoniously to the end of this 360-day period. It is interesting to note that these five added days were not always considered (perhaps for religious reasons) to be a legitimate part of the more preferable 360-day per year cycle.⁴⁰

In addition to this correlation, the number 360 has a direct connection with the sun itself. The sun has an apparent diameter of just over 1/2 of a degree, or about 1/720th of a full circular rotation of the sky. River fog conditions will often allow the sun's disk to be clearly viewed for brief periods with the naked eye, thus making the task of measuring of the sun's relative apparent diameter a fairly simple undertaking along a river such as the Nile. (The apparent diameter of the full moon, though somewhat variable, is also almost exactly 1/2 of a degree.) On a daily basis, due to the Earth's progress in its orbit, the sun appears to move the equivalence of two of its own diameters (i.e., about 1/360th of a full rotation) eastward through the heavens relative to the fixed stars. The Egyptians were very concerned with recording the first visibility immediately before sunrise of various stars, and so would certainly have been well aware of the sun's daily eastward displacement relative to these stars.

None of the above observations are difficult to make, and each would have again brought up numbers related both to circular contexts and to the number 360. It may have been understood, however, that numbers as measured in the exterior world should not be expected to be exactly the same as a particular "ideal" number, but only to represent, or point the way to, this ideal. It is perhaps this approach that Plato had in mind when he has Socrates say:

These sparks that paint the sky.....we must recognize that they fall far short of the truth, the movements namely, of real speed and real slowness in true number and in all true figures both in relation to one another and as vehicles of the things they carry and contain. These can be apprehended only by reason and thought, but not by sight.⁴¹

Corroboration for the choice of 360 could have been taken from the fact that it is wholly divisible by all of the first 12 numbers except for 7 and 11, an attribute that greatly facilitates the further partitioning of the circle into whole number sections. It is also the smallest number that is divisible by 10, 20, 30, 40, 90 and 60.

The concept of dividing a whole unit into 60 parts, and then dividing each of these parts into 60ths, and so on, originated in the Mesopotamian region. There is evidence (drawn from clay tablets excavated at a site known as Jemdet Nasr, located in present day Iraq) that the workings of a "base sixty" system was already in use by about 3,000 B.C.⁴² There is also evidence of substantial Mesopotamian influence taking place in Egypt at precisely this same point in time.⁴³ In fact, some of the evidence of such contact is based on findings unearthed at this same Jemdet Nasr site. With the Mesopotamian impact of this period having affected Egyptian architectural and artistic designs choices, it would seem reasonable to suppose that there was coincident Egyptian exposure to base sixty counting methods as well.

If, as can be construed by the length of the perimeter of the Great Pyramid, Egyptian architects were aware of a base sixty system, and chose to divide a circle into 360 degrees, then why is there no demonstration of either usage in the surviving written historical record of ancient Egypt? The answer may be due to a combination of factors. It may have been thought that since the use of this knowledge allowed access to such intrinsically powerful results (i.e., trigonometry), and then perhaps this knowledge should be closely held by only a select few.

It may also have been found that the use of a base sixty, and a 360 system, ran counter to methods that were already well entrenched and in daily use. Cultural inertia, then, may have prevented their widespread adoption. (This would be similar to the present day resistance in the United States to the adoption of the metric system). And lastly, it may be due to the very limited amount of material that has endured through the ages until the present time. It has been wisely noted in regard to ancient Egyptian capabilities, "it would be rash to assume that no advance was made beyond what can be found in the scanty and mostly fragmentary surviving texts".⁴⁴

In addition to the length of each side of the Great Pyramid, there is one other Old Kingdom design choice that may possibly offer confirmation not only of the issues discussed in the preceding paragraphs, but also of the initial assumptions stated at the beginning of this essay. I refer to the Old Kingdom choice for the length of the ancient Egyptian standard unit of measure, the Royal Cubit.

The Giza Vanishing Point

Here is a link to Stephen Goodfellow's Giza Vanishing Point web site.

Back in 1979, just as I was about to have my findings on the Giza Site Plan published for the first time, the Detroit artist Stephen Goodfellow contacted me. Evidently, Stephen had attended a lecture given in the US by the Egyptologist Robert Anderson, who was then Honorary Secretary of the Egyptian Exploration Society in London. For reasons best known to himself at that time, Stephen had asked Robert whether he knew of anyone who could supply him with accurate information about the dimensions and relative positions of the three major pyramids on the Giza plateau. It so happened that through the EES, I had recently shown some of my work on the Giza plan to Mr. Anderson, who was therefore aware that I had carried out a detailed analysis of Petrie's survey results for the Giza monuments, and had calculated the exact dimensions and relative positions of the three pyramids in terms of the customary Egyptian units of measurement.

By this rather far-fetched coincidence, Stephen was put into contact with one of the few people anywhere who could provide him with precisely the information he was looking for, in order to investigate his concept of a "vanishing point" for these three famous pyramids. Stephen had the idea that the three pyramids together could be encompassed by inner and outer circles which would intersect at a single point comparable to the vanishing point used by artists to define the visual convergence of parallel straight lines. Whereas the successively diminishing dimensions of the three pyramids could not be explained by the convergence of straight lines, any three points in a plane can be shown to fall on the circumference of a circle, and hence it was possible to construct circles which would pass through the SE and NW corners of the three pyramids, and intersect at a single point comparable to the artist's conception of a vanishing point.

As I recall, it was seven or eight years before I agreed to carry out the detailed calculations which Stephen had asked for. I had always taken the view that the plans of the Giza pyramids had been based primarily upon rectilinear geometry - the use of straight lines and right angles - and it didn't seem to me that there was much evidence to suggest that the architects had made use of circles in their designs. Furthermore, the encompassing circles of the Giza pyramids are of very large dimensions, and it was absolutely inconceivable that these circles could have been laid out on the ground. If Stephen's concept of a vanishing point for the Giza pyramids had any validity whatsoever, then it was only by calculation that the pyramid-architects could have determined this position on the plateau, and for this they would have required some basic knowledge of the geometry of the circle, as well as an understanding of Pythagoras' theorem.

Perhaps it was my developing interest in computer programming that made me respond to Stephen's renewed request that I should calculate the exact position of the Giza vanishing point. With the fairly new-fangled contraption of the home computer, I could compute the dimensions and points of intersection of the various circles that can be passed through the corners and centers of the three pyramids, in a relatively painless manner.

The results of these calculations proved to be rather more interesting than I had anticipated. For the vanishing point itself turned out to be located not on some empty patch of desert, but very close to the rubble enclosure wall situated to the south of the Third Pyramid. At first glance, this position seemed to rule out any deliberate correlation; yet I knew that the most significant archaeological discovery at Giza over the past seventy years had only come about after a similar rubble-built enclosure wall had been demolished - this wall having been built on the south side of the Great Pyramid, right over the roofing-beams of the rock-cut pit in which the disassembled wooden components of the fabulous Cheops boat were discovered.

I was also puzzled by some curious features of the enclosure wall to the south of the Third Pyramid. Whereas the walls around the Second and Great Pyramids had been laid out in a plan conforming rigorously to the cardinal directions from east to west and from north to south, the wall to the south of the Third Pyramid veered away towards the south, and had a curve of enormous radius, reminiscent of the curves which encompass the three pyramids. Furthermore, a branch wall had been constructed towards the north in two segments, which very much brought to mind the two chords which must be struck across the circumference of a given circle, in order to fix the position of the centre.

These factors, taken together, have led me to the conclusion that there might be some merit in Stephen's vanishing point theory. Although I still maintain that the dimensions and relative positions of the Giza pyramids were determined by the design which I first described in 1979, the concept of the vanishing point could be said to represent the summation of this design, since the exact spot depends upon the positions and dimensions of all three pyramids together. Perhaps one day, the enclosure wall will be properly excavated and something of interest will come to light. Or perhaps not. Who knows?

John Legon, 27/5/2000
Next Section: The Royal Cubit

16.  As explained in the Square Root Derivation section in the Appendix, I believe the scribe would have found the 5 to be 2+1/8+1/9. Hence, in his notation he would find 5/2 -1/2 to be equal to 1/2+1/16+1/18. 
17.  CB is in fact the correct length for the side of an equal-sided inscribed pentagon, and OB is the correct length for an equal-sided inscribed decagon. Using the Pythagorean Theorem, CB computes to be 1.17557 the size of the radius. The Egyptians could have expressed this value as 1+1/9+1/18+1/112. See Euclid's derivation of the 36 angle (and hence of the inscribed equal-sided pentagon) in the Appendix.
18.  Numerically, this statement means that: 1 ÷.618034 = .618034 ÷ (1-.618034) = 1.618034.
19.  Via trial and error persistence, a near perfect nonagon can be derived, again using the pentagon. See Appendix.
20.  This is usually written: Sine(A-B) = CosB x SineA - SineB x CosA, and is known as the Sine Subtraction Formula.
21.  R. Gillings, op. cit., p. 233.

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