- EGYPT 131
then move up and mark the next row whose right-hand column contains an entry
not larger than this remainder (in this case the second row), subtract the entry in
its right-hand column (2), from the previous remainder to get a smaller remainder
(in this case 1), and so forth.
We shall refer to this general process of doubling and adding as calculating.
What we call division is carried out in the same way, by reversing the roles of the
two columns. For example, what we would call the problem of dividing 873 by 97
amounts to calculating with 97 so as to obtain 873. We can write it out as follows:
- 97 1
194 2
388 4 - 776 8
873 9 Result.
The process, including the rules for creating the rows and deciding which ones
to mark with an asterisk, is exactly the same as in the case of multiplication, except
that now it is the left-hand column that is used rather than the right-hand column.
We create rows until the next entry in the left-hand column would be larger than
- We then mark the last row, subtract the entry in its left-hand column from
873 to obtain the remainder of 97, then look for the next row above whose left-hand
entry contains a number not larger than 97, mark that row, and so on.
1.2. "Parts". Obviously, the second use of the two-column system can lead to
complications. While in the first problem we can always express any positive integer
as a sum of powers of 2, the second problem is a different matter. We were just
lucky that we happened to find multiples of 97 that add up to 873. If we hadn't
found them, we would have had to deal with those parts that have already been
discussed. For example, if the problem were "calculate with 12 so as to obtain 28,"
it might have been handled as follows:
12 1
* 24 28 I
* 4 3_
28 2 3 Result.
What is happening in this computation is the following. We stop creating rows
after 24 because the next entry in the left-hand column (48) would be bigger than- Subtracting 24 from 28, we find that we still need 4, yet no 4 is to be found.
We therefore go back to the first row and multiply by |, getting the row containing
8 and 3. Dividing by 2 again gets a 4 in the left-hand column. We then have the
numbers we need to get 28, and the answer is expressed as 2 3. Quite often the
first multiplication by a part involves the two-thirds part 3. The scribes probably
began with this part instead of one-half for the same reason that a carpenter uses
a plane before sandpaper: the work goes faster if you take bigger "bites."
The parts that are negative powers of 2 play a special role. When applied to
a hekat of grain, they are referred to as the Horus-eye parts.^1 Since 1/2 + 1/4 +
(^1) According to Egyptian legend, the god Horus lost an eye in a fight with his uncle, and the eye
was restored by the god Thoth. Each of these fractions was associated with a particular part of
Horus' eye.