132 6. CALCULATION1/8 + 1/16 + 1/32 + 1/64 = 63/64, the scribes apparently saw that unity could
be restored (approximately), as Horus' eye was restored, by using these parts.
The fact that (in our terms) 63 occurs as a numerator, shows that division by
3, 7, and 9 is facilitated by the use of the Horus-eye series. In particular, since
1/7 = (1/7) · ((63/64) + 1/64) = 9/64 + 1/448 = 8/64 + 1/64 + 1/448, the seventh
part could have been written as 8 64 448. In this way, the awkward seventh part
gets replaced by the better-behaved Horus-eye fractions, plus a corrective term (in
this case 448, which might well be negligible in practice. Five such replacements are
implied, though not given in detail, in the Akhmim Wooden Tablet.^2 As another
example, since 64 = 4-13 + 8 + 4, we find that T3 becomes 16 Ú04 208.
There are two more complications that arise in doing arithmetic the Egyptian
way. The first complication is obvious. Since the procedure is based on doubling,
but the double of a part may not be expressible as a part, how does one "calculate"
with parts? It is easy to double, say, the twenty-sixth part: The double of the
twenty-sixth part is the thirteenth part. If we try to double again, however, we are
faced with the problem of doubling a part involving an odd number. The table at
the beginning of the papyrus gives the answer: The double of the thirteenth part
is the eighth part plus the fifty-second part plus the one hundred fourth part. In
our terms this tabular entry expresses the fact that
I-I4-I4--L
13 8 ~ 52 T 104'
Gillings (1972, p. 49) lists five precepts apparently followed by the compiler of
this table in order to make it maximally efficient for use. The most important of
these are the following three. One would like each double (1) to have as few terms
as possible, (2) with each term as small as possible (that is, the "denominators"
as small as possible), and (3) with even "denominators" rather than odd ones.
These principles have to be balanced against one another, and the table in Fig.
1 represents the resulting compromise. However, Gillings' principles are purely
negative ones, telling what not to do. The positive side of creating such a table
is to find simple patterns in the numbers. One pattern that occurs frequently is
illustrated by the double of 5, and amounts to the identity 2/p = l/((p + l)/2) +
l/(p(p + l)/2). Another, illustrated by the double of 13, probably arises from the
Horus-eye representation of the original part.
With this table, which gives the doubles of all parts involving an odd number
up to 101, calculations involving parts become feasible. There remains, however,
one final complication before one can set out to solve problems. The calculation
process described above requires subtraction at each stage in order to find what
is lacking in a given column. When the column already contains parts, this leads
to the second complication: the problem of subtracting parts. (Adding parts is no
problem. The author merely writes them one after another. The sum is condensed
if, for example, the author knows that the sum of 3 and 6 is 2.) This technique,
which is harder than the simple procedures discussed above, is explained in the
papyrus itself in Problems 21 to 23. As mentioned above, Problem 21 asks for the
parts that must be added to the sum of 3 and 15 to obtain 1. The procedure used
to solve this problem is as follows. Begin with the two parts in the first row:
(^2) See http://www.inathworld.com/AkhmimWoodenTablet.html. In a post to the history of mathe-
matics mailing list in December 2004 the author of that article, Milo Gardner, noted that recent
analysis of this tablet has upset a long-held belief about the meaning of a certain term in these
equations.