134 6. CALCULATION(subtract) parts, what are called hau (or aha) computations by the author, one can
confidently attack any arithmetic problem whatsoever. Although there is no single
way of doing these problems, specialists in this area have detected systematic pro-
cedures by which the table of doubles was generated and patterns in the solution of
problems that indicate, if not an algorithmic procedure, at least a certain habitual
approach to such problems.
Let us now consider how these principles are used to solve a problem from the
papyrus. The one we pick is Problem 35, which, translated literally and mislead-
ingly, reads as follows:
Go down I times 3. My third part is added to me. It is filled. What
is the quantity saying this? ·
To clarify: This problem asks for a number that yields 1 when it is tripled
and the result is then increased by the third part of the original number. In other
words, "calculate with 3 3 so as to obtain 1." The solution is as follows:33 1
10 3 [multiplied by 3]
5 \2_
1 5 10 Result.1.3. Practical problems. One obvious application of calculation in everyday life
is in surveying, where one needs some numerical way of comparing the sizes of
areas of different shapes. This application is discussed in Chapter 9. The papyrus
also contains several problems that involve proportion in the guise of the slope of
pyramids and the strength of beer. Both of these concepts involve what we think
of as a ratio, and the technique of finding the fourth element in a proportion by
the procedure once commonly taught to grade-school students and known as the
Rule of Three. It is best explained by a sample question. If three bananas cost 69
cents, what is the cost of five bananas? Here we have three numbers: 3, 69, and- We need a fourth number that has the same ratio to 69 that 5 has to 3, or,
equivalently, the same ratio to 5 that 69 has to 3. The rule says that such a number
is 69 ÷ 5 -ô- 3 = 105. Since the Egyptian procedure for multiplication was based
on an implicit notion of proportion, such problems yield easily to the Egyptian
techniques. We shall reserve the discussion of pyramid slope problems until we
examine Egyptian geometry in Chapter 9. Several units of weight are mentioned
in these problems, but the measurement we shall pay particular attention to is a
measure of the dilution of bread or beer. It is called a pesu and defined as the
number of loaves of bread or jugs of beer obtained from one hekat of grain. A hekat
was slightly larger than a gallon, 4.8 liters to be precise. Just how much beer or
bread it would produce under various circumstances is a technical matter that need
not concern us. The thing we need to remember is that the number of loaves of
bread or jugs of beer produced by a given amount of grain equals the pesu times
the number of hekats of grain. A large pesu indicates weak beer or bread. In the
problems in the Ahmose Papyrus the pesu of beer varies from 1 to 4, while that for
bread varies from 5 to 45.
Problem 71 tells of a jug of beer produced from half a hekat of grain (thus its
pesu was 2). One-fourth of the beer is poured off and the jug is topped up with
water. The problem asks for the new pesu. The author reasons that the eighth part
of a hekat of grain was removed, leaving (in his terms) 4 8, that is, what we would