Talk:Lahun Mathematical Papyri

Untitled

The following text was added to the page. It needs to be seriously cleaned up for as far as the writing goes. The writing style is very confusing. And some of the material duplicates the text already on the page.

What may have been attempted is a summary of the mathematical contents of the papyri. That is probably worthwhile to pursue, but the writing needs to be cleaned up. Undefined terms need to be explained, excessive numbers of examples should be avoided. Examples should just be used to clarify a point. Some issues:

• The introduction is already on the page.
• Part A is already treated and should not be included again.
• Arithmetic progressions need to be defined (in a brief concise manner)
• Part B needs to be rewritten. Does not make sense.
• The formatting of part C makes is almost impossible to follow
• D refers to the geometric problems I think. That should be written in a much more concise manner. The large number of formulas and their formatting again make it almost impossible to follow.
• E through G should be summarized and made much more concise.
• H should probably be omitted
• Inline references need to be added for verifiability. A list of articles/books in the middle of the text is inappropriate. --AnnekeBart (talk) 01:08, 2 September 2010 (UTC)

Proposed text

The Kahun Papyrus(KP) datedsto 1825 BCE Egypt. The fragmented text was discovered by Flinders Petrie in 1889. Its fragments are kept at the University College London. Most of the fragments date to the reign of Amenemhat III. One of the fragments, referred to as the Kahun Gynaecological Papyrus, dealt with gynecological illnesses and conditions.

A. A second fragment of the KP began with a traditional 2/n table, a Middle Kingdom scribe's method of defining a rational number as an exact unit fraction series. The KP 2/n table was abbreviated version, converting 2/3 to 2/21 (with attached proofs). The Rhind Mathematical Papyrus (RMP) 2/n table converted 51 rational numbers, 2/3 to 2/101. Considering the KP arithmetic topics, arithmetic progressions was likely the highest form of Egyptian arithmetic. The KP scribe defined a 10-term arithmetic progression summed to 100, with a difference (d) of 5/6. The KP arithmetic progression was discussed in two RMP problems.

A generalized form of arithmetic progressions has been found in the RMP. Ahmes listed two columns of data (published by Gillings in 1972). Ahmes's thinking is shown in Gillings' column 11 by multiplying 5/12 times 9, a fact that was needed to find the largest term of the RMP progression. Ahmes then added 10 and wrote out the correct largest term of the arithmetic progression, and subtracted 5/6, nine times. Gillings found the remaining terms of the progressions by using the KP's method. To understand the KP method, readers must make arithmetic calculations as the Middle Kingdom scribes wrote down in their three problems, double and triple checking your work with several tools.

Gillings' 1972 analysis of both RMP versions of Middle Kingdom arithmetic progression failed to parse the method in a manner that was comparable, in every respect, to the KP method. For example, Gillings noticed similar problems in the RMP (RMP 40, 64) yet Gillings muddled three pages of his analysis, reaching no definitive conclusions on the topic.

B. In 1987, Egyptologist Gay Robins, and Charles Shute, wrote on the Rhind Mathematical Papyrus(RMP). Five years later, Egyptologist John Legon wrote on the KP, and closely related arithmetic proportions in the RMP. The KP and RMP report scribal uses the same method to find the largest term of closely related arithmetic progressions. The method: take 1/2 of the difference, 1/2 of 5/6 (5/12 in the KP) times the number of differences (nine times 5/12 = 15/4 in the KP) plus the sum of the A.P progression (100 in the KP) divided by the number of terms (10 , meaning 100/10 = 10 in the KP). Finally add column 11's result, 3 3/4, to 10, and the largest term, 13 3/4.

To repeat, add column 11, 5/12 times 9, or 45/12, or 3 3/4, 3 2/3 1/12 in Egyptian fractions to 10 in column 12 beginning with the largest term 13 2/3 1/12. The scribe subtracted 5/6 nine times, creating the remaining terms of the progression.

Robins-Shute confused an aspect of the problem by omitting the sum divided by the number of terms parameter in the RMP. An algebraic statement could have been created by Robins-Shute from matched pairs that added to 20, five pairs summing to 100, as potentially related to RMP 40.

The KP method found the largest term, and used other facts that have been reported in RMP 64, and RMP 40, by John Legon in 1992. Scholars, at other times, have attempted to parse Rhind Mathematical Papyrus 40, a problem that asks 100 loaves of bread to be shared between five men by finding the smallest term of an arithmetic progression.

C. A confirmation of the Kahun Paprus method is reported in RMP 64, and RMP 40. In RMP 64 Ahmes asked 10 men to share 10 hekats of barley, with a differential of 1/8, by using an arithmetical progression? Robins and Shute reported, "the scribe knew the rule that, to find the largest term of the arithmetical progression, he must add half the difference to the average number of terms as many times as there are common differences, that is, one less than the number of terms" (note that Robins-Shutre omitted the sum divided by the number of terms), as noted by:

1. number of terms: 10

2. arithmetical progression difference: 1/8

3. arithmetic progression sum: 10

The scribe used the following facts to find the largest term.

1. one-half of differences, 1/16, times number of terms minus one, 9,

1/16 times 9 = 9/16

2. The computed parameter(1), was found by 10, the sum, divided by 10, the number of terms. It was inserted by Robins-Shute, but had not been high-lighted, citing 1 + 1/2 + 1/16, or 1 9/16, the largest term. The remaining nine terms were found by subtracting 1/8 nine times to obtain the remaining barley shares.

That is, the KP scribe used formula 1.0:

(1/2)d(n-1) + S/n = Xn (formula 1.0)

with,

d = differential, n = number of terms in the series, S = sum of the series, Xn = largest term in the series allowed three(of the four) parameters: d, n, S and Xn, to algebraically find the fourth parameter.

When n was odd, x (n/2) = S/n,

and x 1 + xn = x2 + x(n -1) = x3 + x(n -2) = ... = x(n/2) = S/n,

a paired data set that Carl Friedrich Gauss implemented as a grammar school student solved the n = even case. Ahmes and Gauss found the sum of 1 to 100, using d = 1, by following the same rule. Both reached 5050 based on 50 pairs of 101 (1 + 101 = 2 + 99 = 3 + 98 = ...) .

D. A four level review of a Kahun Papyrus problem that reported 1365 1/3 khar as the volume of cylinder with a diameter of 12 cubit and height of 8 cubits summarized by:

1. Level 1 shows that pi was set to 256/81, and knowing one khar equaled 3/2 of a hekat, the scribe computed 1365 1/3 khar by beginning with the area of a circle, A=(pi)r2 , and input pi = 256/81 and D = 2, considering:

a. A = (256/81)(D/2)(D/2) = (64/81)(D)(D) solved for D

b. Sqrt A = (8/9)D (algebraic formula 1.0)

This formula was used in MMP 10, RMP 41 and RMP 42. In RMP 42 Ahmes adding height (H) and created two volume formulas.

c. V = h(8/9)(D)(8/9)D cubits squared (algebraic geometry formula 1.1)

d. V = (3/2)(8/9)D(8/9)D khar (algebraic geometry formula 1.2)

and in the Kahun Papyrus and RMP 43, algebraic geometry formula 1.0 was scaled by 3/2 considering

e.(3/2)Sqrt (A) =(3/2[(8/9)(D)(D)] = (4/3)(D)(D)

f. V = (2/3)(H)[(4/3)(D)(4/3)(D)] (algebraic geometry formula 1.3)

Input D = 12, H = 8, in the Kahun Papyrus

g. (4/3)12 meant to the scribe (16 x)(16) = 256, such that

h. V = (2/3)8(256)=(1365 + 1/3)khar

exactly as RMP 43 input D = 8 and H = 6

i. V = (2/3)(6)[(4/3)(8)(4/3)(8)] = (4)(32/3)(32/3) = 4096/9 = (455 + 1/9) khar

2. Level 2 reported the value of 12 fowls in terms of a set-duck unit paid in the following problem by:

a. 3 re-geese unit value 8 set-ducks = 24

b. 3 terp-geese unit value 4 set-ducks = 12

c. 3 Dj. Cranes unit value 2 set-ducks = 6

d. 3 set-duck unit value 1 set-duck = 3

total value 45 set-ducks. Not calculated, but included in the valuation of 12 - 1 = 11 with 100 - 45 = 55, cited 55/11 as the total value as 5 times the value of one set-duck.

E. The RMP includes a more complex Egyptian bird feeding problem:

that links MK bird valuations in hekats in a manner that confirms Egyptian fraction arithmetic was primarily focused upon economic issues.

3. Level 3 showed that Ahmes in RMP 38 may have corrected the KP scribe's over-estimated the grain volume in the cylinder. Ahmes could have down-sized the volume of a 12 cubit diameter and 8 cubit high cylinder to 1356 3/14 khar by approximating pi to 22/7, improving the volume estimate by 9 5/42 khar.

4. Level four consider Greek arithmetic when Plato spoke of mathematics by:

"How do you mean?

I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.

That is very true.

Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, -what would they answer? "

from Chapter 7. "The Republic" (Jowell translation)."

and the work of Archimedes.

that includes Archimedes showing that pi was an irrational number limited to well-defined rational number limits smaller than Egyptians recorded by 256/81 and 22/7 approximations.

F. The Kahun Papyrus calculation of the 1365 1/3 khar volume would likely have been corrected by Ahmes with pi set at 22/7, 175 years later, and improved by Archimedes including calculus, with Egyptian and Greek businessmen freed from the abstract number of Egyptians.

G. The Kahun Papyrus contains other numerical information. One data set, eight lines of large quotient and remainder rational numbers was preceded by 14 lines of missing data. The fragmented data may relate to a calculation ending with 1/12. The historical context of the data, cited below is unclear.

15. 925157 + 1/3

16. 708453 + 1/3

17. 709533 + 1/3

18. 508098 + 2/3 + 1/8 + 1/16

19. 407042 + 2/3

20. 440003 + 1/6

21 209200

22. 1/12

As a wild guess, a prime number analysis may offer a few hints to decoding aspects of the ancient data:

15. 2775460 divided by 3, factors (2, 2, 5, 73, 1901) divided by 3

16. 21283600 divided by 3, factors 2, 2, 2 ,5 ,13, 4093) divided by 3

17. 2128600 divided by 3, factors (2, 2, 2, 5, 5, 29, 367) divided by 3

18. 508098 + 41/48 = 243887050 divided by 48, or

121943525 divided by 24, factors (5, 5, 11, 443431) divided by 24

19. 1221128 divided by 3, factors (2, 2, 2, 152641) divided by 3

20. 2640019 divided by 6, factors (61, 113, 383) divided by 6

21. factors (2, 2, 2, 2, 5, 5, 523)

22. 209200 times 12 = 25700900

The data was converted to rational numbers and prime factors to consider astronomical cycles as a possible ancient context. More on this data when, or ever, reliable data becomes available.

H. In summary, the Kahun Papyrus (KP) was notable for a 2/n table, an arithmetic progression problem, the calculation of the volume of a cylinder, a valuation of four classes of birds by the lowest valued bird, an economic building block that was passed down to the Greeks, and a large number problem that discussed an upper numerical range of ancient scribal calculations.

Ancient Egyptian fractions represented positive rational numbers 2/n scaled to 2m/mn unit fraction series by an LCM m/m. Rational numbers 2/n were optimized, but not optimal, as well as a large number problem. The unit fraction sums were typically 5-terms 1/a + 1/b + 1/c + 1/d + 1/e, or less, scaled in a manner that confused 19th and 20th century scholars. The Egyptian fraction Middle Kingdom notation was continuously used for about 3,600 years. Scholars in the 21st century AD have parsed the 4,000 year old notation that fell into disuse after 1454 AD. With the dominance of the algorithmic base 10 decimal arithmetic, approved by the Paris Academy in 1585 AD, and updated by Napier and others, replaced the Egyptian fraction notation.

The KP and RMP scribes used identical methods to calculate the largest term in arithmetic progressions, and identical methods to find one of four unknown variables. Formula 1.0 defines the four variables (d, n, S and xn). Note that formula 1.0 did not rely on rational number differences (d) being converted to Egyptian fraction series. Agreement on the larger questions, i.e., what were the beginning, and intermediate arithmetic steps of the arithmetic progression formula, have been resolved. Other common calculations were used in the KP and the RMP. The KP scribe created Egyptian fractions as final statements based on Egyptian fraction arithmetic, wrote beginnings, and intermediate vulgar fractions in identical ways.

Research continues with respect to the large number problem, and other meta mathematics issues cited in the text. Bibliography

1

   Richard Gillings, "Mathematics in the Time of the Pharaohs", pages 176-180, MIT Press, Cambridge, 1972

2

   John Legon, "A Kahun Papyrus Fragment", pages 21-24, Discussions in Egyptology 24, 1992.

3

   Luca Miatello, "The difference 5 1/2 in a problem of rations from the Rhind mathematical papyrus", Historia Mathematica, vol 34, issue 4, pages 277-284, Nov. 2008.

4

   Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", pages 41-43, British Museum Press, Dover Reprint, 1987.

External links

a naive question

Lahun IV.3 appears to approximate π by a value between 4 and 5 (closer to 5). Am I missing something? Tkuvho (talk) 04:42, 19 September 2010 (UTC)

No, you are right. I had only looked at the arithmetic which was wrong. In my haste I did not look at the overall formulas, which are incorrect as well.--AnnekeBart (talk) 11:54, 19 September 2010 (UTC)

RMP 41, 42, 43, MMP 10, and the Kahun Papyrus set pi = 256/81, an easy to use number, filled with highly composite numbers, applied to four formulas.

1. MMP 10 and RMP 41 wrote A = (256/81)(D/2)^2 = 64/81(D)^2 = [(8/9)D]^2 (corrected by Clagett, 1999)

2. RMP 41 wrote V = (H)[(8/9)(D)]^2 CC and V = 3/2[(8/9)(D)]^2 khar. The khar unit was divided by 20 to calculate a 100-hekat unit and multiplied by 100 to obtain hekat units. Scholars continue to struggle with 'quadruple hekat' issues left by Ahmes (corrected by Clagett, 1999).

3. RMP 42 repeated RMP 41 formulas, showing additional scribal techniques for converting khar into 100- hekat and hekat units. Despite this information scholars continue to struggle with 'quadruple hekat' fragmented left by Ahmes (corrected by Clagett, 1999)

4. RMP 43 and the Kahun Papyrus wrote V = (2/3)(H)[(4/3)(D)]^2 khar and converted to 100-hekat 22 (1/2 + 1/32 + 1/64 + 1/180) 100-hekat, corrrected by Clagett in 1999. Line 5 was not completed by any scholar. Easy to read raw data was reported by Clagett citing 2275 heat. The 5/9 hekat remainder was converted to binary hekat and ro remainders by a hard to read scribal method that Hana Vymazalova in 2002 clarified by defining a hekat unity as (64/64). I'll leave a major portion line 5 conversion of 5/9 hekat to readers that wish to consider Ahmes' actual arithmetic. Clagett reported Schack-Schackenberg's conversion of 5/9 by three bits of information

/1 144 /2 288 /4 576

data that reports the first term (1/2 hekat) of (288 + 18 + 9 + 5)/576 = (1/2 + 1/32 + 1/64)hekat + (5/9)ro

leaving adequate work for anyone that wises to calculate a binary quotient and ro remaidner as Ahmes wrote in line 5.

Best Regards, Milo Gardner Milogardner (talk) 13:01, 24 September 2010 (UTC)

The formula used in Lahun IV.3 is the same as the one used in RPM 42. It expresses the volume directly in khar. It is equivalent to the formula found in RPM 41 and so gives the same approximation for π (namely 3.16...). I tried to give just enough explanation in the article to give the interesting tidbits, but also aimed to keep it short and easy to read. Hope this part works for everyone. --AnnekeBart (talk) 21:41, 25 September 2010 (UTC)

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