**The RHIND PAPYRUS**

The Rhind Mathematical
Papyrus, which is also known as the Ahmes Papyrus, is the major source of our
knowledge of the mathematics of ancient Egypt. Although the many ceremonial
objects which have been discovered in tombs and elsewhere in Egypt during
archaeological explorations provide examples of numerals written in
hieroglyphic and hieratic script and while many non-mathematical papyri contain
hieratic and demotic writings involving numerals, these usually contain little
or no actual mathematics as such.

The Rhind Papyrus dates
from approximately 1650 B.C.E. Mr. A. Henry Rhind, a Scottish lawyer, visited
Egypt in the mid-nineteenth century on the advice of his physician in hopes
that its dry climate would be beneficial to his poor health, and he became very
interested in Egyptian antiquities. This interest led Rhind to participate in
archaeological excavations in Thebes, an ancient Pharaohnic capital. Rhind
purchased the papyrus in Luxor, Egypt, in 1858. In later years, it was willed
to the British Museum, where it remains today. A piece missing from the center
of the papyrus was located in New York City many years later and was restored
to the Rhind Papyrus after 1922.

Recall that Champollion
had deciphered Egyptian hieroglyphic script in the period from 1810 to 1830 and
that the key to unlocking the secret of this ancient writing, the Rosetta
Stone, also contained a large passage written in demotic script, which enabled
scholars to interpolate between the two systems and decode hieratic script as
well. In 1877, Professor A. Eisenlohr published the first translation of the
Rhind Papyrus into a modern language, in his book * Ein matematisches
handbuch der älten Ägypter (Papyrus Rhind der British Museum), übersetzt und
erklärt.* T.E. Peet issued an English edition in 1923.

The papyrus is in the
form of a scroll about 12-13 inches wide and 18 feet long, written from right
to left in hieratic script on both sides of the sheet, in black and red inks.
After identifying himself as the writer, the scribe Ahmes (or Ahmose -- since the
Egyptians seldom indicated vowels, there is some ambiguity about the
pronunciation) begins by saying that he has copied this work from a very old
scroll from the period of the Middle Kingdom, a couple of hundred years
earlier. Thus the mathematics in this papyrus dates from the same time period
as that in another important mathematical scroll, the Golenishev or Moscow
Papyrus. Before getting down to business, Ahmes begins with an announcement,
that he will provide a "complete and thorough study of all things"
and will reveal "the knowledge of all secrets." Then he goes on to
the content, consisting of two tables of fractions followed by 84 worked
problems.

The first fraction table
occupies a large part of the manuscript. For each odd integer *n* from 5
through 101, it gives a decomposition of twice the unit fraction 1/n into a sum
of distinct UNIT FRACTIONS, fractions whose numerator is 1. This table is not
just a list of facts; every entry is either derived from scratch or is verified
in detail. Notationally, the hieratic symbol for 1/k consisted of the symbol
for k with a dot over it. The hieroglyphic equivalent used a small oval over
the numeral for the integer. To facilitate working with unit fractions, Otto
Neugebauer introduced the notation which is now in general use, n denoting 1/n. (Similarly to Ahmes' double over-dots, Neugebauer used a 3 with two overbars for the important special fraction that we write as 2/3.)

The second fraction table
decomposes one tenth of n as a sum of distinct unit fractions, for n = 1, 2,
..., 9. These representations will be referred to several times in the rest of
his text.

Evidently, the ancient
Egyptians did not consider a fraction whose numerator is greater than 1 to be a
number -- only unit fractions, the number 2/3 and a few other fractions of the
special form n/(n + 1) were felt to be completed processes. For example, 3/7 is
felt to be incomplete, not exactly a number, and so it must be expressed as a
sum of genuine fractions when it appears either in the answer to a problem or in
a related calculation.

Why is n odd in the
doubles table? These people of long ago realized that the case of *even* n's was
easy. As Neugebauer says in his * The Exact Sciences in Antiquity*,
"Here we find that twice 2,
4,
6,
8, etc. are always replaced by 1,
2,
3,
4, respectively."
Some of the table decompositions seem reasonable, given the
unstated rule that no term in the sum can be repeated. For instance, 2/3 and
1/2 1/6 (addition is understood) were recognized as being interchangeable, and
that 2/5 is 1/3 1/15 is understandable to us. But some of the table entries are
hard to believe, such as: twice 1/83 is 1/60 1/332 1/415 1/498.

Not only were 2/3 and the
1/n's essentially the only fractions used as numbers, but repetitions were not
allowed in the representations of fractions as sums. E.g., 2/7 could not be
presented as 1/7 1/7. But, using the twice 1/n table one can express any proper
fraction in terms of the basic ones. E.g., 5/7 = 1/7 + 4/7 = 1/7 + 2(2/7). From
Table 1, 2&lowast 7 = 4 28;
so 2&lowast (2&lowast 7) = 2&lowast 4 28
= 2 14. Hence 5/7 is 2 7 14
, if we list the parts in decreasing size (increasing order of
denominators).

Historians of mathematics
have devised many explanations of how the twice 1/n and one-tenth n tables were
derived. For more information, you may consult the books ESA (by Neugebauer,
cited above), Mathematics in Civilization (Resnikoff and Wells), The Rhind
Mathematical Papyrus: an ancient Egyptian text (G. Robins and C. Shute; Dover
Publications reprint of the 1987 edition published by the Trustees of the
British Museum), Mathematics in the Time of the Pharaohs (R.J. Gillings; Dover
Publications 1982 reprint of the 1972 edition published by the MIT Press).

Some of Ahmes' problems were
straightforward mathematics, such as seeking the solution of a linear equation.
E.g., Problem 24 asks to find a quantity if it plus one seventh of it equals
nineteen. This was stated partly verbally and partly symbolically, since Ahmes
had symbols for "plus" and "minus" and used a word meaning
"heap" or "quantity" (pronounced something like 'aha') for
the unknown. Some of the linear equations problems are stated as word problems
involving the division of commodities such as grain, bread, or beer into equal
parts. (Important for those responsible for feeding the construction crews or
overseers who built pyramids.) The presentation is usually in three parts: the
problem is stated, a brief calculation is done which leads to a solution -- or
a solution is pulled out of the air, and a check is done to show that the
number found really satisfies the condition. The arithmetical calculations are
presented in detail.

When one reads Ahmes'
calculations one sees very quickly that MULTIPLICATION of whole numbers was
done by a process (DUPLATION) consisting of repeated doublings and one
addition. For instance to multiply 43 by 19 (i.e., to find 19 times 43), Ahmes
would have done the work in two columns more or less as follows. (To find 43
times 19, the roles would have been interchanged.)

43 1/ Keep doubling until you reach the last power

` 86 2/ of two which is below the multiplier. `

` 172 4 Check the rows needed to add up to the multiplier, `

` 344 8 in this case 1 + 2 + 16 = 19, and then add the `

` 688 16/ corresponding doubles in the other column, in this`

` ---- --- case 43 + 86 + 688 (which was easy to do using `

` 817 19 the hieroglyphic notation). `

Multiplying
by a fraction was done the same way, which explains the presence of the first
table in the Rhind Papyrus. Neugebauer writes "Every multiplication and
division which involves fractions leads to the problem of how to double unit
fractions."

DIVISION was carried out by a
variation of the duplation process. (If the quotient was an integer, the
division was very much like duplation; otherwise, adjustments were necessary.)
See examples in the text, and note David Burton's remark, "It is extraordinary
that to get one third of a number, the Egyptians first found two-thirds of the
number and then took one-half of the result."

These multiplication and division
processes hinge crucially on the fact that every positive integer is the sum of
distinct powers of two. Indeed, although they didn't have the concept, the
Egyptians of those days were using base two expansions.

In his article "The Rhind
Papyrus" in the four-volume book *The World of Mathematics *, which
he compiled, James R. Newman writes "It is remarkable that the Egyptians,
who attained much skill in their arithmetic manipulations, were unable to
devise a fresh notation and less cumbersome methods. We are forced to realize
how little we understand the circumstances of cultural advance: why societies
move -- or perhaps it is jump -- from one orbit to another of intellectual
energy, why the science of Egypt "ran its course on narrow lines" and
adhered so rigidly to its clumsy rules. Unit fractions continued in use, side
by side with improved methods, even among Greek mathematicians. Archimedes, for
instance, wrote 1/2 1/4 for 3/4, and

Hero[n] 1/2 1/17 1/34 1/51 for 31/51."

Some of the problems in the Rhind
Papyrus translate for us into the form "Find x if x + ax = b." In
four of those problems, a is a unit fraction, and in four others it is a sum of
two or three unit fractions. The equations where a = 1/n are solved by Ahmes by
a method which has become known as Regula Falsi (in Latin), or FALSE POSITION
(or False Supposition). In this method, an incorrect but convenient GUESS of
the value of the unknown is made, the left side of the equation is evaluated at
the guessed value, and then the guess is adjusted by multiplying by a suitable
amount. In a sense, Ahmes knew that multiplication distributes over addition.
So he understood that solving the equation amounts to dividing b by (1 + a) --
e.g., he knew that "x + (1/n) x = b" is equivalent to [(n + 1)/n] x =
b and thus x is b multiplied by n/(n + 1). But he almost never solved a linear
equation by a process that we could consider as equivalent to doing that.

Consider the solution of Problem
24, which is to find heap if heap and its one-seventh added together become 19.
Ahmes states a linearity or proportionality principle: "As many times as 8
must be multiplied to give 19, just as many times must 7 be multiplied to give
the correct answer." That is, guess that heap is 7; then the left side of
the equation evaluates to 7 + one-seventh of 7, or 8. We want the left side to
be 19, so we change everything proportionally, using the factor which will take
us from 8 to 19. That factor is 2 plus 1/4 1/8, which will give us 16 + 2 + 1 =
19. So the solution is (2 1/4 1/8) times 7; but that's too complicated to find
by duplation, so Ahmes does the simpler calculation, 7 times (2 1/4 1/8) and
finds that heap = 16 1/2 1/8.

The method of false position was
used in Europe from the 1200s until algebra notation was firmly established in
the late sixteenth century, owing largely to its popularization by Leonardo of
Pisa (nowadays known as Fibonacci) as a way to solve problems of commercial
arithmetic.

Ahmes's Problem 6 is to find how
to divide 9 loaves equally among 10 men. Because of the severe limitation on
the ancient Egyptians' use of fractions, this was a difficult problem.

Here is his presentation:

"The doing as it occurs: Make thou the multiplication 2/3 1/5 1/30 times.

` 1 2/3 1/5 1/30 `

` \2 1 2/3 1/10 1/30 `

` 4 3 1/2 1/10 `

` \8 7 1/5 `

Total loaves 9; it, this is."

Ahmes tells the reader the answer
without deriving it. (This may have made for complications if one had to divide
9 loaves among 15 men, e.g., since we're given no clue on how to begin.) Then
he presents a verification that his answer is correct, in the usual duplation
style. In the first step, he doubles

2/3 1/5 1/30
and gets

1 2/3 1/10 1/30
but gives no explanation, not even
a reference to the doubles table.

The next
doubling goes from

` 1 2/3 1/10 1/30 to 3 1/2 1/10, again without explanation.`

(We would get 10/3 1/5 1/15 = 3 1/3 1/5 1/15, but that's not Ahmes's decomposition.) The final doubling gives us 6 + 1 + 1/5 = 7 1/5, which agrees with Ahmes's value.

To verify the correctness of the answer, one adds the multiples corresponding to the checkmarks, namely

1 2/3 1/10 1/30 plus 7 1/5, giving us 8 2/3 1/5 1/10 1/30.

Some simplifying is required to see that this equals 9.