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 strucrture.

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 would 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.

This early 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 directly 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 h
arsh 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 issuee must be judged
by Occam's Razor, and rigorously tested against the contents
of the other hieratic texts.

The simpliest method should win out as the historical method.
The Reisner, following this Occam's Razor rule, says that 10
workmen units were used to divide raw data using a method
that was clearly defined in the text, and the RMP in its
first problem.

It may take several years before the errors of omission
of the last 100 years may be fully corrected in the
professional journals. This blog is being submitted to
hopefully jump start a long overdue debate on the topic
of scribal remainder arithmetic.


author: Milo Gardner