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The Balance of the Two Lands: Ancient Egypt's Division According to the Ratio of Triangular Equilibrium (As immortalized in the Great Pyramid) (cont.)
By William Glyn-Jones

The ratio in question is derived from a process of halving the area of the square using diagonals and a circle, as shown here above. The inner square is half the area of the outer, so that this diagram shows the Great Pyramid from above, where the inner square represents the level on which the King's Chamber is placed but shown on the outer surface of the pyramid.

The ratio of the diagonal of the smaller to that of the larger is, by maths, is 2.41 : 3.41. This makes the ratio of the distance from the centre straight to one of the sides of the smaller square compared to the extra distance from here to the side of the larger square, 3.41 : 1. And, sure enough, this is the ratio formed by the distance from the Memphis down to the Upper Egyptian capital, the second Egyptian centre at Thebes compared to the distance from the north point of the meridian on the coast down to Memphis. This 1 : 3.41 ratio can truly be called a sacred geometric constant because, for example, it is the relative height of the horizontal that divides the areas of all isosceles triangles in half.

Map

The Egyptians themselves said that the Two Lands had been first unified by, and that the location of Memphis had been established by the proto-dynastic king known variously as Men, Min, Menes and so on. The name may well come from the Egyptian for "to establish".

Memphis, (the first part of which name derives from this same root) the Old Kingdom (Pyramid Age) capital continued to be of primary sacred and administrative importance right through Pharaonic Egypt. Another name for Memphis, incidentally, was njwt heh - that word heh meaning infinity - so that that Memphis is The Eternal City. Next the capital, royal household and court, and the Unification of the Two Lands symbolism, were all transferred to Thebes, in Upper Egypt, but Memphis retained capital status over Lower Egypt, and now we see that the twin capitals of the Two Lands of Lower and Upper Egypt, Memphis and Thebes, are latitudinally located within a scheme based on this ratio that can be found from the geometry of the "halving of the square" using the circle and diagonals method.

A look at part of the Egyptian text from the Shabaka Stone, which claims itself to have been copied from an Old Kingdom text, is extremely relevant at this point. It's pure poetry to me because it expresses such resonant ideas:

He judged between Horus and Seth; he ended their quarrel. He made Seth the king of Upper Egypt in the land of Upper Egypt, up to the place in which he was born, which is Su. And Geb made Horus King of Lower Egypt in the land of Lower Egypt, up to the place in which his father was drowned which is "Division-of-the-Two-Lands." Thus Horus stood over one region, and Seth stood over one region. They made peace over the Two Lands at Ayan [location immediately to the north of Memphis]. That was the division of the Two Lands….Then Horus stood over the land. He is the unifier of this land, proclaimed in the great name: Ta-tenen, South-of-his-Wall, Lord of Eternity. Then sprouted the two Great Magicians upon his head. He is Horus who arose as king of Upper and Lower Egypt, who united the Two Lands in the Nome of the Wall, the place in which the Two Lands were united.
Reed and papyrus were placed on the double door of the House of Ptah. That means Horus and Seth, pacified and united. They fraternized so as to cease quarreling in whatever place they might be, being united in the House of Ptah, the "Balance of the Two Lands" in which Upper and Lower Egypt had been weighed.

Notice that the above does not refer only to a political unification, but also to a measuring, a weighing in a balance to find the location of Memphis. "The Balance of the Two Lands" (Mekhat Tawi), was in fact an epithet for the Old Kingdom capital of Memphis (where 'balance' is a set of scales). The Nome of the Wall was a name for the Memphite region of the pyramid builders. Memphis was the home of the god Ptah, and Ta-Tenen or 'Risen Land' is here the Memphite Primordial Mound of Ptah the Creator, which the Saqqara pyramid may well in some sense represent. (The other chief Primordial Mound was the sacred mountain of Thebes.) Horus is equated with this mound, and on the mound in the creation myth the first plants sprouted. In the quote there is a sprouting from this mound, in the form of the two magicians, a reference to a crown, but one which in this context seems therefore to be refer also to the lotus (a symbol of Upper Egypt) and papyrus (a symbol of Lower Egypt), which are placed in the door of the house (Memphite temple) of Ptah, presumably as its two pillars. (We find such a combination of a pairing of lotiform and papyriform pillars extant at Thebes.) Memphis was famous down through the ages for being the location of the Temple of Ptah.

What is clear is that the establishment of Memphis as the Old Kingdom capital is central to this notion of the Balance of the Two Lands, and it makes sense in terms of the geodesy.

To recapitulate this scheme then, a 'meridian' line extends due south from the north point of the Delta. Memphis is upon this central axis line, and also adjacent to the Nile, as all the sites in question are, rather as a matter of necessity in Egypt. Thebes is at the latitude (not longitude) of the centre of the scheme, and the ratio of the latitude difference between Thebes and North Point to the difference between Thebes and Memphis is given by the geometry of halving the area of a square, and also of equivalence of "upper" and "lower" parts of an isosceles triangle, such as is marked in the cross section of the Great Pyramid by the level of the King's Chamber.

But we needn't rely on a diagram alone for confirmation of the accuracy of this, for a simple calculation from latitude figures can be performed. Measuring due north from Memphis we come to the coast at 31'33''N. The latitude line running through the Temple of Amun in Karnak, Thebes, is 25'43''. The 'balance point' between these, using the triangular equilibrium ratio of 1:2.41, is precisely at the latitude of Memphis, at 29'51''. (Convert to decimal for ease of calculation, subtract 25.71 from 31.57= 5.86. Multiply by 1/3.41=1.72. Subtract 1.72 from 31.57 = 29.85, convert back to degrees and minutes = 29'51''.) I find myself extremely convinced by this theory - the Balance of the Two Lands was a profoundly geometric concept, in accordance with Ma'at. This "Balance of the Lands" was something they measured very accurately, from close observations of stellar (or solar) altitude. The word for 'balance' in the Egyptian phrase "Balance-of-the-Two Lands" is even pretty much the same word as the name of the instrument used, along with the palm rib, to measure stellar angles, which by extension can also be a measurement of relative latitude position. As far as I know merkhet andmekhat may indeed be two transliterations of the same Egyptian word, and although I would say don't quote me on that, I do note that in an appendix entitled The Scales of the World in their book Keeper of Genesis Bauval and Hancock tell us that that the hieroglyphic determinative sign for 'to weigh' shows a triangle, or builders' 'square', with a plumb bob suspended from the apex, 'a sign which can also mean to 'balance the Earth'.' The plumb bob was also the chief component of the merkhet, of course, used to measure stellar inclination. They give the name for the great scales of Thoth in the place of Ma'at as Mekhaat, and, citing Wallis Budge's dictionary of hieroglyphs, they say that this name 'means in other contexts 'the balance of the Earth'.'

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