Rhind Mathematical Papyrus 1

Rhind Mathematical Papyrus

Rhind Mathematical PapyrusBritish Museum, London

A portion of the Rhind PapyrusDate Second Intermediate Period of Egypt

Place of origin Thebes

Language(s) Egyptian (Hieratic)

Size Length: 536 centimetres (211 in)Width: 32 centimetres (13 in)

Other BM/Big no. AE 10058, Reg no.  1865,0218.3 [1]

The Rhind Mathematical Papyrus (RMP) (also designated as: papyrus British Museum 10057, and pBM 10058), isthe best example of Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, whopurchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near theRamesseum. It dates to around 1650 BC. The British Museum, where the majority of papyrus is now kept, acquiredit in 1864 along with the Egyptian Mathematical Leather Roll, also owned by Henry Rhind;[2] there are a few smallfragments held by the Brooklyn Museum in New York[3][4] and an 18 cm central section is missing. It is one of thetwo well-known Mathematical Papyri along with the Moscow Mathematical Papyrus. The Rhind Papyrus is largerthan the Moscow Mathematical Papyrus, while the latter is older than the former.[3]

The Rhind Mathematical Papyrus dates to the Second Intermediate Period of Egypt. It was copied by the scribeAhmes (i.e., Ahmose; Ahmes is an older transcription favoured by historians of mathematics), from a now-lost textfrom the reign of king Amenemhat III (12th dynasty). Written in the hieratic script, this Egyptian manuscript is33 cm tall and consists of multiple parts which in total make it over 5m long. The papyrus began to be transliteratedand mathematically translated in the late 19th century. In 2008, the mathematical translation aspect remainsincomplete in several respects. The document is dated to Year 33 of the Hyksos king Apophis and also contains aseparate later Year 11 on its verso likely from his successor, Khamudi.[5]

In the opening paragraphs of the papyrus, Ahmes presents the papyrus as giving "Accurate reckoning for inquiringinto things, and the knowledge of all things, mysteries...all secrets". He continues with:

This book was copied in regnal year 33, month 4 of Akhet, under the majesty of the King of Upper andLower Egypt, Awserre, given life, from an ancient copy made in the time of the King of Upper andLower Egypt Nimaatre (?). The scribe Ahmose writes this copy.[2]

Several books and articles about the Rhind Mathematical Papyrus have been published, and a handful of these standout.[3] The Rhind Papyrus was published in 1923 by Peet and contains a discussion of the text that followedGriffith's Book I, II and III outline [6] Chase published a compendium in 1927/29 which included photographs of thetext.[7] A more recent overview of the Rhind Papyrus was published in 1987 by Robins and Shute. [8]

Rhind Mathematical Papyrus 2

Book IThe first part of the Rhind papyrus consists of reference tables and a collection of 20 arithmetic and 20 algebraicproblems. The problems start out with simple fractional expressions, followed by completion (sekhem) problems andmore involved linear equations (aha problems).[3]

The first part of the papyrus is taken up by the 2/n table. The fractions 2/n for odd n ranging from 3 to 101 areexpressed as sums of unit fractions. For example . The decomposition of 2/n into unitfractions is never more than 4 terms long as in for example .This table is followed by a list of fraction expressions for the numbers 1 through 9 divided by 10. For instance thedivision of 7 by 10 is recorded as:

7 divided by 10 yields 2/3 + 1/30After these two tables, the scribe recorded 84 problems altogether and problems 1 through 40 which belong to BookI are of an algebraic nature.Problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unitfractions. Problems 7–20 show how to multiply the expressions 1 + 1/2 + 1/4 and 1 + 2/3 + 1/3 by different fractions.Problems 21–23 are problems in completion, which in modern notation is simply a subtraction problem. Theproblem is solved by the scribe to multiply the entire problem by a least common multiple of the denominators,solving the problem and then turning the values back into fractions. Problems 24–34 are ‘’aha’’ problems. These arelinear equations. Problem 32 for instance corresponds (in modern notation) to solving x + 1/3 x + 1/4 x = 2 for x.Problems 35–38 involve divisions of the hekat. Problems 39 and 40 compute the division of loaves and usearithmetic progressions.[2]

Book II

A portion of the Rhind Papyrus

The second part of the Rhind papyrus consists of geometry problems.Peet referred to these problems as "mensuration problems". [3]

Volumes

Problems 41 – 46 show how to find the volume of both cylindrical andrectangular based granaries. In problem 41 the scribe computes thevolume of a cylindrical granary. Given the diameter (d) and the height(h), the volume V is given by:

In modern mathematical notation (and using d = 2r) this clearly equalsThe quotient 256/81 approximates

the value of π as being ca. 3.1605.In problem 42 the scribe uses a slightly different formula whichcomputes the volume and expresses it in terms of the unit khar.

In modern mathematical notation this isequal to (measured in khar). This is equivalent to measured incubic-cubits as used in the other problem. [2]

Problem 47 gives a table with equivalent fractions for fractions of 100 quadruple hekat of grain. The quotients areexpressed in terms of Horus eye fractions. The short table gives the values related to the original 100 quadruplehekat; the quantity "ro" here is a standard ancient Egyptian measure equivalent to 1/320 of a hekat.-

1/10 gives 10 quadruple hekat

Rhind Mathematical Papyrus 3

1/20 gives 5 quadruple hekat1/30 gives 3 1/4 1/16 1/64 (quadruple) hekat and 1 2/3 ro

1/40 gives 2 1/2 (quadruple) hekat1/50 gives 2 (quadruple) hekat1/60 gives 1 1/2 1/8 1/32 (quadruple) hekat 3 1/3 ro

1/70 gives 1 1/4 1/8 1/32 1/64 (quadruple) hekat 2 1/14 1/21 ro

1/80 gives 1 1/4 (quadruple) hekat1/90 gives 1 1/16 1/32 1/64 (quadruple) hekat 1/2 1/18 ro

1/100 gives 1 (quadruple) hekat[2]

AreasProblems 48 - 55 show how to compute an assortment of areas. Problem 48 is often commented on as it computesthe area of a circle. The scribe compares the area of a circle (approximated by an octagon) and its circumscribingsquare. Each side is trisected and the corner triangles are then removed. The resulting octagonal figure approximatesthe circle. The area of the octagonal figure is: ; Next we approximate 63 to be 64 and

note that . And we get the approximation . Solving for π, we get the approximation

(the approximation has an error of .0189).

That this octagonal figure, whose area is easily calculated, so accurately approximates the area of the circle is justplain good luck. Obtaining a better approximation to the area using finer divisions of a square and a similar argumentis not simple. [2][9] Other problems show how to find the area of rectangles, triangles and trapezoids.

PyramidsThe final five problems are related to the slopes of pyramids. A seked problem is reported by :[]

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 therun-to-rise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the angle to thebase of the pyramid and its face.[]

Book IIIThe third part of the Rhind papyrus consists of a collection of 84 problems. [3] Problem 61 consists of 2 parts. Part 1contains multiplications of fractions. Part b gives a general expression for computing 2/3 of 1/n, where n is odd. Inmodern notation the formula given is

Problems 62- 68 are general problems of an algebraic nature. Problems 69 - 78 are all pefsu problems in some formor another. They involve computations regarding the strength of bread and or beer. [2]

Problem RMP 79' sums five terms in a geometric progression. It is a multiple of 7 riddle, which would have beenwritten in the Medieval era as, "Going to St. Ives" problem.[3] Problems 80 and 81 compute Horus eye fractions ofhenu (or hekats). Problem 81 is followed by a table. The last three problems 82 - 84 compute the amount of feednecessary for fowl and oxen. [2]

Rhind Mathematical Papyrus 4

References[1] http:/ / www. britishmuseum. org/ research/ search_the_collection_database/ search_object_details. aspx?objectid=110036& partid=1[2][2] Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American

Philosophical Society) American Philosophical Society. 1999 ISBN 978-0-87169-232-0[3][3] Anthony Spalinger , The Rhind Mathematical Papyrus as a Historical Document, Studien zur Altägyptischen Kultur, Bd. 17 (1990), pp. 295-,

Helmut Buske Verlag GmbH[5] 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[6] Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool

limited and Hodder & Stoughton limited[7] 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 ofAmerica. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7

[8] Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British MuseumPublications Limited. ISBN 0-7141-0944-4

[9] Don Allen Egyptian and Babylonian Mathematics from (http:/ / www. math. tamu. edu/ ~dallen/ history/ egypt/ egypt. html)

Further reading•• Gillings, Richard J. "Mathematics in the Time of the Pharaohs", 1972, MIT Press, Dover reprint ISBN

0-486-24315-X

External links• Allen, Don. April 2001. The Ahmes Papyrus (http:/ / www. math. tamu. edu/ ~don. allen/ history/ egypt/ node3.

html) and Summary of Egyptian Mathematics (http:/ / www. math. tamu. edu/ ~don. allen/ history/ egypt/ node5.html).

• Egypt/Texts (http:/ / www. dmoz. org/ Society/ History/ By_Time_Period/ Ancient/ Africa/ Egypt/ Texts/ ) at theOpen Directory Project

• British Museum webpage on the Papyrus. (http:/ / www. britishmuseum. org/ explore/ highlights/highlight_objects/ aes/ r/ rhind_mathematical_papyrus. aspx)

• O'Connor and Robertson, 2000. Mathematics in Egyptian Papyri (http:/ / www-history. mcs. st-andrews. ac. uk/history/ HistTopics/ Egyptian_papyri. html).

• Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: TheRhind/Ahmes Papyrus (http:/ / math. truman. edu/ ~thammond/ history/ RhindPapyrus. html).

• "Rhind Papyrus" (http:/ / mathworld. wolfram. com/ RhindPapyrus. html). MathWorld–A Wolfram WebResource.

• Williams, Scott W. Mathematicians of the African Diaspora (http:/ / www. math. buffalo. edu/ mad/ index. html),containing a page on Egyptian Mathematics Papyri (http:/ / www. math. buffalo. edu/ mad/ Ancient-Africa/mad_ancient_egyptpapyrus. html).

• BBC audio file (http:/ / www. bbc. co. uk/ ahistoryoftheworld/ objects/ y1T3knf-T66RwWyEt_cZBw) A Historyof the World in 100 Objects. (15 mins)

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Article Sources and ContributorsRhind Mathematical Papyrus  Source: http://en.wikipedia.org/w/index.php?oldid=550513138  Contributors: 790, APH, Akriasas, Albert1ls, Amicon, Angela, AnnekeBart, Arteum, Auntof6,Bad germ, Bobo192, Broodoobob, Bubuka, CBM, CRGreathouse, Captmondo, Charles Matthews, Chasuble, Chlewey, ChrisG, Cimon Avaro, Claritas, Cmapm, CompositeFan, CryptoDerk,Dave Light, David Eppstein, Dominus, DopefishJustin, Dungodung, EamonnPKeane, Epbr123, Everyking, FeanorStar7, Flying Jazz, Fæ, Ghirlandajo, Giftlite, Gnomon, GrayFullbuster,Hempfel, Hibernian, Hipocrite, Hobe, Homonihilis, InverseHypercube, Jagged 85, Jdforrester, Jheald, Jitse Niesen, John of Reading, JustUser, Kanags, Kelvin Case, Klompje7, KoshVorlon,Leoboudv, Ling.Nut, Lisamh, LittleOldMe, Liujiang, Lupo, M a s, Maashatra11, Man vyi, Migaber, Mild Bill Hiccup, Milogardner, Morgan Leigh, Narayan, NawlinWiki, Nefertum17,Nwbeeson, Oleg Alexandrov, Ottershrew, Paul August, PaulHammond, Pepo13, PericlesofAthens, Philip Trueman, RDBury, Richard Arthur Norton (1958- ), Rktect, Rogper, Roylee, Ruud Koot,Salix alba, Sam Hocevar, Skysmith, Smither, Spanglej, Stepa, Stephen MUFC, Stumps, Sverdrup, Sławomir Biały, Tagus, Thanatosimii, That Guy, From That Show!, Thue, Tm14, Tradnor,Troglodyto, Twthmoses, Ulric1313, Versus22, Vmenkov, Waggers, William Avery, Xdamr, Zentuk, 102 ,עמית רפל anonymous edits

Image Sources, Licenses and ContributorsFile:Rhind Mathematical Papyrus.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Rhind_Mathematical_Papyrus.jpg  License: Public Domain  Contributors: Paul James Cowie(Pjamescowie)File:Egyptian A'h-mosè or Rhind Papyrus (1065x1330).png  Source: http://en.wikipedia.org/w/index.php?title=File:Egyptian_A'h-mosè_or_Rhind_Papyrus_(1065x1330).png  License: PublicDomain  Contributors: Anarkman, G.dallorto, GeorgHH, JMCC1, Luestling, Mdd, Otso Huuska, 4 anonymous edits

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