Date: Dec 7, 2012 2:17 PM
Author: Milo Gardner
Subject: Re: Egyptian and Greek sqare root

Premise: Egyptian numeration has been parsed in ways that scaled rational numbers.The inverse proportion aspect of Egyptian. Greek and medieval square root was the basis of the inverse proportion pesu in Middle Kingdom Egypt. Scribes from 2050 BCC to 1202 AD approximated square root in two and three step square root methods.

A. To explicitly decode Egyptian and square root, LE Siglier, "Liber Abaci" in 2002 transliterated a square root of 10 problem in ways that were used in 2050 BCE Egypt, that needs to be translated as originally written.

1. guess (3 + 1/6)^2, since 1/(3 + 3) offers an estimate 10 + (1/6)^2 = 10 + 1/36

2. divide 1/36 by 2(10 + 1/6) = 1/36 (6/38) = 1/228 .

Sigler stopped at this point, and continued wity a misleading geometric root supposition that included graphs.

The raw data easily extends to:

(3 + 1/6 + 1/228)^2 = [10 + (1/228)^2],

seems to be an accurate translation, based on the following two data sets.

B. Parker reported a Demote square root of 200 as (14 + 1/7)^2. Note that only the first step of Fibonacci's method approximated 200^1/2 by

1. guess (14 + 1/7)^2 since 1/(14 + 14) estimate

200 + (1/7)^2 = 200 + 1/49

Had a more accurate estimate been needed, the Demotic scribe could have

2. divided 1/49 by 2(14 + 1/7) = 1/49(7/198) = 1/698 estimated = 200 + (1/686)*2 = 200 + 1/470596 =

(14 + 1/686)^2

C. Gillings in 1972 in "Mathematics in the Time of the Pharaohs", page 217, cited a 200 BC Greek square root 164 by Archibald:

(12 + 2/3 + 1/15 + 1/26 + 1/32)^2 or ((12 + 2/3 + 1/15 + 1/24 + 1/32)^2,

quoted Archibald, "If 1/26 was an error for 1/24 the approximation would indeed be remarkable".

Let us find the remarkable approximation, if there was one, by applying a scribal three step method.

1. guess (12 + 5/6)^2 since 20/(12 + 1/12) estimated 200 + (5/6)^2

Rather than divide 25/36 by 2(12 + 5/6) an awkward data set a new guess

2. (12 + 4/5)^2 estimated (144 + 48/5 + 48/5 + 16/25) = 163 + 21/25, hence an error of 4/25.

3. divide 4/25 by 2(12 + 4/5) = 4/25(5/128) = 1/160 estimated 164 + (1/160)*2 = 164 + 1/25600 = (12 + 4/5 + 1/160)^2

Summary: for those skilled in Egyptian numeration the scribe scaled 4/5 in 24 parts by

4/5(24/24) = 96/120 = (2/3 + 1/15 + 1/24 + 1/40), add 1/160

since (1/32 -1/40) = 1/160

meant 164^1/2 approx as

(12 + 2/3 + 1/15 + 1/24 + 1/32)^2

was the "obvious" historical unit fraction series.

Comments?