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Cosmic Game © Douglass A. White, 2012 v151207 221 * Simple experiments are needed to measure the Compton wavelength of the electron and the proton in order to verify the approximate rest masses of the electron and proton could be measured by the ancients. * We know that the Egyptians had standard weights and measures, and much of their system was based on the Eye of Horus binary fractions, but we do not know how fine the craftsmen and engineers were able to go in such measurements. One way the ancients achieved fine resolution in measurements was to use instruments built at a large scale. For example, a large Senet Board could be used to get extremely accurate measurements of irrational quantities such as square roots. Large pyramids and obelisks could be used as instruments for measuring astronomical motions and cycles of time. We are only part way there. We have found many cosmic numbers enshrined in the sacred geometry of the Senet Board, and hints at how the ancient Egyptians might have actually understood the physics related to these numbers. Perhaps knowledge of the relation between geometry and physics was a surviving tradition from a bygone era of "Atlantean" civilization that collapsed during some great catastrophe and lost all its higher technology. Perhaps the Egyptians simply celebrated the beauty of the geometry on an intuitive level without any awareness of the underlying physical laws. This paper as it stands is still only a rough draft that requires further research so that we may answer the above questions and further understand what the Egyptians knew. Unfortunately so much of their technical lore has been lost that we are faced with the task of reconstructing from the limited remnants that survive. Most of the perishable artifacts are long gone and we are left with whatever the archaeologists can recover from the shifting sands of Egypt. Douglass A. White October 25, 2012 Appendix A: Taking Square Roots with Egyptian Arithmetic The Egyptians in early times already were quite adept at arithmetic, algebra, trigonometry, and the basics of calculus. Unfortunately only a few documents about mathematics have survived. Wikipedia ("Egyptian mathematics") lists for us the scanty scraps of primary documents in Egyptian that have survived. "The earliest true mathematical documents date to the 12th dynasty (ca 1990–1800 BC). The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (ca 1650 BC) is said to be based on an older mathematical text from the 12th dynasty." The Temple of Man (2 vols) by Schwaller de Lubicz, translated into English from French by Deborah and Robert Lawlor is a good source that discusses in detail Egyptian
Cosmic Game © Douglass A. White, 2012 v151207 222 mathematics and its deeper meaning. The chapters "Foundation of Pharaonic Mathematics", "Pharaonic Calculation", and "Pharaonic Trigonometry" are especially useful and form the basis for Schwaller's analysis of Egyptian megalithic architecture. In this appendix I will suggest a possible scenario by which an Egyptian scribe might calculate the square root of 10 numerically with great accuracy. The formula in modern algebraic form is a ' = ½ (a + N/a), where N is the number you are taking the root of, a is a reasonable guess, and a ' is an average between a and N/a. We discussed Egyptian multiplication earlier in the essay. The procedure we are using for taking the square root requires the operations of division and addition. Addition was simple in Egyptian. nH¡ H¡rQ! Q!rs (Wah kher-k met her-s.) Place [by you] 10 over it. (Egyptian addition is like piling things.) Add 10 to whatever you already have. Division nH¡y`m```r H¡y`m```rv6! (Wah-tep em khemt er gemet met.) Put 3 so that 10 is found. Divide 10 by 3. Another way to announce division: Z#!Vt``` (Nas met khenet khemt.) Call 10 in front of 3. (Divide 10 by 3). !V ``` From here on I will use Arabic numerals to suggest the Egyptian format. Generally speaking the Egyptians preferred to represent all fractions as unit fractions (with numerator = 1) except for occasional use of ⅔. In the first iteration we will start with a guess of 3, which we know is a bit low. Put 3 to find 10. (The first-guess divisor a is 3, and the dividend N is 10.) *1 3 *2 6 *⅓ 1 Remainder Total: 3 + ⅓ 9 + 1 = 10 Quotient is 3 + ⅓ = 3.3333333.... Notes: The column on the right starts with the divisor, and doubles to find the next number. The doubling continues until the next number in the sequence exceeds the dividend. The column on the left is a binary sequence that starts with 1 and doubles to find each succeeding number. The doubling iterates the same number of times in each column. When the right-hand column is added, the sum is 9, which is 1 short of 10. We then divide the remainder of 1 by the divisor 3 to get ⅓, which we place at the bottom of the left-hand column opposite the remainder 1. The stars indicate the numbers in the list that are selected. In this case we use them all, but in other instances the sum that equals or most nearly equals the dividend does not use all components of the list. In this case our answer for the dividend 10 is the quotient 3 + ⅓. The fraction indicates the left-
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