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# definition - Rhind_Mathematical_Papyrus

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# Rhind Mathematical Papyrus

(Redirected from Rhind mathematical papyrus)
A portion of the Rhind Papyrus

The Rhind Mathematical Papyrus (RMP) (also designated as: papyrus British Museum 10057, and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 B.C. The British Museum, where the papyrus is now kept, acquired it in 1864 along with the Egyptian Mathematical Leather Roll, also owned by Henry Rhind; there are a few small fragments held by the Brooklyn Museum in New York. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. [1]

The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and is the best example of Egyptian mathematics. It was copied by the scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics), from a now-lost text from the reign of king Amenemhat III (12th dynasty). Written in the hieratic script, this Egyptian manuscript is 33 cm tall and over 5 meters long, and began to be transliterated and mathematically translated in the late 19th century. In 2008, the mathematical translation aspect is incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later Year 11 on its verso likely from his successor, Khamudi.[2]

In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving “Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets”.

## Mathematical problems

A portion of the Rhind Papyrus

The papyrus, written on both sides, began with a RMP 2/n table, followed by 87 problems. The RMP 2/n table took up one third of the manuscript. The table converted 2 divided by the odd numbers from 3 to 101 by sums of Egyptian fractions using a Egyptian multiplication and division method that 19th and 20th century scholars found hard-to-read.

Alternative scribal 2/n table methods were proposed by 19th and 20th century scholars. A German Egyptologist F. Hultsch (1895) proposed an aliquot part aspect of the $\frac{2}{p}$ table. Hultsch'saliquot part fragments were independently confirmed by E.M. Bruins in 1944. The hard-to-read $\frac{2}{pq}$ conversion method apparently consisted of selecting optimized, but not always optimal least common multiples (LCMs). Ahmes practiced selecting non-optimal LCMs in the Egyptian Mathematical Leather Roll to convert 1/p and 1/pq rational numbers to unit fraction series. An optimized LCM method was used by Ahmes was confirmed in 2005 that converted $\frac{2}{95}$ into $\frac{1}{5} \times \frac{2}{19}$, with $\frac{2}{19}$. Ahmes' LCM method converted 2/95 by selecting the LCM 12, written as 12/12, such that:

$\frac{2}{95}*\frac{12}{12} = \frac{24}{1140} = \frac{19 + 3 + 2}{1140} = \frac{1}{60} + \frac{1}{380} + \frac{1}{570}$

Ahmes' hard-to-read shorthand notes included additive numerators in RMP 36 that were written in red. Ahmes omitted initial and intermediate steps, however, Ahmes' method is clear based on his shorthand notes containing informational links to other RMP problems. Today, Ahmes' 2/n table calculation omissions that had confused math historians began to clear up in 2001 by citing the EMLR's 26 lines of data. The EMLR paper set decoding ground work for a 2005 Akhmim Wooden Tablet paper. The final piece of the 2/n table problem puzzle was confirmed in the 2009 RMP 36 paper.

21st century scholars are increasingly reporting the RMP's 87 problems that began with six division-by-10 problems to be amplified by the contents of the Reisner Papyrus. In the RMP there were 15 problems that dealt with addition, and 19 algebra problems. There were 15 algebra problems, RMP 18-23 and RMP 24- 34, of the same type, followed by RMP 35- 38 written in a hekat context. The 19 algebra problems asked Ahmes to find x and a fraction of x such that the sum of x and its fraction equaled an integer. Problem #24 is the easiest that asked Ahmes to solve the equation, x + 1/7x = 19. Ahmes worked the problem this way:

(8/7)x = 19, or x = 133/8 = 16 + 5/8,

with 133/8 being the initial vulgar fraction find 16 as the quotient and 5/8 as the remainder term. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 1/8, making his final quotient plus remainder based answer x = 16 + 1/2 + 1/8.

The algebra problems, from RMP 21 -34, produced increasingly difficult vulgar fractions. RMP 38 converted a hekat, written 320 ro, by multiplying by 35/110, 7/22, obtaining 101 9/11. The initial 320 ro was obtained by multiplying 101 9/11 by 22/7. RMP 82 partitioned a hekat written as (64/64). Hekat unity problems limited n to 1/64 < n < 64, obtaining quotient (Q) and remainder (R) two-part numbers: Q/64 + (5R/n)ro. Ro answers were converted to a one-part 1/10 hekat hin unit by writing 10/n (29 times). Vulgar fractions were easily converted to an optimal (short and small last term) Egyptian fraction series in all RMP problems.

Two arithmetical progressions (A.P.) were solved, one being RMP 64. The method of solution followed the method defined in the Kahun Papyrus. The problem solved sharing 10 hekats of barley, between 10 men, by a difference of 1/8th of a hekat finding 1 7/16 as the largest term.

The second A.P. was RMP 40, the problem divided 100 loaves of bread between five men such that the smallest two shares (12 1/2) were 1/7 of the largest three shares' sum (87 1/2). The problem asked Ahmes to find the shares for each man, which he did without finding the difference (9 1/6) or the largest term (38 1/3). All five shares 38 1/3, 29 1/6, 20, 10 2/3 1/6, and 1 1/3) were calculated by first finding the five terms from a proportional A.P. that summed to 60. The median and the smallest term, x1, were used to find the differential and each term. Ahmes then multiplied each term by 1 2/3 to obtain the sum to 100 A.P. terms. In reproducing the problem in modern algebra, Ahmes also found the sum of the first two terms by solving x + 7x = 60.

The RMP continues with 5 hekat division problems from the Akhmim Wooden Tablet, 15 problems similar to ones from the Moscow Mathematical Papyrus, 23 problems from practical weights and measures, especially the hekat, and three problems from recreational diversion subjects, the last the famous multiple of 7 riddle, written in the Medieval era as, "Going to St. Ives".

The Rhind Mathematical Papyrus also contains the following problem related to trigonometry:[3]

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[3]

## Mathematical knowledge

Upon closer inspection, modern-day mathematical analyses of Ahmes' problem-solving strategies reveal a basic awareness of composite and prime numbers;[4] arithmetic, geometric and harmonic means;[4] a simplistic understanding of the Sieve of Eratosthenes[4], and perfect numbers.[4][5]

The papyrus also demonstrates knowledge of solving first order linear equations[5] and summing arithmetic and geometric series.[5]

The papyrus calculates π as $(8/9)^2*4 \simeq 3.1605$ (a margin of error of less than 1%). In addition 255/81 was considered (3.1481481...) and 22/7. In RMP 38, Ahmes multiplied a hekat, 320 ro, by 7/22 obtaining 101 9/11. The divisor 7/22 was inverted to 22/7 and multiplied by 101 9/11 obtaining 320 ro as a proof. Ahmes' use of 22/7 may have corrected the hekat's built-in loss based on using 256/81 as pi.

Other problems in the Rhind papyrus demonstrate knowledge of arithmetic progressions (Kahun Papyrus), algebra and geometry.

The papyrus demonstrates knowledge of weights and measures, business distributions of money (paid out in arithmetic progressions, with one group proportionally being paid more than another), and several recreational diversions.

## Influence of the RMP

The Egyptian use of arithmetic proportions in the Rhind Papyrus, problems 40 and 64, and the Kahun Papyrus, are briefly discussed by Gillings. In particular the use of the Remen, which has two values, is reflected in the foot which has two values, (the second being the nibw or ell which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling seems to have been the basis of most of the unit fraction calculations, which it was not (multiples were) up to and including the calculations of circles with dimensions given in khet (see Ancient Egyptian units of measurement), looking at how the remen and seked were used provided many insights to Greek and Roman geometers and architects. The actual and proposed readings/decodings of the RMP and Kahun 2/n tables is required to be fairly interjected.

In the Rhind Papyrus we first encounter the remen which is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. Yet, a singular arithmetic proportion formula reported in the RMP and Kahun Papyrus offer an additional example beyond the remen's diagonal of a square, with its sides a cubit. We also find problems using the seked or unit rise to run proportion. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.

This document is one of the main sources of our knowledge of Egyptian mathematics.

## References

1. ^ Great Soviet Encyclopedia, 3rd edition, entry on "Папирусы математические", available online here
2. ^ cf. Thomas Schneider's paper 'The Relative Chronology of the Middle Kingdom and the Hyksos Period (Dyns. 12-17)' in Erik Hornung, Rolf Krauss & David Warburton (editors), Ancient Egyptian Chronology (Handbook of Oriental Studies), Brill: 2006, p.194-195
3. ^ a b Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 0691095418.
4. ^ a b c d [1] MathPages - Egyptian Unit Fractions.
5. ^ a b c [2] Scott W. Williams, The Mathematics Department of The State University of New York at Buffalo.
• Allen, Don. April 2001. The Ahmes Papyrus and Summary of Egyptian Mathematics.
• Borbola J. Kiralykörök /the Hungarian reading and solving of the Rhind-papyrus/
• Borbola J. Olvassunk együtt magyarul /Hungarian reading and solving of the Moskow Mathematic Papyrus/
• "Rhind Papyrus". MathWorld–A Wolfram Web Resource.
• Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical

Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7

• Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
• Friberg, Joran, 2005, "Unexpected Links between Egyptian and Babylonian Mathematics", World Scientific Books
• Gillings, Richard J. "Mathematics in the Time of the Pharaohs", 1972, MIT Press, Dover reprint ISBN 0-486-24315-x
• Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003.
• Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
• Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: The Rhind/Ahmes Papyrus.

# Rhind Mathematical Papyrus

A portion of the Rhind Papyrus

The Rhind Mathematical Papyrus (RMP) (also designated as: papyrus British Museum 10057, and pBM 10058), is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 B.C. The British Museum, where the papyrus is now kept, acquired it in 1864 along with the Egyptian Mathematical Leather Roll, also owned by Henry Rhind; there are a few small fragments held by the Brooklyn Museum in New York. It is one of the two well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is larger than the Moscow Mathematical Papyrus, while the latter is older than the former. [1]

The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt and is the best example of Egyptian mathematics. It was copied by the scribe Ahmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics), from a now-lost text from the reign of king Amenemhat III (12th dynasty). Written in the hieratic script, this Egyptian manuscript is 33 cm tall and over 5 meters long, and began to be transliterated and mathematically translated in the late 19th century. In 2008, the mathematical translation aspect is incomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains a separate later Year 11 on its verso likely from his successor, Khamudi.[2]

In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving “Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries...all secrets”.

## Mathematical problems

A portion of the Rhind Papyrus

The papyrus, written on both sides, began with a RMP 2/n table, followed by 87 problems. The RMP 2/n table took up one third of the manuscript. The table converted 2 divided by the odd numbers from 3 to 101 by sums of Egyptian fractions using a Egyptian multiplication and division method that 19th and 20th century scholars found hard-to-read.

Alternative scribal 2/n table methods were proposed by 19th and 20th century scholars. A German Egyptologist F. Hultsch (1895) proposed an aliquot part aspect of the $\frac{2}{p}$ table. Hultsch'saliquot part fragments were independently confirmed by E.M. Bruins in 1944. The hard-to-read $\frac{2}{pq}$ conversion method apparently consisted of selecting optimized, but not always optimal least common multiples (LCMs). Ahmes practiced selecting non-optimal LCMs in the Egyptian Mathematical Leather Roll to convert 1/p and 1/pq rational numbers to unit fraction series. An optimized LCM method was used by Ahmes was confirmed in 2005 that converted $\frac{2}{95}$ into $\frac{1}{5} \times \frac{2}{19}$, with $\frac{2}{19}$. Ahmes' LCM method converted 2/95 by selecting the LCM 12, written as 12/12, such that:

$\frac{2}{95}*\frac{12}{12} = \frac{24}{1140} = \frac{19 + 3 + 2}{1140} = \frac{1}{60} + \frac{1}{380} + \frac{1}{570}$

Ahmes' hard-to-read shorthand notes included additive numerators in RMP 36 that were written in red. Ahmes omitted initial and intermediate steps, however, Ahmes' method is clear based on his shorthand notes containing informational links to other RMP problems. Today, Ahmes' 2/n table calculation omissions that had confused math historians began to clear up in 2001 by citing the EMLR's 26 lines of data. The EMLR paper set decoding ground work for a 2005 Akhmim Wooden Tablet paper. The final piece of the 2/n table problem puzzle was confirmed in the 2009 RMP 36 paper.

21st century scholars are increasingly reporting the RMP's 87 problems that began with six division-by-10 problems to be amplified by the contents of the Reisner Papyrus. In the RMP there were 15 problems that dealt with addition, and 19 algebra problems. There were 15 algebra problems, RMP 18-23 and RMP 24- 34, of the same type, followed by RMP 35- 38 written in a hekat context. The 19 algebra problems asked Ahmes to find x and a fraction of x such that the sum of x and its fraction equaled an integer. Problem #24 is the easiest that asked Ahmes to solve the equation, x + 1/7x = 19. Ahmes worked the problem this way:

(8/7)x = 19, or x = 133/8 = 16 + 5/8,

with 133/8 being the initial vulgar fraction find 16 as the quotient and 5/8 as the remainder term. Ahmes converted 5/8 to an Egyptian fraction series by (4 + 1)/8 = 1/2 + 1/8, making his final quotient plus remainder based answer x = 16 + 1/2 + 1/8.

The algebra problems, from RMP 21 -34, produced increasingly difficult vulgar fractions. RMP 38 converted a hekat, written 320 ro, by multiplying by 35/110, 7/22, obtaining 101 9/11. The initial 320 ro was obtained by multiplying 101 9/11 by 22/7. RMP 82 partitioned a hekat written as (64/64). Hekat unity problems limited n to 1/64 < n < 64, obtaining quotient (Q) and remainder (R) two-part numbers: Q/64 + (5R/n)ro. Ro answers were converted to a one-part 1/10 hekat hin unit by writing 10/n (29 times). Vulgar fractions were easily converted to an optimal (short and small last term) Egyptian fraction series in all RMP problems.

Two arithmetical progressions (A.P.) were solved, one being RMP 64. The method of solution followed the method defined in the Kahun Papyrus. The problem solved sharing 10 hekats of barley, between 10 men, by a difference of 1/8th of a hekat finding 1 7/16 as the largest term.

The second A.P. was RMP 40, the problem divided 100 loaves of bread between five men such that the smallest two shares (12 1/2) were 1/7 of the largest three shares' sum (87 1/2). The problem asked Ahmes to find the shares for each man, which he did without finding the difference (9 1/6) or the largest term (38 1/3). All five shares 38 1/3, 29 1/6, 20, 10 2/3 1/6, and 1 1/3) were calculated by first finding the five terms from a proportional A.P. that summed to 60. The median and the smallest term, x1, were used to find the differential and each term. Ahmes then multiplied each term by 1 2/3 to obtain the sum to 100 A.P. terms. In reproducing the problem in modern algebra, Ahmes also found the sum of the first two terms by solving x + 7x = 60.

The RMP continues with 5 hekat division problems from the Akhmim Wooden Tablet, 15 problems similar to ones from the Moscow Mathematical Papyrus, 23 problems from practical weights and measures, especially the hekat, and three problems from recreational diversion subjects, the last the famous multiple of 7 riddle, written in the Medieval era as, "Going to St. Ives".

The Rhind Mathematical Papyrus also contains the following problem related to trigonometry:[3]

"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"

The solution to the problem is given as the ratio of half the side of the base of the pyramid to its height, or the run-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to the base of the pyramid and its face.[3]

## Mathematical knowledge

Upon closer inspection, modern-day mathematical analyses of Ahmes' problem-solving strategies reveal a basic awareness of composite and prime numbers;[4] arithmetic, geometric and harmonic means;[4] a simplistic understanding of the Sieve of Eratosthenes[4], and perfect numbers.[4][5]

The papyrus also demonstrates knowledge of solving first order linear equations[5] and summing arithmetic and geometric series.[5]

The papyrus calculates π as $(8/9)^2*4 \simeq 3.1605$ (a margin of error of less than 1%). In addition 255/81 was considered (3.1481481...) and 22/7. In RMP 38, Ahmes multiplied a hekat, 320 ro, by 7/22 obtaining 101 9/11. The divisor 7/22 was inverted to 22/7 and multiplied by 101 9/11 obtaining 320 ro as a proof. Ahmes' use of 22/7 may have corrected the hekat's built-in loss based on using 256/81 as pi.

Other problems in the Rhind papyrus demonstrate knowledge of arithmetic progressions (Kahun Papyrus), algebra and geometry.

The papyrus demonstrates knowledge of weights and measures, business distributions of money (paid out in arithmetic progressions, with one group proportionally being paid more than another), and several recreational diversions.

## Influence of the RMP

The Egyptian use of arithmetic proportions in the Rhind Papyrus, problems 40 and 64, and the Kahun Papyrus, are briefly discussed by Gillings. In particular the use of the Remen, which has two values, is reflected in the foot which has two values, (the second being the nibw or ell which is two feet), and the cubit which has two values. Doubling is also seen in the subdivisions such as fingers and palms. Since doubling seems to have been the basis of most of the unit fraction calculations, which it was not (multiples were) up to and including the calculations of circles with dimensions given in khet (see Ancient Egyptian units of measurement), looking at how the remen and seked were used provided many insights to Greek and Roman geometers and architects. The actual and proposed readings/decodings of the RMP and Kahun 2/n tables is required to be fairly interjected.

In the Rhind Papyrus we first encounter the remen which is defined as the proportion of the diagonal of a rectangle to its sides when its other sides are whole units. Yet, a singular arithmetic proportion formula reported in the RMP and Kahun Papyrus offer an additional example beyond the remen's diagonal of a square, with its sides a cubit. We also find problems using the seked or unit rise to run proportion. Typical of the Classical orders of the Greeks and Romans, it was built upon the canon of proportions derived from the inscription grids of the Egyptians.

This document is one of the main sources of our knowledge of Egyptian mathematics.

## References

1. ^ Great Soviet Encyclopedia, 3rd edition, entry on "Папирусы математические", available online here
2. ^ cf. Thomas Schneider's paper 'The Relative Chronology of the Middle Kingdom and the Hyksos Period (Dyns. 12-17)' in Erik Hornung, Rolf Krauss & David Warburton (editors), Ancient Egyptian Chronology (Handbook of Oriental Studies), Brill: 2006, p.194-195
3. ^ a b Maor, Eli (1998). Trigonometric Delights. Princeton University Press. p. 20. ISBN 0691095418.
4. ^ a b c d [1] MathPages - Egyptian Unit Fractions.
5. ^ a b c [2] Scott W. Williams, The Mathematics Department of The State University of New York at Buffalo.
• Allen, Don. April 2001. The Ahmes Papyrus and Summary of Egyptian Mathematics.
• Borbola J. Kiralykörök /the Hungarian reading and solving of the Rhind-papyrus/
• Borbola J. Olvassunk együtt magyarul /Hungarian reading and solving of the Moskow Mathematic Papyrus/
• "Rhind Papyrus". MathWorld–A Wolfram Web Resource.
• Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical

Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7

• Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
• Friberg, Joran, 2005, "Unexpected Links between Egyptian and Babylonian Mathematics", World Scientific Books
• Gillings, Richard J. "Mathematics in the Time of the Pharaohs", 1972, MIT Press, Dover reprint ISBN 0-486-24315-x
• Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003.
• Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
• Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
• Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: The Rhind/Ahmes Papyrus.

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