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Cosmic Game © Douglass A. White, 2012 v151207 209 . . . . Given a number x and its logarithm log b (x) to an unknown base b, the base is given by: . . . . Among all choices for the base b, three are particularly common. These are b = 10, b = e (the irrational mathematical constant ≈ 2.71828), and b = 2. In mathematical analysis, the logarithm to base e is widespread because of its particular analytic properties explained below. On the other hand, base-10 logarithms are easy to use for manual calculations in the decimal number system: Thus, log 10 (x) is related to the number of decimal digits of a positive integer x: the number of digits is the smallest integer strictly bigger than log 10 (x). For example, log 10 (1430) is approximately 3.15. The next integer is 4, which is the number of digits of 1430. The logarithm to base two is used in computer science, where the binary system is ubiquitous." Here is Feynman's brief list that holds the key to the elegant palace of logarithms. Power s 1024 s 10 s (10 s - 1)/s 1 1024 10.00000 9.00 1/2 0512 03.16228 4.32 1/4 0256 01.77828 3.113 1/8 0128 01.33352 2.668 1/16 0064 01.15478 2.476 1/32 0032 01.074607 2.3874 1/64 0016 01.036633 2.3445 1/128 0008 01.018152 2.3234 211 1/256 0004 01.0090350 2.3130 104 1/512 0002 01.0045073 2.3077 053 1/1024 0001 01.0022511 2.3051 026 ↓ 026 ∆/1024 ∆ 1 + 00.0022486∆ ← 2.3025 (∆ → 0) In our discussion we, as Feynman does, will focus on base 10, and its iterated square roots, which are binary in nature. You may notice right off the bat that Egyptian mathematics simultaneously used base 10 on the Senet Board combined with the binary system derived from the Eye of Horus fractions and exponents. They apparently knew something special about these two systems that prevented them from using the
Cosmic Game © Douglass A. White, 2012 v151207 210 duodecimal (base 12) system except for time-keeping purposes. Other ancient cultures such as the Babylonians and Chinese also used duodecimal and/or sexagesimal (base 60) systems for tracking time, as we still do today. However, the Egyptians were special in their emphasis on the binary system, which they used widely in their standard weights and measures -- and which we in the conservative U.S. stubbornly continue to use in our liquid measures (gallon, half-gallon, quart, pint, cup, gill, half-gill, ounce, tablespoon). Feynman reviews the way in which a generalized system of logarithms is established. He uses base 10 for our ease in following the calculations. He creates the simple table of numbers that make up the "Successive Square Roots of Ten" chart that I transcribed on the previous page. I added some zeroes in front of numbers to keep the columns neat. Feynman points out that the value of 2.3025 actually is closer to 2.3026, but that is not critical to his discussion. The column on the left contains the components of the Eye of Horus, carrying the series four more steps. The second column from the left simply inverts the first column and shows us the reciprocal of each component of the Eye of Horus with respect to the arbitrary limit Feynman chose. He chose that limit because after ten steps the numbers in the third column become very close to unity, and they also converge on a fixed pattern that is also binary halving at each step, which means that we can then predict the whole continuing pattern in the same way we can predict the continuing series in the Benben pyramidion of the Great Pyramid. The column on the far right gives the difference between each succeeding entry in column four. After 26 the sum of all the other possible differences if we continue down the list is another 26, just like the sum of all the finer gradations of the Eye of Horus is 1/64, because the series becomes the same; each succeeding result is half the previous one. You can see this on your calculator if you pick any number and then start to hit the √ button over and over. The decimal portion after the 1 will start to become approximately 1/2 the previous value at each iteration, showing that the Eye of Horus is everywhere and sees all with its penetrating gaze. The 1 shows that, no matter how small the decimal fragment becomes, the Eye still gazes from Unity. The number 2.3025 becomes the key to establishing the natural logarithms on the base e = 2.7183, because e = 10 1/2.3025 = 10 0.434294 , where 0.434294 = 444.72/1024. The interesting aspect of the "natural" base e is that loge (1 + n) ≈ n. That is e n = 1 + n as n → 0. The number 2.3025 comes from subtracting 26 from 51, the last digits of the last item in column four (2.3051), indicating we have taken the sequence to its Eye of Horus limit. The number 444.72 = 256 + 128 + 32 + 16 + 2 + 0.72, wherein the number .72 is below 1 and comes from knowing that by making ∆ small enough, we end up with 1 + 2.3025∆ for column three. So the final factor is 72% of 22511. The decimal .0022486 below column three corresponds to the 2.3025 below column four and is almost equal to .0022511, allowing for the small remaining discrepancy before the sequence converges onto the Eye of Horus halving principle. So the final factor is 1 plus 72% of .0022511 or 1.00162079. (Feynman uses .73 and gets 1.001643 as his final factor, which gives about the same result.) Look up the factors in the table and multiply all the factors to see if you get 2.7183 or something very close.
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