CubitsEgyptianGeometryAreasCalculatedIn 2009-05-25 16:04:35 2009-11-10 19:05:53 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Scribal areas of triangles and areas of other shapes were reported in 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 followed scribal weights and measures that partitioned one hekat by substituting 64/64, 10 hin, 64 dja or 320 ro. Answers to each problem were multiplied by its divisor to return 64/64, 10 hin, 64 dja, 320 ro), algebra, and/or 2/n tables (converting 2/n by LCM multipliers, recalling 2/97 times 56/56) calculations. 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. The area first triangle, with an altitude of 5 khet, and a base of 9/4 khet, was found by triangle formula: 1/2 the base times the altitude, 5*(9/4)*(1/2)= (45/8) = 5 5/8 setat. The second triangle area, with an altitude of 7 khet, and a base of 9/4 khet, was found by the triangle formula: 1/2 the base times the altitude, Ahmes calculated 7*(9/4)*(1/2) = 63/8 = 7 7/8 setat The third calculation found the area of undefined shape discussed by: 11/8 mh = 110/8 mh + 10 mh = 23 3/4 mh = 1/8 setat + 11 1/4 mh since 12 1/2 mh = 1/8 setat. Alternative views suggest that the third shape may have defined a truncated pyramid (base 6, top 3, height 95/18), or a triangle (base 6, and altitude 95/12). To assist the decoding of the third RMP 53 area RMP 54, and RMP 55 scribal guidelines have been consulted. 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: a. (7/10)*(4/4) = 28/40 = (24 + 3)/40 = 3/8 setat + 300/40 mh = 5/8 setat + 7 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 c. (28/10)*(2/2) = 56/20 = (55 + 1)/20 = 11/4 setat + 100/20 mh = 2 3/4 setat + 5 mh d. (56/10) = (55 + 1)/10 = 11/2 setat + 100/10 COL = 5 1/2 setat + 10 mh Ahmes may have also made calculations thinking in mh unuts. For example, 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: (1/2 + 1/5)setat = (1/2 + 1/8 + (20 - 12 1/2 mh) = 5/8 setat + 7 1/2 mh. RMP 55 takes 3/5 of 5 setat to obtain 3 setat by four steps (a to d): a. 1/2 setat + 10 mh b. (1 + 1/8) setat + (7 + 1/2) mh c. (1 + 3/8) setat + (2 + 1/2) setat d. adding steps a and c, and knowing (12 + 1/2) mh = 1/8 setat (1/2 setat + 10 mh) + [(1 + 3/8) setat + (2+ 1/2)mh] = (2 + 7/8)setat + (1 + 2 + 1/2) mh = 3 setat \begin{thebibliography}{9} \bibitem{1} A.B. Chace, Bull, L, Manning, H.P., Archibald, R.C., \emph{The Rhind Mathematical Papyrus}, Mathematical Association of Amnerica, Vol I, 1927. NCTM reprints available. \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{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{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{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. \end{thebibliography}