The Cosmic Game ( PDFDrive.com )

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.