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| A scribal area of triangles and other shapes area calculation method is reported by three \PMlinkexternal{Rhind Mathematical Papyrus}{http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html} problems \PMlinkexternal{RMP 53-55}{http://rmp50-60.blogspot.com/}. The scribal geometry utilized quotients and remainders in an arithmetic context that was also applied in scribal weights and measures, algebra, and 2/n tables. |
A scribal area of triangles and other shapes area calculation method is reported by three \PMlinkexternal{Rhind Mathematical Papyrus}{http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html} problems \PMlinkexternal{RMP 53-55}{http://rmp50-60.blogspot.com/}. The scribal geometry utilized quotients and remainders in an arithmetic context that was also applied in scribal weights and measures, algebra, and 2/n tables. |
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| RMP 53 calculated the area of two triangles, of 45/8 setat and 63/8 setat, and a third area of an undefined shape by the note, 1/10 of 1 3/8 mh added to 10 cubits of land (COL) is the desired area. A setat was 100 cubit by 100 cubit, or 10,000 square cubits. A cubit of land (COL) was one cubit wide by 100 cubits long, or 1/100 setat. A mh unit may have been 10 setat (or less likely 1/10 setat). |
RMP 53 calculated the area of two triangles, of 45/8 setat and 63/8 setat, and a third area of an undefined shape by the note, 1/10 of 1 3/8 mh added to 10 cubits of land (COL) is the desired area. A setat was 100 cubit by 100 cubit, or 10,000 square cubits. A cubit of land (COL) was one cubit wide by 100 cubits long, or 1/100 setat. A mh unit may have been 10 setat (or less likely 1/10 setat). |
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| The first triangle had an altitude of 5 khet and a base of 9/4 khet. Using the area of a triangle formula, 1/2 the base times the altitude, 5 x 9/4 x 1/2= 45/8 setat. |
The first triangle had an altitude of 5 khet and a base of 9/4 khet. Using the area of a triangle formula, 1/2 the base times the altitude, 5 x 9/4 x 1/2= 45/8 setat. |
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| The second triangle had an altitude of 7 khet and a base of 9/4 khet. Using the area of a triangle formula, 1/2 the base times the altitude, Ahmes calculated 7 x 9/4 x 1/2 = 63/8 setat |
The second triangle had an altitude of 7 khet and a base of 9/4 khet. Using the area of a triangle formula, 1/2 the base times the altitude, Ahmes calculated 7 x 9/4 x 1/2 = 63/8 setat |
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| The third calculation found the area of undefined shape was discussed by: |
The third calculation found the area of undefined shape was discussed by: |
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| 11/8 mh = 110/8 setat + 10 COL = (110/8 + 10/100)setat = (1100/80 + 10/80) setat = 111/8 setat = 13 7/8 setat, |
11/8 mh = 110/8 setat + 10 COL = (110/8 + 10/100)setat = (1100/80 + 10/80) setat = 111/8 setat = 13 7/8 setat, |
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| or, less likely, |
or, less likely, |
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11/8 mh = 110/8 COL + 10 COL = (1100 + 800)/80 COL = 190/8 = 23 3/4 COL
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11/8 mh = 110/8 COL + 10 COL = (1100 + 80)/80 COL = 118/8 = 14 3/4 COL
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| As a triangle, as implied by Ahmes' annotated diagram, the third shape had a base 6 khet, an altitude of 37/8 khet and an area of 111/8 setat. The area calculation would have been 6 x 37/8 x 1/2 = 111/8 setat. |
As a triangle, as implied by Ahmes' annotated diagram, the third shape had a base 6 khet, an altitude of 37/8 khet and an area of 111/8 setat. The area calculation would have been 6 x 37/8 x 1/2 = 111/8 setat. |
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| Others have suggested that a trazazoid may define the area. |
To assist the proper decoding of the third RMP 53 area, RMP 54 and RMP 55 will be consulted. |
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| To assist the decoding of the third RMP 53 area, RMP 54, and RMP 55 details will be consulted. |
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| RMP 54 segmented 7/10 setat by 10, 5, 2 1/2 and 1 1/4 fields. Proof was provided by finding setat + COL areas of 7/10, 14/10, 28/10 and 56/10 setats fields within quotient and remainder data: |
RMP 54 segmented 7/10 setat by 10, 5, 2 1/2 and 1 1/4 fields. Proof was provided by finding setat + COL areas of 7/10, 14/10, 28/10 and 56/10 setats fields within quotient and remainder data: |
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| a. 7/10*(4/4) = 28/40 = (24 + 3)/40 = 3/8 setat + 300/40 COL = 5/8 setat + 7 1/2 COL |
a. 7/10*(4/4) = 28/40 = (24 + 3)/40 = 3/8 setat + 300/40 COL = 5/8 setat + 7 1/2 COL |
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| b. 14/10*(4/4) = 56/10 = (55 + 1)/40 = 11/8 setat + 100/4 COL = 1 3/8 setat + 2 1/2 COL |
b. 14/10*(4/4) = 56/10 = (55 + 1)/40 = 11/8 setat + 100/4 COL = 1 3/8 setat + 2 1/2 COL |
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| c. 28/10*(2/2) = 56/20 = (55 + 1)/20 = 11/4 setat + 100/20 COL = 2 3/4 setat + 5 COL |
c. 28/10*(2/2) = 56/20 = (55 + 1)/20 = 11/4 setat + 100/20 COL = 2 3/4 setat + 5 COL |
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| d. 56/10 = (55 + 1)/10 = 11/2 setat + 100/10 COL = 5 1/2 setat + 10 COL |
d. 56/10 = (55 + 1)/10 = 11/2 setat + 100/10 COL = 5 1/2 setat + 10 COL |
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| RMP 55 is a work-in-progress. A translation is nearing completion. |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{1} Milo Gardner, \emph{The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences}, Hindustan Book Company, 2002. |
\bibitem{1} Milo Gardner, \emph{The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences}, Hindustan Book Company, 2002. |
| \bibitem{2} Milo Gardner, \emph{An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati}, MD Publications Pvt Ltd, 2006. |
\bibitem{2} Milo Gardner, \emph{An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati}, MD Publications Pvt Ltd, 2006. |
| \bibitem{3}Richard Gillings, \emph{Mathematics in the Time of the Pharaohs}, Dover Books, 1992. |
\bibitem{3}Richard Gillings, \emph{Mathematics in the Time of the Pharaohs}, Dover Books, 1992. |
| \bibitem{4} Oystein Ore, \emph{Number Theory and its History}, McGraw-Hill Books, 1948, Dover reprints available. |
\bibitem{4} Oystein Ore, \emph{Number Theory and its History}, McGraw-Hill Books, 1948, Dover reprints available. |
| \bibitem{5} T.E. Peet, \emph{Arithmetic in the Middle Kingdom}, Journal Egyptian Archeology, 1923. |
\bibitem{5} T.E. Peet, \emph{Arithmetic in the Middle Kingdom}, Journal Egyptian Archeology, 1923. |
| \bibitem{6} Tanja Pommerening, \emph{"Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass}, Buske-Verlag, 2005. |
\bibitem{6} Tanja Pommerening, \emph{"Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass}, Buske-Verlag, 2005. |
| \bibitem{7} Gay Robins, and Charles Shute \emph{Rhind Mathematical Papyrus}, British Museum Press, Dover reprint, 1987. |
\bibitem{7} Gay Robins, and Charles Shute \emph{Rhind Mathematical Papyrus}, British Museum Press, Dover reprint, 1987. |
| \bibitem{8} L.E. Sigler, \emph{Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation}, Springer, 2002. |
\bibitem{8} L.E. Sigler, \emph{Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation}, Springer, 2002. |
| \bibitem{9} Hana Vymazalova, \emph{The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai}, Charles U Prague, 2002. |
\bibitem{9} Hana Vymazalova, \emph{The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai}, Charles U Prague, 2002. |
| \end{thebibliography} |
\end{thebibliography} |
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