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The '''Lahun Mathematical Papyri''' (also known as the '''Kahun Mathematical Papyri''') are part of a collection of [[Kahun Papyri]] discovered at [[El-Lahun]] (also known as Lahun, Kahun or Il-Lahun) by [[Flinders Petrie]] during excavations of a worker's town near the pyramid of [[Sesostris II]]. The [[Kahun Papyrus]] are a collection of texts including administrative texts, medical texts, veterinarian texts and six fragments devoted to mathematics.<ref>[http://www.digitalegypt.ucl.ac.uk/lahun/papyri.html The Lahun Papyri] at University College London</ref> |
The '''Lahun Mathematical Papyri''' (also known as the '''Kahun Mathematical Papyri''') are part of a collection of [[Kahun Papyri]] discovered at [[El-Lahun]] (also known as Lahun, Kahun or Il-Lahun) by [[Flinders Petrie]] during excavations of a worker's town near the pyramid of [[Sesostris II]]. The [[Kahun Papyrus]] are a collection of texts including administrative texts, medical texts, veterinarian texts and six fragments devoted to mathematics.<ref>[http://www.digitalegypt.ucl.ac.uk/lahun/papyri.html The Lahun Papyri] at University College London</ref> |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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. |
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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: |
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1. number of terms: 10 |
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2. arithmetical progression difference: 1/8 |
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3. arithmetic progression sum: 10 |
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The scribe used the following facts to find the largest term. |
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1. one-half of differences, 1/16, times number of terms minus one, 9, |
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1/16 times 9 = 9/16 |
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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. |
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That is, the KP scribe used formula 1.0: |
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(1/2)d(n-1) + S/n = Xn (formula 1.0) |
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with, |
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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. |
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When n was odd, x (n/2) = S/n, |
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and x 1 + xn = x2 + x(n -1) = x3 + x(n -2) = ... = x(n/2) = S/n, |
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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 = ...) . |
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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: |
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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, |
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A=(pi)r2 |
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, and input pi = 256/81 and D = 2, considering: |
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a. A = (256/81)(D/2)(D/2) = (64/81)(D)(D) solved for D |
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b. Sqrt A = (8/9)D (algebraic formula 1.0) |
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This formula was used in MMP 10, RMP 41 and RMP 42. In RMP 42 Ahmes adding height (H) and created two volume formulas. |
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c. V = h(8/9)(D)(8/9)D cubits squared (algebraic geometry formula 1.1) |
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d. V = (3/2)(8/9)D(8/9)D khar (algebraic geometry formula 1.2) |
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and in the Kahun Papyrus and RMP 43, algebraic geometry formula 1.0 was scaled by 3/2 considering |
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e.(3/2)Sqrt (A) =(3/2[(8/9)(D)(D)] = (4/3)(D)(D) |
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f. V = (2/3)(H)[(4/3)(D)(4/3)(D)] (algebraic geometry formula 1.3) |
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Input D = 12, H = 8, in the Kahun Papyrus |
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g. (4/3)12 meant to the scribe (16 x)(16) = 256, such that |
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h. V = (2/3)8(256)=(1365 + 1/3)khar |
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exactly as RMP 43 input D = 8 and H = 6 |
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i. V = (2/3)(6)[(4/3)(8)(4/3)(8)] = (4)(32/3)(32/3) = 4096/9 = (455 + 1/9) khar |
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2. Level 2 reported the value of 12 fowls in terms of a set-duck unit paid in the following problem by: |
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a. 3 re-geese unit value 8 set-ducks = 24 |
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b. 3 terp-geese unit value 4 set-ducks = 12 |
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c. 3 Dj. Cranes unit value 2 set-ducks = 6 |
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d. 3 set-duck unit value 1 set-duck = 3 |
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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. |
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E. The RMP includes a more complex Egyptian bird feeding problem: |
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that links MK bird valuations in hekats in a manner that confirms Egyptian fraction arithmetic was primarily focused upon economic issues. |
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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. |
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4. Level four consider Greek arithmetic when Plato spoke of mathematics by: |
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"How do you mean? |
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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. |
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That is very true. |
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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? " |
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from Chapter 7. "The Republic" (Jowell translation)." |
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and the work of Archimedes. |
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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. |
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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. |
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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. |
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15. 925157 + 1/3 |
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16. 708453 + 1/3 |
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17. 709533 + 1/3 |
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18. 508098 + 2/3 + 1/8 + 1/16 |
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19. 407042 + 2/3 |
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20. 440003 + 1/6 |
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21 209200 |
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22. 1/12 |
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As a wild guess, a prime number analysis may offer a few hints to decoding aspects of the ancient data: |
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15. 2775460 divided by 3, factors (2, 2, 5, 73, 1901) divided by 3 |
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16. 21283600 divided by 3, factors 2, 2, 2 ,5 ,13, 4093) divided by 3 |
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17. 2128600 divided by 3, factors (2, 2, 2, 5, 5, 29, 367) divided by 3 |
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18. 508098 + 41/48 = 243887050 divided by 48, or |
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121943525 divided by 24, factors (5, 5, 11, 443431) divided by 24 |
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19. 1221128 divided by 3, factors (2, 2, 2, 152641) divided by 3 |
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20. 2640019 divided by 6, factors (61, 113, 383) divided by 6 |
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21. factors (2, 2, 2, 2, 5, 5, 523) |
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22. 209200 times 12 = 25700900 |
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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. |
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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. |
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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. |
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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. |
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Research continues with respect to the large number problem, and other meta mathematics issues cited in the text. |
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Bibliography |
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1 |
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Richard Gillings, "Mathematics in the Time of the Pharaohs", pages 176-180, MIT Press, Cambridge, 1972 |
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2 |
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John Legon, "A Kahun Papyrus Fragment", pages 21-24, Discussions in Egyptology 24, 1992. |
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3 |
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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. |
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4 |
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Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", pages 41-43, British Museum Press, Dover Reprint, 1987. |
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External links |
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The mathematical texts most commented on are usually named: |
The mathematical texts most commented on are usually named: |
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==The 2/''n'' tables== |
==The 2/''n'' tables== |
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The Lahun papyrus IV.2 |
The Lahun papyrus IV.2 reports a 2/''n'' table for odd ''n'', ''n'' = 1, , 21. The [[Rhind Mathematical Papyrus]] reports an odd ''n'' tabke up to 101. <ref>Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp. 19–27</ref> These fraction tables were related to multiplication problems and the use of [[Egyptian fraction|unit fractions]], namely scaling n/p by LCM m to nm/mp. With the exception of 2/3, all fractions were represented as sums of unit fractions (i.e. of the form 1/n), first in red numbers. Multiplication algorithms and scaling factors involved repeated doubling of numbers, and other operations. Doubling a unit fraction with an even denominator was simple, divided the denominator by 2. Doubling a fraction with an odd denominator however results in a fraction of the form 2/n. The [[RMP 2/n table]] allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve unscalable rational numbers (ie. 28/97 in RMP 31,and 30/53 n RMP 36 by substituting 26/97 + 2/97 and 28/53 + 2/53) and generally n/p by (n - 2)/p + 2/p. Decompositions were unique, and [[red auxiliary numbers]] selected divisors of denominators mp that best summed to numerator mn. |
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==References== |
==References== |
Revision as of 21:41, 1 September 2010
The Lahun Mathematical Papyri (also known as the Kahun Mathematical Papyri) are part of a collection of Kahun Papyri discovered at El-Lahun (also known as Lahun, Kahun or Il-Lahun) by Flinders Petrie during excavations of a worker's town near the pyramid of Sesostris II. The Kahun Papyrus are a collection of texts including administrative texts, medical texts, veterinarian texts and six fragments devoted to mathematics.[1]
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
The mathematical texts most commented on are usually named:
- Lahun IV.2 (or Kahun IV.2) (UC 32159): This fragment contains a segment of a 2/n table. A more complete version of this table of fractions is given in the Rhind Mathematical Papyrus.[2]
- Lahun IV.3 (or Kahun IV.3) (UC 32160) contains numbers in arithmetical progression and a problem very much like problem 40 of the Rhind Mathematical Papyrus. Another problem on this fragment computes the volume of a cylindrical granary and is of a type also found in the Rhind Mathematical Papyrus (problems 41-43).[2][3]
- Lahun XLV.1 (or Kahun XLV.1) (UC 32161) contains a group of very large numbers (hundreds of thousands).[2][4]
- Lahun LV.3 (or Kahun LV.3) (UC 32134A and UC 32134B) contains a so called aha problem which asks one to solve for a certain quantity. The problem resembles ones from the Rhind Mathematical Papyrus (problems 24-29).[2]>[5]
- Lahun LV.4 (or Kahun LV.4) (UC 32162) contains what seems to be an area computation and a problem concerning amounts of ducks, geese and cranes.[2][6]
The 2/n tables
The Lahun papyrus IV.2 reports a 2/n table for odd n, n = 1, , 21. The Rhind Mathematical Papyrus reports an odd n tabke up to 101. [8] These fraction tables were related to multiplication problems and the use of unit fractions, namely scaling n/p by LCM m to nm/mp. With the exception of 2/3, all fractions were represented as sums of unit fractions (i.e. of the form 1/n), first in red numbers. Multiplication algorithms and scaling factors involved repeated doubling of numbers, and other operations. Doubling a unit fraction with an even denominator was simple, divided the denominator by 2. Doubling a fraction with an odd denominator however results in a fraction of the form 2/n. The RMP 2/n table allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve unscalable rational numbers (ie. 28/97 in RMP 31,and 30/53 n RMP 36 by substituting 26/97 + 2/97 and 28/53 + 2/53) and generally n/p by (n - 2)/p + 2/p. Decompositions were unique, and red auxiliary numbers selected divisors of denominators mp that best summed to numerator mn.
References
- ^ The Lahun Papyri at University College London
- ^ a b c d e Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0871692320; Annette Irmhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religous, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1841715727, 92-93
- ^ Annette Irmhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religous, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1841715727, 84-85
- ^ Annette Irmhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religous, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1841715727, 94-95
- ^ Annette Irmhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religous, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1841715727, 74-77
- ^ Annette Irmhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religous, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1841715727, 78-79
- ^ Annette Irmhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religous, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1841715727, 90-91
- ^ Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp. 19–27