Reisner Papyrus

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The Reisner Papyrus is one of the most basic of the hieratic mathematical texts. It was found in 1904 by George Reisner. It dates to the 1800 BCE period and was translated close to its historical form of remainder arithmetic in association with the Boston Museum of Fine Arts. Gillings and other scholars accepted 100 year old views of this document, with several of the views being incomplete and misleading. Two of the documents, reported in Tables 22.2 and 22.2, a detail a division by 10 method, a method that also appears in the Rhind Mathematical Papyrus. Labor efficiencies seemed to have monitored with this method. For example, how deep can 10 workmen dig in one day, seems to have been calculated in the Reisner Papyrus, and by Ahmes 150 years later.

Gillings later repeated a common but incomplete view that analzed lines G10, from table 22.3B, and line 17 from Table 22.2 on page 221, as reported in the "Mathematics in the Time of the Pharaohs", with these facts:

divide 39 by 10 = 4,

a poor approximation to the correct value, reported Gillings.

Gillings fairly reported that the scribe should have stated the problem and data as:

39/10 = (30 + 9)/10 = 3 + 1/2 + 1/3 + 1/15

Yet, all other the division by 10 problems and answers were correctly stated. Table 22.2 data describing work done in the Eastern Chapel. Additional raw data was listed on lines G5, G6/H32, G14, G15, G16, G17/H33 and G18/H34, as follows:

12/10 = 1 + 1/5 (G5)

10/10 = 1 (G6 & H32)

8/10 = 1/2 + 1/4 + 1/20 (G14)

48/10 = 4 + 1/2 + 1/4 + 1/20 (G15)

16/10 = 1 + 1/2 + 1/10 (G16)

64/10 = 6 + 1/4 + 1/10 + 1/20 (G17 & H33)

36/10 = 3 + 1/2 + 1/10 (G18 & H34)

Chace and Shute had also noted this division by 10 method, though not clearly using quotients and remainders. Other scholars have also muddled reading the first 6 problems of the Rhind Mathematical Papyrus, missing its clear use of quotient and remainders.

Gillings, Chace and Shute apparently did not analyze the RMP data in a broader context, and report its structure, thereby missing a major fragment of scribal remainder arithmetic. That is, Gillings' citation in Reisner and RMP "Mathematics in the Time of the Pharaohs" only scratched the surface of this information. Had scholars dug a little deeper, as cited, academics may have found other reasons for the 39/10 error, the most important one being quotient and remainders.

The Reisner error could have been noted by Gillings by using a single statement quotient (Q) and remainder (R), an answer that Ahmes and other Middle Kingdom scribes had used to write this type of information in other texts. Gillings may have simply forgotten to summarize his findings in a rigorous manner, showing that many scribes, writing in many Middle Kingdom texts, used the same quotient and remainder structure in the same manner.

Seen in a broader sense the data, can be noted as:

39/10 = (Q' + R)/10 with Q' = (Q*10), Q = 3 and R = 9

such that:

39/10 = 3 + 9/10 = 3 + 1/2 + 1/3 + 1/15

with 9/10 being converted to a unit fraction series following rules set down in the AWT, and followed in RMP and other texts.

Confirmation of the scribal remainder arithmetic is found in other hieratic texts. The most important one is the Akhmim Wooden Tablet (AWT) since it defines scribal remainder arithmetic in term of another context, a hekat (volume unit). Oddly, Gillings did not cite the AWT in his main book, nor has any other serious Egyptian math scholar, thereby Gillings and the earlier 1920's scholars had missed a major opportunity to have been first to point out a multiple use of scribal remainder arithmetic.

The modern looking remainder arithmetic was later found by others by taking a broader view of the 39/10 error, as corrected as the actual Easten Chapel data reports.

Gillings and the academic community therefore had inadvertently omitted a critically important discussion of fragments of remainder airthmetic. Remainder arithmetic, as used in many ancient cultures to solve astronomy and time problems, is one of several plausible historical division methods that may have allowed a full restoration of scribal division around 1906.

In summary, the Reisner Papyri was built upon a method described in the Akhmim Wooden Tablet, and later followed by Ahmes writing the RMP. The Reisner calculations apparently follows our modern Occam's Razor rule, that the simplest method was the historical method; in this case remainder arithmetic, such that:

n/10 = Q + R/10

where Q was a quotient and R was a remainder.

The Reisner, following this Occam's Razor rule, says that 10 workmen units were used to divide raw data using a method that was defined in the text, a method that also begins the Rhind Mathematical Papyrus, as noted in its first six problems.

[edit] References

Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7

Gillings, Richard J., "Mathematics in the Time of the Pharaohs", Dover, New York, 1971, ISBN 0-480-24515-X

Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4