Date: Nov 29, 2012 12:30 PM
Author: Milo Gardner
Subject: Re: Paper on: The Geometric Grids of The Hieratic Numeral Signs

Hossam,

Thank you for consistently analyzing hieratic arithmetic as a geometric metaphor, your special interest.

Reading ancient math texts as recorded by scribes were discussed by Gillings and Archibald in terms of an Egyptian and Greek square root of 164, and other problems.

The square root of 164 problem recorded a unit fraction answer, and was approximated as

12 + 2/3 + 1/15 + 1/24 + 1/32

correct within (1/160)^2, a term that was irrational, and hence never zero.

Given that the 1/24 term was mistranslated as 1/26, as discussed by:

http://planetmath.org/encyclopedia/EgyptianAndGreekSquareRoot.html

led my research to a 1900 BCE scribal solution to two second degree equations, with rational roots.

The Egyptian and Greek square root method that was likely used from 1900 BCE to 200 BCE,

http://planetmath.org/encyclopedia/BerlinPapyrusAndSecondDegreeEquations.html

if not 800 AD (when the Egyptian and Greek unit fraction system was modified by Arabs that brought Indian numerical symbols and a Babylonian algorithm).

Both classes of Greek and Egyptian square root were more algebraic and arithmetic. Of course scholars have viewed these problems within modern and Greek geometry, in my view dead-end points of views, that have left many interesting aspects of unit fraction based square root problems unresolved for over 100 years.

Best Regards,

Milo Gardner