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Topic: Reading Egyptian texts three times
Replies: 7   Last Post: Jul 9, 2009 5:51 PM

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Milo Gardner

Posts: 1,105
Registered: 12/3/04
Reading Egyptian texts three times
Posted: Jun 7, 2009 8:54 AM
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Planetmath offers an updated approach to parse ancient texts, read each text three times, from three different points of view.

That is: "Reading and translating the Rhind Mathematical Papyrus (RMP), and other hieratic texts dating to 2,000 BCE and 1500 BCE requires decoding keys. Following a well known adage, to fully read any good book, in any era, hieratic texts should be read on three levels. Garbled hieratic texts, reported in the 20th century, can be corrected by reading each text from three different points of view.

First level readings gain over views of potential theoretical contents, and actual practical contents. Keeping notes and considering possible abstract points of view, RMP 53, 54 and 55 are offered as examples.

Second level readings formalize two possible theoretical points of view. As background, a 2008 study reported Rhind Mathematical Papyrus (RMP) and the Kahun Papyrus 2/n table within an aliquot part method that had been suggested by F. Hultsch in 1895 (and confirmed by E.M. Bruins in 1944). Numerators and divisors of final Egyptian fraction answers were likely created from either a subtraction context, written as:

1. (2/n - 1/A) = (2A -p)/An

i.e. (2/43 - 1/42) = (84-43)/(42*43) = (21 + 14 + 6)/(42*43) = 1/42 + 1/86 + 1/129 + 1/301

or a LCM multiplication context, written as:

2. (2/n)*(A/A) = 2A/An

i.e. 2/73*(60/60) = 120/(60*73) = (73 + 20+ 15+ 12)/(60*73) = 1/60 + 1/219 + 1/292 + 1/365

The 114 year old Hultsch-Bruin aliquot part story-line implicitly reports divisors of A = 42 (by 21. 14, 7, 6, 3, 2, and 1), and divisors of A= 60 ( by 30, 20, 15, 12, 6, 4, 3, 2, and 1) used in an vivid manner. Ahmes almost always selected additive divisors, written in red, to calculate final Egyptian fraction series.

Third level readings test practical contents against possible theoretical backgrounds. In parsing 2/n tables Ahmes' red auxiliary numbers, used in several problems, point out additive numerators within a multiplication context. In RMP 21-23 Ahmes practiced selecting LCM multipliers in a manner that pointed to the second method, 2/n =2n*(A/A) = 2A/nA, as the likely historical method.


First level readings: Ahmes' raw data included cubit and khet units, and areas, recorded in setat and mh units. A substitution of one setat by the LCM multiplier, 4 and 2, allowed the setat to be partitioned into 1/8 setat and mh units.

The LCM multiplication data reported cubit and khet units written into setat areas, 100 cubit by 100 cubits. Setats were divided into 1/100 setat strips, or mh units. A first reading of RMP 54 included Ahmes' implicit use of the LCM 2/n table conversion method. Ahmes scaled a setat to (4/4) and (2/2) before multiplying by 7/10, 14/10 and 28/10, respectively.

For example:

1. (7/10)*(4/4) setat = 28/40 setat = (25 + 3)/40 setat

allows the 2/n table LCM conversion method to confirm that

2. 5/8 setat + 300/40 mh = 5/8 setat + 7 1/2 mh.

describes the correct method and answer.

The first reading of RMP 55 implies that Ahmes computed in mh units, as 5 setat times 3/5 was solved and written out.

Second readings of RMP 53, 54 and 55 consider closely related theoretical contents of the Akhmim Wooden Tablet, as reported by Hana Vymazalova in 2002. Vymazalova reported that one hekat was divided by 3, 7, 10, 11 and 13, was exactly returned to (64/64) (as Daressy had not seen in 1906). The RMP also reported the AWT method as an implicit initial substitution over 36 times. As one theoretical method, the AWT and RMP replaced a hekat by (64/64), a hekat unity, to allow the division by n, limited to 1/64 < n < 64 over 40 times.

The theoretical AWT substitution method appeared in RMP 47. Ahmes reported 100 hekat written as 6400/64, divided by 70 to compute a quotient 91/64 and a remainder 30/(70*64), scaled to a ro unit, writing (150/70)*1/320 as the answer. An intermediate step included:

[(64 + 16 + 8 + 2 + 1)/64]hekat + (2 + 1/7)*1/320, or

since Ahmes' answer was written as:

[1 + 1/4 + 1/8 + 1/32 + 1/64]hekat + [2 + 1/7]ro

A second theoretical substitution method replaced one hekat with 320 ro. This method may have replaced the "awkward" 6400/64 substitution of 100 hekat method reported in RMP 47.

In RMP 38, the division of one hekat, written as 320 ro, was multiplied by 7/22, and returned by 320 ro when multiplied by 22/7. A confirmation of the RMP 38 method is provided by RMP 35. RMP 35 dividing 10 hekat of fat, written as 3200 ro, by 365. Ahmes reported a rate (8 + 280/365)ro, which was returned to 3200 ro by multiplying the answer by 365.

Third level readings of RMP 53, 54 and 55 reach implicit conclusions that Ahmes' division and multiplication methods were inverse operations, a feature of modern multiplication and division operations.

SUMMARY: Three level readings of hieratic mathematical problems, such as RMP 53, 54, and 53, can be parsed in terms of 2/n tables and other attested ancient mathematical methods. Confirming third level readings follow Occam's Razor considerations, the simplest version was the historical method. Third level 21st century readings of hieratic arithmetic, algebra, geometry and weights and measures methods are documenting hieratic mathematical texts, like the RMP, in ways that 20th century translators had not expected."

Best Regards,

Milo Gardner

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