There is evidence that the measurement unit known as the Royal Cubit was already in use as much as one hundred years prior to the building of the Great Pyramid, and perhaps even somewhat earlier.⁴⁵ It was the standard length that was used to measure building dimensions, land holdings, grain quantities, etc., throughout Egyptian antiquity. It would appear, then, that the unification of Upper and Lower Egypt precipitated not only a standardization of the length of the year to 365 days, but also the standardization of the foremost Egyptian unit of measurement.

It would stand to reason that given the central importance in ancient Egypt of this unit of measure, a fitting rationale would have been sought for the length assigned to it. A variety of different theories for this have indeed been proposed, with many having as their basis the proportions inherent in the human body.⁴⁶ While there is much that is plausible in these approaches, I believe they all fall short as a rationale should it be the case that trigonometric, and Earth size, knowledge had been gained by the Fourth Dynastic period of the Old Kingdom. Since this essay is predicated on these latter assumptions, we will now explore a theory for a Royal Cubit derivation method that flows from the existence of trigonometric knowledge in ancient times.

The Royal Cubit, as employed during the Old Kingdom, is generally understood to have been 524 millimeters +/- 2 mm ( 20.63 inches) in length.⁴⁷ It can be assumed that prior to the unification of Upper and Lower Egypt a number of different standards of measure would have been found in use as one traveled up and down the Nile. Some of these, if not all, were certainly based on hand, forearm, or foot length and some undoubtedly provided lengths either near to the Royal Cubit length or near to equal subdivisions of it. Seeking to standardize these preexisting measures, and having made a determination of the full circumference of the Earth, it would have been eminently fitting to make the new unit a specific fraction of this circumference. (The meter was originally defined in exactly this way in the late 18th century of the present era.)

If, as we have presupposed, the distance of 6,046 feet had been determined to be the length of a minute of latitude (i.e., a 60th of a degree), then it would have been appropriate to have further divided this amount by 60ths in search of a unit standard. Dividing 6,046 feet by 60, one gets 100.76 feet. Dividing by 60 again, one gets 1.6794 feet, the equivalent of 20.1528 inches. If this was the route taken, then why was the Royal Cubit not made equal to this length?

I think the reason would have been that the 20.1528 inch length holds relevance to the circumference of the Earth as a whole, but does not have any immediate relevance to unified Egypt's actual and distinct placement on the planet. One way to obtain this relevance would have been to consider using the length of a degree of longitude, as determined at some Egyptian latitude, for their unit. Let me explain.

Latitude, as explained before, is a locality's angular distance north or south of the equator. A parallel of latitude is therefore an imaginary circle at or near the surface of the earth that maintains this angular distance around the globe. Longitude, on the other hand, is angular distance measured east or west from some chosen point of beginning. In Diagram 19, point Z is a location situated at some latitude X, and line OZ is a radius from the center of the Earth extending to point Z. OD is also a radius, and it extends along the equator. ZG is a line that we can intentionally draw to be perpendicular to OD, thus creating the rectangle OAZG. With this being done, we can see that AZ is equal in length to OG, which in turn is the cosine of angle ZOG. (The cosine of any angle is defined as the adjacent ÷ hypotenuse. Since OZ is a radius, it can be seen as equaling 1, hence OG ÷ OZ = OG). If the angle of point Z's latitude is known, then through the use of a trigonometric table the relative lengths of OG and ZG can be found.

Line OD, being a radius at the equator, obviously defines the full circumference of the Earth (from C = 2R). Line AZ can also be seen as being the radius of a circle, but it defines the circumference of the circle that is the parallel of latitude at latitude X. Since AZ = OG (= the cosine of angle ZOG), the circle having AZ as its radius will then have a circumference length that is also this same cosine percentage of the Earth's full circumference. Knowing the size of the circle at latitude X, one could then derive a unit of measurement that was a subdivision of it, and was therefore not only based on the size of the Earth, but was also relevant to the latitude at which Egypt is located.

If the Egyptian priests had gotten this far, they would then have been faced with the task of deciding which Egyptian latitude was the most suitable and advantageous one to use. I believe that since this decision would have been occurring not long after the unification of the two separate fiefdoms of Upper and Lower Egypt into one newly created entity, there would have been a number of strong reasons to have chosen the latitude that best defined the border area between these two realms.

Such a boundary would have been at latitude 2927', the line between the precincts of Memphis and Medum, border districts of Lower Egypt and Upper Egypt respectively.⁴⁸ The length of a minute of latitude of full circumference would, as a result, be multiplied by the cosine of 2927' (=.8708), before being divided by 60ths. Thus we have: 6046 feet times .8708, divided by 60 and divided by 60 again, yielding 1.46246 feet, or 17.5495 inches for the new unit.

But would this translation have been sufficient? The derivation of 17.5495 inches as a standardized unit has relevance for the reasons cited, but the process does not contain any acknowledgement of the diagrammatic breakthroughs that made these later discoveries possible. I believe there would have been a strong desire to also have the derivation process include a step that infused the Royal Cubit with the "magical" properties of the golden ratio.

Repeated here (again) is Diagram 11, the diagram that made possible the construction of an inscribed pentagon.

In Diagram 11 it was found that if line AC was swung down to meet radius OH, it would establish a point B. Line CB then turned out to be exactly the right length (1.17557 times the radius) to inscribe a perfect pentagon within the circle, thus paving the way for the formulation of a complete trigonometric table.

The incorporation of this diagram into the standard unit derivation process would clearly have carried great symbolic meaning.

If the radius in Diagram 11 is given the length of 17.5495 inches (see derivation above), then CB will be 1.17557 times this amount, or 20.6306 inches, and each side of the resulting inscribed pentagon will then be 20.6306 inches in length. As pointed out earlier, 20.63 inches is the precise length that the ancient Egyptians chose to be the length of their ubiquitous standard unit of measurement, the Royal Cubit.⁴⁹

From the perspective of the Egyptian priests, a standardized unit derived in this way would have rather elegantly fulfilled the following criteria:

1) It would have established a standardized unit of measurement having a basis in the measured size of the Earth.

2) The standard would further be defined by the location, relative to the whole Earth, of unified Egypt's political and cultural balance point.

3) The standard would also have been tempered by one of the most profound secrets of the physical world that Egyptian geometers may have thus far discovered, the diagrammatic derivation route to the inscribed pentagon.

The amount of knowledge quite possibly locked inside this seemingly innocuous unit of length is remarkable. Knowing but the length of the Royal Cubit and the code of its derivation, one can work backwards not only to an accurate measure of the Earth's size, but also to the diagram that opens the door to the trigonometric table.

If the theory presented in this paper is essentially correct, then the Great Pyramid was designed to celebrate, and perhaps memorialize, the attainment of this knowledge. In concordance with this sense of memorialization, I mention the findings of Richard Proctor, an astronomer of the last century, who makes the compelling case that the Grand Gallery feature of the Great Pyramid, prior to completion of the pyramid, would have served as a perfect sighting device with which to have made precise measurements of the stars, moon, and planets.⁵⁰ Proctor points out that after having been used for this purpose, the Grand Gallery was then obviously sealed and enshrined.

The question then comes to the fore as to why the means to inscribe an equal-sided pentagon does not enter the surviving written record until Euclid's Elements in 300 B.C., and why the development of trigonometry doesn't surface until 200 B.C.⁵¹ Sir Thomas Heath details evidence to support the probability that Euclid's treatment of the pentagon was not his own, but rather had come to him from earlier Pythagorean sources.⁵² If this is so, then how did the Pythagoreans come by this knowledge?

There are numerous references by ancient authors to the sojourns in Egypt by Thales (ca. 600 B.C.), Pythagoras (ca. 530 B.C.), Plato (ca. 380 B.C.), and other Greek philosophers who traveled to the Nile seeking inroads into Egyptian knowledge.⁵³ There is little doubt that Egypt was a source of inspiration to the development of Greek mathematics and science, and it may be that the secrets of the inscribed pentagon found its way to Greek soil as a result of these pilgr_images.

However, it must be recognized that the possibility exists that the geometrical knowledge received by Thales, Pythagoras and others from Egyptian sources may have contained only glimmers of the whole of the trigonometry derivation technique. Egyptian priests may have felt it wise to show these seekers only selective parts of the ancient knowledge, or perhaps, the priests themselves no longer recognized the full significance and context of what surviving information they themselves had access to. It may therefore be that the Greek geometers, having been given their lead by the Egyptians, were then able to independently construct and reforge the route to trigonometry.

I end this presentation with the statement that I believe this essay has done what it set out to do, namely to show that it was within ancient Egyptian capability to both discover the mysteries unlocked by the golden ratio, and to accurately measure the size of the Earth. Whether it is probable that these feats were actually accomplished, and whether they were then intentionally designed into the Great Pyramid, I now leave to the reader to ponder. It is to be hoped that future discoveries will clarify these issues.

Next Section: Appendix

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An ancient unit of linear measure, originally equal to the length of the forearm from the tip of the middle finger to the elbow, or about 17 to 22 inches (43 to 56 centimeters).

cubit = noun

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There were different kinds of cubits. The common cubit, called the cubit of a man, was about eighteen inches (Deut. 3:11). The king's cubit was three inches longer than the common one. The holy cubit was a yard, or two common ones.
The Royal Cubit, as employed during the Old Kingdom, is generally understood to have been 524 millimeters +/- 2 mm ( 20.63 inches) in length.

A long time ago, when people wanted to tell each other how long something was, they would compare it to the length of their arms, from the elbow to the tip of the longest finger. If you imagine telling your friends that your little sister is two "arms" tall, you get an idea of how the Ancient Egyptians used to talk about length. The word cubit was originally used to mean the length of one arm.

Naturally, not everyone has the same size arms. If a tall person uses his arm to measure a cow, and a short person later measures the same cow using his arm, they will each have a different size in mind for the cow.

In order to overcome problems like this, the people of Ancient Egypt got together and measured their King's arm. They made sticks that were the same length as the King's arm, then agreed that the sticks would be their official way of measuring length. So from then on, a cubit meant the length of one of these measuring sticks.

45. John A.R. Legon, "The Cubit and The Egyptian Canon of Art", DE 35, 1996, pp. 61-76.
46. For a list of these references see the bibliography at the end of Legon's paper, DE 35, 1996
47. See W.M. Flinders Petrie, The Pyramids and Temples of Giza, p. 28.
48. Baines and Malek, Atlas of Ancient Egypt. Refer to maps on p.16 and 121.
49. The Egyptians, of course, were not working in feet or inches, but in some unit that they were familiar with. However, irrespective of the units in which they may have been measured, the actual lengths of the 'thirds' of longitude would have been found to be the same as described here. In addition, computation is not necessary if CO (the radius) is in fact made equal to the length of 17.5495 inches. The length of CB will then follow from the completion of the diagrammatic construction.
50. Richard A. Proctor, The Great Pyramid: Observatory, Tomb and Temple, 1880. Proctor states that a trained observer using the Grand Gallery "would be able to make observations only inferior in accuracy to those made in our own time with telescopic adjuncts." p. 141.
51. For reference to the earliest written evidence for the development of a trigonometric table see Asger Aaboe, Episodes From The Early History of Mathematics, p. 111. Note that Badawy, op. cit., presents evidence of the use of the golden ratio in various design motifs throughout much of Egyptian antiquity.
52. Sir Thomas Heath, Euclid's Elements, Vol. II, pp.97-99. See also Heath, A History of Greek Mathematics V.1, pp.160-162. In this latter citation he references the fact that the Pythagoreans used the pentagram as a "symbol of recognition between the members" of their sect.
53. See Heath, A History of Greek Mathematics, Vol. I, pp. 121, 128-131 for many of these sources. The earliest surviving reference to Pythagoras' studies in Egypt appears in the work Busiris, written ca. 390 B.C. by the Greek philosopher Isocrates. See Isocrates, Vol. lll, p. 119. Harvard University Press, 1961.

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