The Reisner Papyrus

THE REISNER PAPYRUS, tables 22.3 and 22.2

The most basic of the hieratic mathematical texts may be the Reisner Papyrus. 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 many other scholars have accepted 100 year old views, some of which are incomplete and misleading.

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

divide 39 by 10 = 4,

a poor approximation to the correct value.

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 division by 10 answers were correctly stated, as cited on other lines in table 22.2 for the Eastern Chapel are listed on lines G5, G6/H32, G14, G15, G16, G17/H33 and G18/H34. The raw information follows, per G5, and so forth:

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)

Gillings did not analyze this correct data, and report its structure created by a scribe, thereby missing a major fragment of scribal remainder arithmetic. That is, Gillings' citation in "Mathematics in the Time of the Pharaohs" went into shallow depth concerning it non-analysis of two-part data. Gillings had discussed none of three possible alternative historical structures of the data. Had he done so, as cited above, Gillings may have found other reasons for the 39/10 error.

The error had been correctly solved by Gillings using a single statement quotient (Q) and remainder (R) answer as Ahmes and other Middle Kingdom scribes written this type of information. Gillings may have forgotten to summarize his findings in a rigorous manner, showing that all the data was written in a quotient and remainder structure.

Had Gillings looked closely at the correct data, as noted by:

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

he may have seen,

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 RMP and elsewhere.

In conclusion, Gillings' single sided view of the information, showing only the scribal error, nowhere did he note or suspect that the Reisner scribe had left several easy to read clues that pointed directly at a modern two-part form of arithmetic in table 22.2. The modern form of arithmetic outlines aspects
of scribal remainder arithmetic.

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, hekat (volume) division.

Oddly, Gillings did not cite the AWT in his main book, nor has other serious Egyptian math scholars. Oddly, Gillings and the earlier 1920's scholars missed major opportunities 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.

To reduce a critique of Gillings, the Reisner data may have been inadvertently hidden, since there was a deeper codified basis to the modern remainder arithmetic. The deeper remainder arithmetic a clearly written into the Akhmim Wooden Tablet .

The Akhmim Wooden Tablet was first translated in 1906, in French, so the translators of the Reisner Papyri would have little or no chance to compare notes. Possibly this is the main reason that Gillings missed the citation of the AWT.

Returning to Gillings, in retrospect he failed to investigate and openly discuss the basic (per Occam's Razor) methods, and comparisons of the Reisner data with other hieratic text, namely the Akhmim Wooden Tablet, the RMP and other texts that report fragments and generalized uses of scribal remainder arithmetic.

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.

An oversight of excluding the Akhmim Wooden Tablet from the Egyptian math debate seemed to have taken place because Gillings, and earlier several scholars may have
concluded in the 1920's that Egyptian division had been exclusively based on an inverse connection with Egyptian multiplication. But, did Egyptian division actually follow a reverse relationship to Egyptian multiplication?

This blog shows that Middle Kingdom scribes had omitted critical steps within their shorthand, with each scribe tending to use his/her own method of leaving out
certain logical steps. In retrospect, it may be overly harsh to conclude that modern scholars should have taken greater care in 1906, and later times, to review all of the Egyptian fraction texts, before closely off debates.

But I think not.

In retrospect, in appears to this writer, that modern scholars, by the 1920's, apparently had jumped to a complex 'inverse multiplication' conclusion, one that had incorrectly and confusingly structured a hard and fast -one and only -
division rule for all Egyptian scribes. Less complex scribal division alternatives were and are available. And one that calls out to be rigorously studied is remainder arithmetic.

Again, using the Akhmim Wooden Tablet, and the RMP as additional reference guides, all of the above Reisner calculations apparently followed the more basic formal
of remainder arithmetic, one that Occam's Razor tends to say was preferred to Gillings', Peet, et al per inverse multiplication view, as given by:

n/10 = Q + R/10

where Q was a quotient and R was a remainder.

The important issue is the the simple form of remainder arithmetic apparently has been found, and connected to several hieratic texts. On each Reisner line, for example,are quotient and remainder values, several listed more than once. The scribe then mentally converted the pertinent vulgar fraction created by the division by 10 operation into quotients and remainders, as noted above.

A wider discussion of the RMP and its remainder arithmetic can be found on:

http://egyptianmath.blogspot.com

This writer's view, therefore, is that the Reisner 'raw data' must be directly considered to assist in shedding light on the wider scope of scribal thought, no more and no less. In the end, the modern analysis of the scribal division, or any other scribal math issue should be rigorously tested against the contents of the other hieratic texts.

The simplest method should win out as the historical method.

It has taken several years before the errors of omission of the last 100 years to begin to be corrected in the professional journals. This blog jump starts a long overdue debate on the topic of scribal remainder arithmetic.

references

author: Milo Gardner