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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 looked like scribal weights and measures, algebra, and/or 2/n tables calculations. |
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 looked like scribal weights and measures, algebra, and/or 2/n tables calculations. |
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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), or mh, was one cubit wide by 100 cubits long, or 1/100 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), or mh, was one cubit wide by 100 cubits long, or 1/100 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*(9/4)*(1/2)= (45/8) = 5 5/8 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*(9/4)*(1/2)= (45/8) = 5 5/8 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*(9/4)*(1/2) = 63/8 = 7 7/8 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*(9/4)*(1/2) = 63/8 = 7 7/8 setat |
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| The third calculation found the area of undefined shape discussed by: |
The third calculation found the area of undefined shape discussed by: |
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| 11/8 mh = 110/8 mh + 10 mh = 23 3/4 mh = 1/8 setat + 11 1/4 mh |
11/8 mh = 110/8 mh + 10 mh = 23 3/4 mh = 1/8 setat + 11 1/4 mh |
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| since 12 1/2 mh = 1/8 setat. |
since 12 1/2 mh = 1/8 setat. |
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| Scholars have suggested that a truncated pyramid or a triangle defined the third shape. |
Scholars have suggested that a truncated pyramid or a triangle defined the third shape. |
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To assist the decoding of the third RMP 53 area RMP 54, and RMP 55 setat and mh data have been consulted.
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To assist the decoding of the third RMP 53 area, RMP 54, and RMP 55 details have been consulted.
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RMP 54 partitioned 7/10 setat by 10, 5, 2 1/2 and 1 1/4 segments. Proof was provided by multiplying one setat by 7/10, 14/10, 28/10 and 56/10 within a quotient and remainder context. A quotient setat and a scaled remainder mh were scaled as the 2/n table and a ro unit in hekat (volume unit) were scaled, by writing:
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RMP 54 segmented 7/10 setat by 10, 5, 2 1/2 and 1 1/4 segments. Proof was provided by finding setat + COL areas of 7/10, 14/10, 28/10 and 56/10 setats segments within a quotient and remainder, that looked like a 2/n table context:
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| a. (7/10)*(4/4) = 28/40 = (24 + 3)/40 = 3/8 setat + 300/40 mh = 5/8 setat + 7 1/2 mh |
a. (7/10)*(4/4) = 28/40 = (24 + 3)/40 = 3/8 setat + 300/40 mh = 5/8 setat + 7 1/2 mh |
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| b. (14/10)*(4/4) = 56/10 = (55 + 1)/40 = 11/8 setat + 100/4 mh = 1 3/8 setat + 2 1/2 mh |
b. (14/10)*(4/4) = 56/10 = (55 + 1)/40 = 11/8 setat + 100/4 mh = 1 3/8 setat + 2 1/2 mh |
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| c. (28/10)*(2/2) = 56/20 = (55 + 1)/20 = 11/4 setat + 100/20 mh = 2 3/4 setat + 5 mh |
c. (28/10)*(2/2) = 56/20 = (55 + 1)/20 = 11/4 setat + 100/20 mh = 2 3/4 setat + 5 mh |
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| d. (56/10) = (55 + 1)/10 = 11/2 setat + 100/10 COL = 5 1/2 setat + 10 mh |
d. (56/10) = (55 + 1)/10 = 11/2 setat + 100/10 COL = 5 1/2 setat + 10 mh |
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Ahmes may have also made calculations thinking in mh unuts. For example,
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Ahmes may have made calculations in mh unuts, or by following the 2/n table scaling method.
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| Ahmes shorthand partition of 7/10 setat, (1/2 + 1/5) setat, may have focused upon 1/5 setat written as 20 mh. Knowing 12 1/2 mh was 1/8 setat, an answer may have been recorded by: |
For example: |
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| (1/2 + 1/5)setat = (1/2 + 1/8 + (20 - 12 1/2 mh) = 5/8 setat + 7 1/2 mh. |
7/10 setat, written as (1/2 + 1/5) setat, may have written 1/5 setat as 20 mh. Knowning that 12 1/2 mh = 1/8 setat |
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(1/2 + 1/5)setat = (1/2 + 1/8) = 5/8 setat + (20 - 12 1/2 mh) = 7 1/2 mh = 5/8 setat + 7 1/2 mh. |
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| RMP 55 takes 3/5 of 5 setat to obtain 3 setat by three steps: |
RMP 55 takes 3/5 of 5 setat to obtain 3 setat by three steps: |
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| a. 1/2 setat + 10 mh |
a. 1/2 setat + 10 mh |
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| b. 1 1/8 setat + 7 1/2 mh |
b. 1 1/8 setat + 7 1/2 mh |
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| c. 1 3/8 setat + 2 1/2 setat |
c. 1 3/8 setat + 2 1/2 setat |
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| d. adding steps a. and c, knowing that 12 1/2 mh = 1/8 setat |
d. adding steps a. and c, knowing that 12 1/2 mh = 1/8 setat |
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| (1/2 setat + 10 mb) + (1 3/8 setat + 2 1/2 mh) = 2 7/8 setat + 12 1/2 mh = 3 setat |
(1/2 setat + 10 mb) + (1 3/8 setat + 2 1/2 mh) = 2 7/8 setat + 12 1/2 mh = 3 setat |
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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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