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.

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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.

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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.

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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.

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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.

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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,

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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).