Title:
THE EGYPTIAN MATHEMATICAL LEATHER ROLL,

ATTESTED SHORT TERM AND LONG TERM.

By: MILO GARDNER

SACRAMENTO, CALIFORNIA

Date: Feb. 16, 2002

1.
ABSTRACT

The
Egyptian Mathematical Leather Roll (EMLR), housed at the British Museum,
contains 26 unique Egyptian fraction series. In summary, five attested methods
converted 1/p and 1/pq to the 26 series. A breakdown

includes
three classes of identities (18 series), a class of remainders (three series), and a class of algebraic identities
(five-eight series). The paper discusses clues to Middle Kingdom scribal school
teaching

methods
that are hidden in the EMLR. Here is a mystery worthy of Conan Doyle and
Sherlock Holmes. The EMLR indicates the student scribe was introduced to Egyptian
fraction methodologies that were an early

form
of abstract mathematics, as needed to work RMP problems.

2.
INTRODUCTION

A
complete EMLR translation is included
as Appendix I, written in reverse order, as the EMLR student actually wrote.
Note that Appendix I contains no questions or applications, only answers.
Horus-Eye numbers

(1/2,
1/4, 1/8, 1/16, 1/32, 1/64) stand out along with 1/3, 1/5, 1/7, 1/9, 1/10, 1/11, 1/13 and two ways of stating 1/15th, amidst other 1/p and 1/(p x
q) conversions, to exact Egyptian fraction series. The EMLR series gave exact
answers with no remainder, showing that the round-off problems of the older
infinite series were attempted to be resolved by Middle Kingdom scribes. This
innovation arguably introduced

an
improved perspective of rational numbers.

This
paper departs from the usual history of Egyptian mathematics, where a system of
multiplication was connected to a base 2 decimal fraction, duplation, infinite
series numeration (Robins-Shute, MacTutor).

This
is a critical point, since using the Old Kingdom duplation methodology to
explain Middle Kingdom texts confuses this writer, and hopefully the reader.
This paper, therefore, stresses a flip side of Horus-Eye math, opening a long
lost door into the development of exact Egyptian mathematics, known as Egyptian
fractions.

Appendix
II discusses the Horus-Eye fraction method in one older sense, awkwardly
computing the units of weights and measures, even when the easier Egyptian
fractions are hidden within the calculation. The new

Middle
Kingdom hieratic system apparently was an improvement that endured in the
Western Tradition for over 3,500 years, from 2,000 BC to 1585 AD, ending
with the formalization of base 10
decimals (Ore).

One
obvious reading of the EMLR, as a test paper, infers that all Egyptian fraction
methods eliminated rounding-off practices, except when associated with Old
Kingdom weights and measures units. Similar

round-off
practices existed in the base 60 Babylon numeration system (van der Waerden).
Middle Kingdom Egyptians learned to eliminate
round-off within a rational number system that used exact unit fraction
series, thereby greatly improving mathematical accuracy related to our pre-base
ten decimal system.

Analysis
of the EMLR unit fraction series introduces five different methods, as the
student test paper is read. It is hypothesized that the RMP duplation methods,
suggested for 75 years, from 1927 to the present, and said to have been the
primary RMP 2/n table method (Robins-Shute), was only a secondary method in the
EMLR.

The
intellectual content of the EMLR reveals that five conversion methods were
plausibly the first and possibly the best, of the new Middle Kingdom methods.
It is also suggested that Robins-Shute's
duplation conversion method was primarily Old Kingdom in origin. The
simplest EMLR and RMP 2/p conversion method is a subtle issue, one
worthy of its own paper, to be written at a later time.

Strong
hints of the actual intellectual contents of the EMLR is suggested by reading
backwards the 26 unique unit fraction series, summarized by five methods:

A. 1/n
= 1/2n + 1/2n, used to calculate four EMLR series

B. 1/2p = 1/p x (1/2) = 1/p x (1/3 + 1/6),
used to calculate ten EMLR series

C. 1/p
= 1/p x (1) = 1/p x (1/2 + 1/3 + 1/6), used to calculate four EMLR
series

D. 1/p
= [1/(p+1)] + 1/[p x (p+1)], used to the calculate three EMLRseries

E. 1/pq = 1/A x A/pq, used to calculate five
- eight EMLR series.

Attestation
of each method relies on the inner consistency between the sub-sets of the 26
series, as well as the closely related contents of 45-50% of the RMP 2/n table. One overlying assumption of this
analysis is the utility of Occam's Razor, which seeks the simplest method
(Sarton), as the historical method(s).

THE
EGYPTIAN MATHEMATICAL LEATHER ROLL, AN OVERVIEW

The
EMLR is an under appreciated document. Since its unrolling in 1927, over 50
years after its sister RMP document, the EMLR has been reported as containing
only simple additive arithmetic (Gillings). This current

status
is misleading, in that a major aspect of the EMLR was ignored to reach this
oversimplified conclusion. A different, more subtle view is suggested.

Mathematicians,
Egyptologists and historians of various disciplines in 2002 tend to work within
mutually exclusive disciplines, as was the case in 1927. That is, scholars in one field tend to be
unfamiliar with the

specialized
language and practices of the other, often accepting the conclusions of the
other without critical examination. One consequence is that errors introduced
by members of one discipline are often not

pointed
out by the members of another.

Scholars
first attempted to decode the EMLR contents in 1927. In presenting their
conclusions, the algebraic aspects were omitted. Perhaps one reason was that classical scholars had previously
reported that Egyptian

fractions
showed signs of intellectual decline from
Middle Kingdom mathematics (Neugebauer).

There
are signs of improvement, refuting Neugebaur's issue of intellectual decline. A
debate has begun on the Pythagorean side
of the Babylonian Plimpton 322 (Robson), that introduces a fresh set of historical methods. This regional
debate extends to the evaluation of Egyptian fractions, beyond issues of
additive mathematics.

One
fresh analysis of the EMLR reveals five methods that were used to build the first member of 1/p, 1/pq, of n/p and n/pq tables, where n stands
for any positive integer:

A. Method one (Identity):

1/n = 1/2n + 1/2n, or (Identity 1.0)

1/n = 1/3n + 1/3n + 1/3n (Identity 1.1)

EMLR Examples:

1. 1/5 = 1/10 + 1/10

2. 1/3 = 1/6 + 1/6

3. 1/2 = 1/6 + 1/6 + 1/6

4. 2/3 = 1/3 + 1/3

None
of these four fractional series technically define a true Egyptian fraction
series, since units could not be repeated. However, the EMLR student was asked
to note these fractional relationships for a reason. Was an introduction to the
properties of numbers, that numbers can be dissected, trisected, and more being asked? Does the EMLR suggest that the
fundamental theorem of arithmetic, that every positive integer can be expressed
as a unique product of primes (and powers of primes), was hidden to the Middle
Kingdom student?

Clearly
The EMLR student was asked to decompose 1/n and less frequently 1/p (where p =
prime) by various means, odds and evens being the first. What else was taught
and discussed in the scribal school, as
hidden in the EMLR answers? What else will be revealed by this Holmes-like
mystery, as EMLR series are studied?

B. Method Two (Identity): 1/2n = 1/n x (1/3 +
1/6), (Identity 2.0)

EMLR examples:

5.
1/6 = 1/9 + 1/18 = 1/3 x (1/2) = 1/3 x (1/3 +
1/6) (n=3)

6.
1/8 = 1/12 + 1/24 = 1/4 x (1/2) = 1/4 x (1/3 +
1/6) (n=4)

7.
1/10 = 1/15 + 1/30 = 1/5 x
(1/2) = 1/5 x (1/3 + 1/6) (n=5)

8.
1/12 = 1/18 + 1/38 = 1/6 x
(1/2) = 1/6 x (1/3 + 1/6) (n=6)

9.
1/14 = 1/21 + 1/42 = 1/7 x
(1/2) = 1/7 x (1/3 + 1/6) (n=7)

10. 1/16 = 1/24 + 1/48 = 1/8 x (1/2) = 1/8 x (1/3 + 1/6) (n=8)

11. 1/20 = 1/30 + 1/60 = 1/10 x (1/2) = 1/10
x (1/3 + 1/6 (n=10)

12. 1/30 = 1/45 + 1/90 = 1/15 x (1/2) = 1/15
x (1/3 + 1/6) (n=15)

13. 1/32 = 1/48 + 1/96 = 1/16 x (1/2) = 1/16
x (1/3 + 1/6) (n=16)

14. 1/64 = 1/96 + 1/92 = 1/32 x (1/2) = 1/32
x (1/3 + 1/6) (n=32)

This
list of even number conversions appears to have stopped at the last Horus-Eye
unit, a limit that Egyptians knew extended to six significant digits (1/128,
1/256, 1/512, 1/1024, 1/2048, and 1/4096). Method Two

could
be indefinitely extended by the scribes so long as smaller units could be
found, a wonderful property of even numbers.

By
contrast, the older Horus-eye notation 1 = 1/2 + 1/4 + 1/8 + 1/16 +1/32 + 1/64
meant that the 1/64th unit was thrown away as a round-off. Rounding-off also
took place with other significant digits, sometimes with

the
1/4096 unit being thrown away. By contrast, Method Two of the EMLR, as was also
true for the other four EMLR methods, avoided rounding-off. It was exact, not
requiring the throwing away of any unit. This shows that in the EMLR all
Egyptian fraction series were calculated to the highest accuracy, with no remainder, whenever rational numbers were
considered.

C.
Method Three: 1/p = 1/p x (1) = 1/p x (1/2 + 1/3 + 1/6) (Identity 3.0)

EMLR examples:

15. 1/7
= 1/14 + 1/21 + 1/42 = 1(1/7) =
1/7 x (1/2 + 1/3 + 1/6)

16. 1/9
= 1/18 + 1/27 + 1/54 = 1(1/9) =
1/9 x (1/2 + 1/3 + 1/6)

17. 1/11 = 1/22 + 1/33 + 1/66 = 1(1/11) =
1/11 x (1/2 + 1/3 + 1/6)

18. 1/15 = 1/30 + 1/45 + 1/90 = 1(1/15) =
1/15 x (1/2 + 1/3 + 1/6)

A
short list of odd numbers does not prove that the scribes intended that all 1/p
conversions would be written by Method Three, as Gillings and others have
implied. Methods Four and Five point out that 1/p and 1/(p x q) were exactly
converted by either a remainder method and/or algebraic identity method.

Before
leaving Method Three, an error appears on line 17, needs to be discussed, where
1/13 = 1/28 + 1/49 + 1/196. Gillings attempted to use Method Three to correct
for this error.

However,
1/14 = 1/28 + 1/49 +1/98 + 1/196, with the succeeding terms omitted in
round-off (1/394 + 1/788) may have been
involved. Considering the fact that 1/13 - 1/14 = 1/182, the student may have dropped
the 1/98 term as an approximate round-off, not knowing how an exact 1/13th
series could be found.

D. Method Four (Remainder),

1/p -1/(p + 1) = 1/[p x (p + 1)]
(Remainder 1.0)

1/n - 1(n + 1)) = 1/[n x (n + 1)] (Remainder 1.1)

EMLR examples:

19. 1/3 - 1/4 = 1/12 or 1/3 = 1/4 + 1/12

20. 1/4 - 1/5 = 1/20 or 1/4 = 1/5 + 1/20

21. 1/8 - 1/9 = 1/72 or 1/8 = 1/9 + 1/72

Attestation
is found in two sources, the EMLR itself, in the other two series, and the RMP
2/n table, with four fractional series, as shown by these examples:

RMP
examples:

2/5
= 1/3 + 1/15 or
2/5 - 1/3 = 1/15

2/7
= 1/4 + 1/28 or
2/7 - 1/4 =
1/28

2/11 = 1/6
+ 1/66 or 2/11 - 1/6
= 1/66

2/23 = 1/12 + 1/276 or 2/23 - 1/12 = 1/276

It
appears that the Remainder 1.0 method may have been extended to include four
2/n table conversions by using calculations from the Remainder 2.0 and
Remainder 2.1 calculations:

RMP example forms:

2/p - 1/((p+1)/2) = 1/(p x (p+1)/2) (Remainder 2.0)

2/n - 1/((n+1)/2) = 1/(n x (n+1)/2) (Remainder 2.1)

Remainder
2.0 is offered as the simplest form that explains the appearance of these three RMP series in the EMLR. There is
room, of course, to differ; a simpler method can be discussed, if one should be
pointed out. Whatever one's views, each remainder method can numerically be
extended indefinitely, to include any p and n.

Stated
another way, the last term, usually a least common multiple (LCM), would be a
large number, and it would not generally be optimal, at least in the eyes of a
Middle Kingdom scribe.

Interestingly,
Method Four, of the *19, *20, and *21 series, is intellectually contained
within Method Five, as shown below.

E.
Method Five (algebraic identity)

1/p = 1/A x (A/p) (Algebraic Identity 1.0)

1/n = 1/A x (A/n) (Algebraic Identity 1.1)

where A = 4, 5, 7, or 25, are operated on in
interesting ways, as noted below:

EMLR examples:

*19.
1/3 = 1/4 + 1/12 = 1/4 x 4/3 = 1/4 x (1/1 + 1/3) (A = 4)

22. 1/4 = 1/7 + 1/14 + 1/28 = 1/7 x (1/1 +
1/2 + 1/4) = 1/7 x 7/4 (A = 7)

*20.
1/4 = 1/5 + 1/40 = 1/5 x (5/4) = 1/5 x (1/1 + 1/8) (A = 5)

23. 1/8 = 1/25 + 1/15 + 1/75 + 1/200 = 1/25
x 25/8 = 1/5 x 25/40, (A = 25)

*21 1/8
= 1/10 + 1/40 = 1/5 x (5/8) = 1/5 x (1/2 + 1/8),
(A = 5)

24. 1/16 = 1/30 + 1/50 + 1/150 + 1/400 =
1/25 x 25/8 = 1/5 x 25/40 (A = 25)

25. 1/13 = 1/7 x 7/13 = 1/7 x (1/2 + 1/14),
[contains a student error] (A = 7)

26. 1/15 = 1/25 + 1/50 + 1/150 = 1/25 x (1/1
+1/2 + 1/6) = 1/25 x (10/6) = 1/5 x 1/3 (A = 25)

One
subtle attestation avenue may be associated with the selection of the
partitioning value A. The well known "false position" method of
guessing a trial number to attempt to solve an Egyptian algebra, may have
originated in this type of thinking process. Given that background, the selection
of a value for A, say 25, operated on by
different inner processes was NOT arbitrary. The goal was to partition
1/p and 1/pq into a concise series, amidst a range of alternatives. The EMLR
scribal school certainly introduced several alternatives.

_______________________________

CONNECTIONS
TO THE RMP 2/n TABLE

A. One EMLR-RMP connection is in the use of
Method Three, where 2/101is the last
RMP series. The general equation 1/p = 1/p x (1/2 + 1/3 + 1/6), as noted in
Method Three, was used to construct 2/p series. All that is

needed
to equate the two series is to add 1/1 to both sides:

2/p = 1/p x (1/1 + 1/2 + 1/3 + 1/6)

In the RMP example:

2/101 = 1/101 x (1/1 + 1/2 + 1/3 + 1/6)

One
must ask whether the fact that the last 2/n entry of the RMP can be written in
a form that is close to an earlier EMLR method provides minimal or reasonable
attestation?

B. Another EMLR-RMP connection may be found
in the fact that 3/5 and 3/7 were converted by the EMLR student. Note that 3/5
was written out as a one or two out-of-order series in the EMLR (depending on
which translation is used). However, 3/7 may have been converted to an
incorrect 1/13 conversion, stated as a properly ordered series. Reviewing EMLR
arithmetical facts, consider that:

1/8 = 1/25 + 1/15 + 1/75 + 1/200

= 1/5 x (1/5 + 1/3 + 1/15 + 1/40)

= 1/5 x (3/5 + 1/40) = 1/5 x (25/40)

= 1/5 x (5/8), revealing A = 5

and

1/16
= 1/50 + 1/30 + 1/150 + 1/400

= 1/10 + (1/5 + 1/3 + 15 + 1/40)

= 1/10 x (3/5 + 1/40)

= 1/10 x (25/40)

= 1/2 x (1/5) x (5/8), revealing A =
5.

1/13
= 1/28 + 1/49 + 1/198, student error with 3/49 = 1/7 x 3/7 being used, rather
than a correct identity = 3/39. That
is, was an attempt at 3/7 = 1/4 + 1/7 + 1/28 being made?

Another
way to examine the origins of the student's error is to look at the method used
to find 1/13 by first computing 1/14. Did the student use 1 = (1/2 + 1/3 + 1/6)
to find 1/14 = 1/28 + 1/42 + 1/84, and then ask what number needs to be added
to obtain 1/13? If so, the student
would have known 1/14 - 1/13 =

1/182,
but perhaps then became confused.

C.
One final EMLR-RMP connection, the algebraic identity, is contrasted by:

EMLR: 1/(p x q) = 1/A x (A/(p x q), (Algebraic Identity 1.0)

RMP:
2/(p x q) = 1/A x (A/(p x q).

One
reason this relationship was not previously recognized may be related to the
EMLR partitioning value A= (4, 5, 7, 25), which differed from the RMP constant
A = (p + 1). One asks whether this list of 18 RMP examples

attests
to an historical use of 'A' in both the EMLR and the RMP.

RMP Examples:

1. 2/9
= 1/6 + 1/18 = 1/2 x (1/3 +
1/9) = 2/4 x (4/9), (p = 3, q = 3)

2. 2/15 = 1/10 + 1/30 = 1/2 x (1/5+ 1/15) = 2/4 x (4/15) (p = 3, q = 5)

3. 2/21 = 1/14 + 1/42 = 1/2 x (1/7 + 1/21) = 2/4 x (4/21), (p = 3, q = 7)

4. 2/25 = 1/15 + 1/75 = 1/5 x (1/3 + 1/15) = 1/5 x (2/5), simple
factors

5. 2/27 = 1/18 + 1/54 = 1/9 x (1/2 + 1/6) = 1/9 x (2/3), simple factors

6. 2/33 = 1/22 + 1/66 = 1/2 x (1/11 + 1/33)= 2/4 x (4/33), (p = 3,
q = 11)

7. 2/39 = 1/26 + 1/78 = 1/2 x (1/13 + 1/39)= 2/4 x (4/39), (p = 3, q = 13)

8. 2/45 = 1/30 + 1/90 = 1/2 x (1/15 + 1/45)= 2/4 x (4/45), (p = 3,
q = 15)

9. 2/49 = 1/28 + 1/196 = 1/7 x (1/4 + 1/28)
= 1/7 x (2/7), simple factors

10. 2/51 = 1/34 + 1/102 = 1/2 x (1/17 +
1/51)= 2/4 x (4/51), (p = 3, q = 17)

11. 2/55 = 1/30 + 1/330 = 1/6 x (1/5 + 1/55)
= 2/6 x (6/55), (p = 5, q = 11)

12. 2/57 = 1/38 + 1/114 = 1/2 x (1/19 + 1/57
= 2/4 x (4/57), (p = 3, q = 17)

13. 2/63 = 1/42 + 1/126 = 1/2 x (1/21 + 1/63
= 2/4 x (4/63), (p = 3, q = 21)

14. 2/65 = 1/39 + 1/195 = 1/3 x (1/13 +
1/65)= 2/6 x (6/65), (p = 5, q =
13)

15. 2/69 = 1/46 + 1/138 = 1/2 x (1/23 +
1/69)= 2/4 x (4/69), (p = 3, q = 23)

16. 2/75 = 1/50 + 1/150 = 1/2 x (1/25 +
1/75)= 2/4 x (4/75), (p = 3, q = 25)

17. 2/77 = 1/44 + 1/308 = 1/4 x (1/11 +
1/77)= 2/8 x (8/77), (p = 7, q = 11)

18. 2/81 = 1/54 + 1/162 = 1/9 x (1/6 + 1/18)
= 1/9 x (2/9), simple factors

19. 2/85 = 1/51 + 1/255 = 1/3 x (1/17 +
1/85)= 2/6 x (6/85), (p = 5, q = 17)

20. 2/87 = 1/58 + 1/174 = 1/2 x (1/29 +
1/87)= 2/4 x (4/87), (p = 3, q = 29)

21. 2/93 = 1/62 + 1/186 = 1/2 x (1/31 +
1/93)= 2/4 x (4/93), (p = 3, q = 31)

22. 2/99 = 1/66 + 1/198 = 1/6 x (1/11 +
1/33)= 2/12 x (12/33), (p = 11, q = 3)

Four
of these simple RMP 2/(p x q) conversions were retained to show that the EMLR
and the RMP both factored its rational
number before converting to an Egyptian fraction series. This point is
significant for mathematicians and Egyptologists. At present, knowledgeable
number theorists apply post-Islamic algorithms to unfactored vulgar fractions,
ending up with awkward results (Klee-Wagon). Math historians should first
factor vulgar fractions, during medieval and earlier periods, parsing out the smallest working units, as
the historical texts have long suggested.

_________________________________

OCCAM'S
RAZOR AND ALTERNATIVE EMLR CONVERSION METHODS

The
EMLR student was introduced to identities in the forms of 1/3 = 1/6 +1/6, 1/2 = 1/6 + 1/6 + 1/6, 2/3 = 1/3 + 1/3 and 1/5 = 1/10 + 1/10.
Technically none of these relationships
are Egyptian fraction series. Identical unit

fractions
could not be repeated in a series. So, why was
this class of answers given by the EMLR student? Was it to show that
numbers could be generally parsed into
a series of similar subunits? As a further discussion of plausible Egyptian
fraction methods, an introduction to
odds, evens, composite and primes, and
a little more may have been offered to the EMLR student. Historians have
guessed at this point, as written in
journal articles and posted on the Internet (Brown).

There
is little argument between historians concerning 1/2 = 1/3 + 1/6 and 1 = 1/2 +
1/3 + 1/6 being EMLR Egyptian fraction parsing identities. Concerning 1/2, it
was used ten times, to write even
denominators:

1/6,
1/8, 1/10, 1/12, 1/14, 1/16, 1/20, 1/30, 1/32, and 1/64. Concerning the second,
1 = 1/2 + 1/3 + 1/6, it

was used four times for odd denominators: 1/7,
1/9, 1/11, 1/15. This fact may mean
that odd denominators were converted by this method. But was that the actual
conclusion or technique taught to the student?

Again,
historians often speculate on this point.

EMLR
historical debates sometimes begin with

1/4
= 1/7 + 1/14 + 1/28 = 1/7 x (1/1 + 1/2 + 1/4),

suggesting
a partitioning value A = 7 relationship, stated as:

1/4
= 1/7 x (7/4) = 1/7 x (1/1 + 1/2 + 1/4).

A
related method to convert 1/15 using A = 25, can be shown by:

1/15
= 1/25 x (1/1 + 1/2 + 1/6) = 1/25 x (10/6)
= 1/5 x 1/3.

Returning
to the A = 7 partitioning pattern, it may have been used as an aspect
of the Method Five form to improperly
convert 1/13 to a mod 7 series. Was the
EMLR student asked to write

1/13 = 1/7 x (7/13),

but
did not know how to convert 7/13 to a unit fraction series? One view is that
the student guessed at 3/49 rather than a correct 3/39 conversion, such as
following a near form: 1/13 = 1/3 x
(3/39). There is

no
proof that this was the case. An alternative 1/14th Horus-Eye question may have
been asked, exposing a problem with Old Kingdom conversion methods that
rounded-off rational numbers.

Whatever
the actual question the EMLR student was asked to solve, Gillings' suggestion
that 1/13 = 1/26 + 1/39 + 1/78 was the desired answer, is unappealing.
Historically the fragmentary EMLR seems to be saying something more abstract.

_____________________________

OTHER
ATTESTATION CONSIDERATIONS

It
is important to mention the many citations of n/p and n/pq tables. The various
tables were continuously computed and used over a period of 2,500 years, both
inside and outside of Egypt. The surviving records are

therefore
very numerous. They show that generally the 1/p and 1/pq tables were used as a
foundation (see Appendix II), and these were then used as a general format to
construct any n/p or n/pq unit fraction series table.

Once
the student scribe understood the general method, he/she could then construct
any n/p or n/pq table as needed.

The
EMLR shows that 1/p, 1/pq, and a few limited 2/pq series conversion methods
were studied by a student in a scribal school. However, the student seemed not
to be asked to generally calculate any of the 2/p or higher series, of the type
shown in the RMP. In the EMLR, only one table entry was constructed for 1/p or
1/pq tables.

It
is plausible this was a first course of study. The methods of constructing a
higher fraction series were probably taught as a prerequisite for a more
advanced course. The material shown in the RMP would serve as a typical
curriculum for such a course, where methods for constructing any size table
were taught, such as the n/11 table shown in Appendix II.

In
contrast, post-Islamic methods like Mahavira-Fibonacci have been overlooked by
historians as closely related to Middle Kingdom thinking (Gupta). For example,
Gupta documents in 850 AD, without using n/p and

n/pq
tables, by using only vulgar fractions, Mahavira computed any rational number
p/q by first letting r = (q + x)/p, meaning that p divides (q + x) such that:

p/q
= 1/r + x/pr , and also letting x = 1, 2, ..., as needed.

Mahavira's
method is associated with Fibonacci's 1202 AD work, as documented by best
selling German author (Lueneburg). Lueneburg shows that two basic methods were
known by Leonardi Pisani, one for n/p series

and
one for n/pq series, one of which is near to the Mahavira approach.

In
passing, it should be noted that no modern algorithmic method, be it the
Fibonacci greedy one, or any other one, has been found to compute the 51
concise series in the RMP table. The greedy algorithm, for example,

can
only compute four of the 51 RMP 2/n tables. Yet, algorithmic methods of various
types continue to be associated with ancient methods of making Egyptian
fraction calculations (Eppstein). The application of these newer methods to
ancient Egyptian mathematical materials tends to obscure the beautiful history
of the fragmented subject of ancient Egyptian fractions.

_____________________________

CONCLUSIONS

1. The EMLR explores elementary 1/p and 1/pq
conversion methods, beginning with odd and even rational numbers, ending up
with a new Egyptian fraction numeration system. This new numeration system
ciphered

hieroglyphic
numbers (1:1 basis), phasing out hieroglyphic many-to-one codes, except the
lowest mathematical problems, (Boyer).

2. The Egyptian fraction notation system was
a sign of enlightenment, and NOT of intellectual decline (Neugebauer).

3. The Old Kingdom's duplation method
continued in use long after the introduction of Hieratic script's Egyptian
fractions. The RMP solves 84 problems but does not directly explain the
contents of its 2/n table. The

RMP
appears to write Old Kingdom "quick and dirty proofs" (Robins-Shute
duplation method) to explain 1/p, 1/pq, 2/p, 2/pq and several n/p and n/pq
series. However another paper needs to be written, along the lines that Brown suggests, that reveal
simpler methods, as Medievel scholars also developed.

4. Middle Kingdom tabular methods began to be
phased out after the time of Diophantus (100 AD). Proof is provided by the
existence of an indeterminate method
used in India by Mahavira (850 AD), that looks
very

much
like the method that Fibonacci used (1202 AD) in Liber Abbaci, where vulgar
fractions of any p/q were computed in the concise manner of the EMLR and RMP
2/nth table.

___________________________________

BIBLIOGRAPHY

Boyer,
Carl B. (1968) History of Mathematics, John Wiley, Reprint

Princeton U. Press (1985).

Brown,
Kevin (2001) personal web page,

http://www.mathpages.com/home/rhind.htm.

Eppstein,
David (2001) personal web page,

http://www.ics.uci.edu/~eppstein/numth/egypt/.

Eves,
Howard, (1961) An Introduction to the History of Mathematics, Holt,

Rinehart & Winston.

Gillings,
Richard J, (1972) Mathematics in the Time of the Pharaohs, MIT

Press, Dover reprint available.

Gupta,
RC, (1993), HPM Newsletter, # 29 July 1993, Editor, Victor Katz.,

U. of District of Columbia

Klee,
Victor and Wagon, Stan (1991) Old and New Unsolved Problems in

Plane
Geometry and Number Theory, Mathematical Association of America,

Dolciani Mathematical Expo. #11.

Knorr,
Wilbur Richard (1982) "Fractions in Ancient Egypt and Greece",

Historia Mathematica, HM 9 (a journal
article).

Leuneburg,
Heinz (1993), Leonardi Pisani Liber Abbaci Oder Lesevergneugen

Eines Mathematikers, Wissenschaftsverlag,
pages 81-85.

Neugebauer,
Otto (1962) Exact Science in Antiquity, Harper & Row,

Dover reprint (1969).

McTutor,
O'Connor, J.J, and Roberston, E.F. (2001) St. Andrews University

http://www-groups.dcs.st-andrews.ac.uk~history/HistTopics/

Egyptian_papyri.html.

Ore,
Oystein (1948) Number Theory and its History, McGraw-Hill, Dover

reprint available.

Robins,
Gay and Shute, Charles (1987) The Rhind Mathematical Papyrus,

British Museum Publications Ltd,, Dover
reprint available.

Robson,
Eleanor (August, 2001) "Neither Sherlock Holmes nor Babylon:

A Reassessment of Plimpton 322", in Historia
Mathematica 28:3, pages

167-206.

Sarton,
George (1927) Introduction to the History of Science, Vol. I,

Williams & Williams.

Smith,
David Eugene (1927) History of Mathematics, Vol. 1, Ginn & Co.,

Dover reprint (1958).

Van
der Waerden, B.L. (1961) Science Awakening, Oxford U. Press.

____________________________________

APPENDIX
I

EMLR translation, by Bruce Friedman is found on his personal web site:

http://members.aol.com/brucefriedmandcg/page5.html

by
paging down four-five times. Friedman's summary of the EMLR is read from

right
to left, as the student wrote:

1/12
= 1/36 1/18; 1/8 = 1/40 1/10; 1/15 = 1/90 1/45 1/30

1/14
= 1/42 1/2; 1/4 = 1/20 1/5; 1/16 = 1/48 1/24

1/30
= 1/90 1/45 1/3
= 1/12 1/4; 1/12 = 1/36
1/18

1/10
= 1/30 1/15; 1/5 = 1/10 1/10; 1/14 = 1/42 1/21

1/10
= 1/30 1/15; 1/3 = 1/6 1/6; 1/30 = 1/90 1/45

1/32
= 1/96 1/48 1/2 = 1/6 1/6 1/6; 1/20 = 1/30 1/60

1/64
= 1/192 1/96; 2/3 = 1/3 1/3; 1/10 = 1/30 1/15

1/8
= 1/200 1/75 1/15 1/25

1/16
= 1/400 1/150 1/50 1/30

1/15
= 1/150 1/50 1/25

1/6
= 1/18 1/9

1/4
= 1/28 1/14 1/7

1/8
= 1/24 1/12

1/7
= 1/42 1/21 1/14

1/9
= 1/55 1/27 1/18

1/11 = 1/66 1/33 1/22

1/13 = 1/96 1/49 1/28

1/15 = 1/90 1/45 1/30

Note
there are 26 unique series. The original hieratic text is found at:

http://members.aol.com/brucefriedmandcg/page2.html,
where Bruce Friedman calls the EMLR the 'Big Ugly' because it is so hard to read.

__________________________________________

APPENDIX
II

A
selection from the hieratic, ~ 2000 BCE, Akhmim wood tablet. A conversion of
8/11 hkt (hekat) to an Horus-Eye series [ Reference: http://catnyp.nypl.org,
search on author: Daressy, Georges , look for Ostraca] (Cairo Museo des
Antiquities Egyptiennes. Catalogue General Ostraca hierariques, see 1901 volume
with [item] Numbers: 25001-25385 par M.G. Daressy)

Data: 8/11 = 1/2 + 1/8 + 1/16 + 1/192 +
(1/192 x (2/3 + 1/22 + 1/66)) (The above identity, among many others, is
demonstrated in hieratic, in black ink on a plastered wooden plank, about 10
inches by 18 inches. The

4224
denominator is not shown but appears below for demonstration and clarification.
Note that (2/3 + 1/22 + 1/66) is shown in this identity, which of itself is
exactly 8/11. Unfortunately for the scribe this exactness does not conform to
his hkt divisions. The tablet entry discussed below is known as C.G. 25.368)

<8/11> 3072/4224

<1/2> 2112/4224

<1/8> 528/4224

<1/16> 264/4224

<1/192> 22/4224

subtotal 3058/4224 <versus 3072/4224
actual)

<(1/192) x (2/3 + 1/22 + 1/66) = 1/192 x
(2816+ 1/192 + 64)/4224>

<(1/192) x (8/11) = 16/4224

3048/4224 + 16/4224 = 3074/4224 [versus 3072
actual]

Errors
associated with the hekat round-offs were avoided when scribes used n/p and
n/pq Egyptian fraction tables. It appears that the EMLR student would later
have learned to work from 1/11 and build an n/11

table,
such as:

1/11
= 1/66 + 1/33 + 1/22, from the EMLR

2/11
= 1/6 + 1/66, EMLR Remainder
1.0, or the RMP 2/nth table

3/11
= 1/6 + 1/11 + 1/66, adding 2/11 + 1/11

4/11
= 1/3 + 1/33, adding 2/11 +
2/11

5/11
= 1/3 + 1/11 + 1/33, adding 4/11 + 1/11

6/11
= 1/3 + 1/6 + 1/22, adding 4/11 + 2/11

7/11
= 1/2 + 1/11 + 1/22, adding 6/11 + 1/11

8/11
= 2/3 + 1/22 + 1/66, adding 6/11 + 2/11

9/11
= 2/3 + 1/11 + 1/22 + 1/66, adding 8/11 + 1/11

10/11 = 2/3
+ 1/6 + 1/22 + 1/33, adding 8/11 + 2/11

and
avoided working in 1/4224 units whenever possible. The n/11 table was widely
used over a very long period of time (Knorr, Brown).