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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 mathematical texts most commented on are usually named:
* '''Lahun IV.2''' (or '''Kahun IV.2''') ([http://www.digitalegypt.ucl.ac.uk/lahun/uc32159.html UC 32159]): This fragment contains a [[mathematical table|table]] of [[Egyptian fraction]] representations of numbers of the form 2/''n''. A more complete version of this table of fractions is given in the [[Rhind Mathematical Papyrus]].<ref name="Clagett">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; Annette Imhausen, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, ISBN 1-84171-572-7, 92-93</ref>
* '''Lahun IV.3''' (or '''Kahun IV.3''') ([http://www.digitalegypt.ucl.ac.uk/lahun/uc32160.html UC 32160]) contains numbers in [[arithmetical progression]] and a problem very much like problem 40 of the Rhind Mathematical Papyrus.<ref name="Clagett"/><ref>Annette Imhausen, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, ISBN 1-84171-572-7, 84-85</ref><ref>Legon, J., A Kahun mathematical fragment, retrieved from [http://www.legon.demon.co.uk/kahun.htm], based on Discussions in Egyptology 24 (1992), p.21-24</ref> Another problem on this fragment computes the volume of a cylindrical granary.<ref>Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", British Museum Press, Dover Reprint, 1987.</ref> In this problem the scribe uses a formula which takes measurements in ''cubits'' and computes the volume and expresses it in terms of the unit ''khar''. Given the diameter (d) and height (h) of the cylindrical granary:
:<math> V = ((1+1/3)d)^2 \ ((2/3) h)</math>.
: In modern mathematical notation this is equal to
:<math> V = \frac{32}{27} d^2\ h = \frac{128}{27} r^2\ h </math> (measured in khar).
: This problem resembles problem 42 of the [[Rhind Mathematical Papyrus]]. The formula is equivalent to <math> V =  \frac{256}{81}  r^2\ h </math> measured in cubic-cubits as used in the other problems.<ref>Katz, Victor J. (editor),Imhausen, Annette et al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007 ISBN 978-0-691-11485-9</ref>
* '''Lahun XLV.1''' (or '''Kahun XLV.1''')  ([http://www.digitalegypt.ucl.ac.uk/lahun/uc32161.html UC 32161]) contains a group of very large numbers (hundreds of thousands).<ref name="Clagett"/><ref>Annette Imhausen, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, ISBN 1-84171-572-7, 94-95</ref>
 
* '''Lahun LV.3''' (or '''Kahun LV.3''') ([http://www.digitalegypt.ucl.ac.uk/lahun/uc32134a.html UC 32134A] and [http://www.digitalegypt.ucl.ac.uk/lahun/uc32134b.html 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).<ref name="Clagett"/><ref>Annette Imhausen, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, ISBN 1-84171-572-7, 74-77</ref>
 
* '''Lahun LV.4''' (or '''Kahun LV.4''') ([http://www.digitalegypt.ucl.ac.uk/lahun/uc32162.html UC 32162]) contains what seems to be an area computation and a problem concerning the value  of ducks, geese and cranes.<ref name="Clagett"/><ref>Annette Imhausen, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, ISBN 1-84171-572-7, 78-79</ref> The problem concerning fowl is a [[Moscow Mathematical Papyrus#Baku problems|baku problem]] and most closely resembles problem 69 in the [[Rhind Mathematical Papyrus]] and problems 11 and 21 in the [[Moscow Mathematical Papyrus]].<ref>[http://www.digitalegypt.ucl.ac.uk/lahun/uc32162.html UC 32162 Lahun LV.4]</ref>
 
* Unnamed fragment ([http://www.digitalegypt.ucl.ac.uk/lahun/uc32118b.html UC 32118B]). This is a fragmentary piece.<ref>Annette Imhausen, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, ISBN 1-84171-572-7, 90-91</ref>
 
==The 2/''n'' tables==
The Lahun papyrus IV.2 reports a 2/''n'' table for odd ''n'', ''n''&nbsp;=&nbsp;1,&nbsp;,&nbsp;21. The [[Rhind Mathematical Papyrus]] reports an odd ''n'' table up to 101.<ref>Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp.&nbsp;19–27</ref> These fraction tables were related to multiplication problems and the use of [[Egyptian fraction|unit fractions]], namely n/p scaled by LCM m to mn/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]] and RMP 36 rules allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve otherwise un-scalable rational numbers (i.e. 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. [[Red auxiliary numbers]] selected divisors of denominators mp that best summed to numerator mn.
 
==References==
{{reflist|30em}}
 
==External links==
*[http://www.legon.demon.co.uk/kahun.htm John Legon: A Kahun Mathematical Fragment]
*http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
*[http://mathforum.org/kb/thread.jspa?threadID=1543145&tstart=0 Math-History-List]
*[http://www.mathorigins.com/image%20grid/BRUCE%20OLD_010.htm MathOrigins.com]
*[http://webdigg.net/History/History-of-Egyptian-fractions/ History of Egyptian fractions]
*[http://www.digitalegypt.ucl.ac.uk/med/birthpapyrus.html Medical Papyrus, UCL website]
*[http://www.reshafim.org.il/ad/egypt/timelines/topics/kahunpapyrus.htm The Kahun Gynaecological Papyrus]
*{{planetmath reference|id=11008|title=Kahun papyrus}}
*{{planetmath reference|id=10080|title=Egyptian fraction}}
 
{{use dmy dates|date=February 2012}}
[[Category:Egyptian mathematics]]
[[Category:Egyptian fractions]]
[[Category:Egyptian papyri]]
[[Category:Ancient Egyptian literature]]
[[Category:Papyrus]]
[[Category:Mathematics manuscripts]]
[[Category:Medical literature]]
 
[[de:Medizinische Papyri aus Lahun]]
[[es:Papiro Kahun]]
[[fr:Papyrus Kahun]]
[[sv:Kahun-papyrusen]]

Revision as of 15:08, 21 March 2013

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 mathematical texts most commented on are usually named:

  • Lahun IV.2 (or Kahun IV.2) (UC 32159): This fragment contains a table of Egyptian fraction representations of numbers of the form 2/n. 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.[2][3][4] Another problem on this fragment computes the volume of a cylindrical granary.[5] In this problem the scribe uses a formula which takes measurements in cubits and computes the volume and expresses it in terms of the unit khar. Given the diameter (d) and height (h) of the cylindrical granary:
.
In modern mathematical notation this is equal to
(measured in khar).
This problem resembles problem 42 of the Rhind Mathematical Papyrus. The formula is equivalent to measured in cubic-cubits as used in the other problems.[6]
  • Lahun XLV.1 (or Kahun XLV.1) (UC 32161) contains a group of very large numbers (hundreds of thousands).[2][7]
  • 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][8]

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 table up to 101.[12] These fraction tables were related to multiplication problems and the use of unit fractions, namely n/p scaled by LCM m to mn/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 and RMP 36 rules allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve otherwise un-scalable rational numbers (i.e. 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. Red auxiliary numbers selected divisors of denominators mp that best summed to numerator mn.

References

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de:Medizinische Papyri aus Lahun es:Papiro Kahun fr:Papyrus Kahun sv:Kahun-papyrusen

  1. ↑ The Lahun Papyri at University College London
  2. ↑ 2.0 2.1 2.2 2.3 2.4 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; Annette Imhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1-84171-572-7, 92-93
  3. ↑ Annette Imhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1-84171-572-7, 84-85
  4. ↑ Legon, J., A Kahun mathematical fragment, retrieved from [1], based on Discussions in Egyptology 24 (1992), p.21-24
  5. ↑ Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", British Museum Press, Dover Reprint, 1987.
  6. ↑ Katz, Victor J. (editor),Imhausen, Annette et al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007 ISBN 978-0-691-11485-9
  7. ↑ Annette Imhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1-84171-572-7, 94-95
  8. ↑ Annette Imhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1-84171-572-7, 74-77
  9. ↑ Annette Imhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1-84171-572-7, 78-79
  10. ↑ UC 32162 Lahun LV.4
  11. ↑ Annette Imhausen, Jim Ritter: Mathematical Fragments, In: Marc Collier, Stephen Quirke: The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical, Oxford 2004, ISBN 1-84171-572-7, 90-91
  12. ↑ Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp. 19–27
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