
Re: Egyptian mathematics: an over view
Posted:
Feb 15, 2007 4:50 PM


Dear Forum members,
Reponding to a friend's comment on the Akhmim Papyrus, that Kevin Brown and I jointly analyzed about 12 years ago, the old data is summarized by: http://www.ecst.csuchico.edu/~atman/Misc/horuseye.html
A few updates need to be made related to Silger's 2002 publication of Fibonacci's 1202 AD Liber Abaci, which points out that a first and a second subtraction were allowed, thereby explaining several hard to evaluate data points in the n/17 table, as had been suggested by Brown and myself. The Akhmim Papyrus was written around 500 AD. It is a Coptic text that used two forms of the HultschBruins method, a first and second subtraction. The first method: 2/p  1/A = (2A p)/AP was cited be Fibonacci per his methods 4, 5 and 6, as listed at the end of his seventh chapter, as aptly translated by Sigler. Note that Fibonacci's methods 4 and 5 stressed unit fraction first partitions, the style followed by Ahmes, while a medieval method 6 allowed vulgar fractions, such as 3/8 raised to a multiple of 6 or 18/48, to be a first partition.
The second HultschBruins method appearing in the Akhmim Papyrus is a modified form. The modification was read by Sylvester in 1891 as only an early version of the greedy algorithm. The Coptic version, and the data provided by Fibonacci, clearly show a second subtaction, as Leonardo (Fibonacci) lists as his 7th method, of 7, was used for 4/17 and several other n/17 table entries (i.e 8/17 and 13/17), and that nsubtractions as a greedy algorithm would require, were not mentioned by Fibonacci, nor by the Coptic scribe.
As a summary, beginning with Wilbur Knorr's 1982 Historia Mathematica data, the following analysis is pertinent, and therefore the data has been updated to be consistent with the RMP and the Liber Abaci. n/17 Akhmim P. Value 1650 BC Egyptian Frac    2/17 12' 51' 68' ; 2/17  1/12 =(4 + 3)/(12*17) 3/17 12' 17' 51' 68'; 2/17 + 1/17 4/17 12' 15' 17' 68' 85'; 4/171/12 =(3+12+16)/(12*17) = 68' + 17' + 16/(12*17), or 4/511/15 = 9/(15*51) = 85'
5/17 4' 34' 68'; 5/17  1/4 = (2 + 1)/(4*17) 6/17 3' 51'; 6/17  1/3 = 51', trivial 7/17 3' 17' 51'; 6/17 + 1/17 8/17 3' 15' 17' 85'; 8/17  1/3 = (3+4)/51, 4/511/15=(51+9)/(51*15) = 15' + 85'
9/17 1/2 34'; 9/17  1/2 = 34', trivial 10/17 1/2 17' 34'; 9/17 + 1/17 11/17 1/2 12' 34' 51' 68'; 10/17 + 2/17 12/17 1/2 12' 17' 34' 51' 68'; 11/17 + 1/17 13/17 1/2 4' 68'; 13/171/2 = 9/34, 9/341/4 =2/(4*34)= 1/68
14/17 1/2 4' 17' 68'; 13/17 + 1/17 15/17 1/2 3' 34' 51'; 9/17 + 6/17 16/17 1/2 3' 17' 34' 51'; 15/17 + 1/17
Legend: 4' = 1/4 following the Greek style of writing Egyptian fractions.
Best Regards to all,
Milo Gardner

