Transliterating, and fully translating the hieratic Rhind Mathematical Papyrus (RMP) and other Egyptian Middle Kingdom texts requires decoding keys. Following a well known adage, to read any good book, in any era, hieratic texts should be read three times on three levels.

Garbled hieratic math texts, reported in the 20th century, can be corrected by reading each text from at least three different points of view. For example, overviews can include possible theoretical and practical topics. That is, on first level readings overviews of potential theoretical contents, and apparent practical contents can be read into one story line. Considering RMP 53, 54 and 55 as a grouped sample, specific theoretical and applied practical applications can be noted in three closely related story-lines.

Second level readings of RMP 53, 54 and 55 identifies at two theoretical points of view. As background, a 2008 study reported RMP and the Kahun Papyrus 2/n tables parsed within an aliquot part method that had been suggested by F. Hultsch in 1895 (and confirmed by E.M. Bruins in 1944 that appears in RMP 53, 54 and 55. Numerators and divisors of 2/n table answers were written by Ahmes in multiplication context(2) and/or a subtraction context(2)

1. (2/n)*(m/m) = 2m/mn (As Ahmes clearly used, over and over again)

a. 2/53*(30/30) = 60/1590 = (53 + 5 + 2)/1590 = 1/30 + 1/318 + 1/795 b. 2/73*(60/60) = 120/(60*73) = (73 + 20+ 15+ 12)/(60*73) = 1/60 + 1/219 + 1/292 + 1/365

2. (2/n - 1/A) = (2A -p)/An as Ahmes may have know. (However Fibonacci used the method in 1202 AD)

a. (2/43 - 1/42) = (84-43)/(42*43) = (21 + 14 + 6)/(42*43) = 1/42 + 1/86 + 1/129 + 1/301 b. (26/97 - 1/4) = (104 - 97)/388 = (97 + 4 + 2 + 1)/388 = 1/4 + 1/97 + 1/194 + 1/388

The 114 year old Hultsch-Bruin decoding key suggesting divisors of an A = 42 (by 21, 14, 7, 6, 3, 2, and 1), and divisors of another A = 60 (by 30, 20, 15, 12, 6, 4, 3, 2, and 1). Ahmes selected additive divisors in red ink to write optimized, but not always optimal, Egyptian fraction series.

Third level readings test the practical contents of 5MP 53, 54 and 55 against alternative theoretical methods. In background 2/n tables Ahmes' red auxiliary numbers pointed out additive numerators written within a multiplication context. In RMP 21-23, additional background shows that Ahmes practiced selecting LCM multipliers closer to the first method, 2/n =2/n*(m/m) = 2/mn.

FORMALLY READING RMP 53, 54, and 55 THREE TIMES

First level readings show Ahmes' raw data included cubit, khet units, and setat and mh areas. One substitution of one setat by two LCM multipliers, 4 and 2, faciliated the setat to be partitioned into 1/8 setat and mh units as 2/n table members were partitioned.

The LCM data reported cubit and khet units written in setat areas, 100 cubit by 100 cubits. Setats were sub-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.

Another view of cubit partitions is found in a fragmented text, which may have found a portion of a cubit defined by 1/3 of a palm: Given 1/14 a cubit is 1/2 a palm [aka 2 fingers]. Operate on 1/2 palm to find 1/3 palm to find 2/3 of 1/2 palm = 2/6 = 1/3 palm and find 2/3 of 1/14 = 2/42 = 1/21 cubits.

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.

Bibliography

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10

Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.