
Re: RMP # 48
Posted:
Oct 1, 2007 9:47 AM


Another view of this topic relates to Archimedes and Eudoxus's jump forward. It is given by:
http://books.google.com/books?id=97jwg1Xwpj0C&pg=PA239&dq=rmp+48&sig=MDH_qlnvTg7LEb5egglteFI37m0
There are, of course, several researchers that only parse RMP 48, and its basis of Middle Kingdom methods for approximating area, and volume (hekat). One approximation was used for the hekat for the measurement of grain, beer and much more, the view that I take on this topic, as discussed on
http://en.wikipedia.org/wiki/Akhmim_Wooden_Tablet
and several Wikipedia sites, one being on the hekat, per:
Hekat (volume unit) From Wikipedia, the free encyclopedia
The hekat or heqat (transcribed HqA.t) was an ancient Egyptian volume unit, used to measure grain, bread, and beer. Until the New Kingdom the hekat was one tenth of a khar, later one sixteenth; while the New Kingdom oipe (transcribed ip.t) contained 4 hekat. It was subdivided into other units  some for medical prescriptions  the hin (1/10), dja (1/64) and ro (1/320). The dja was recently evaluated by Tanja Pemmerening in 2002 to 1/64th of a hekat (75 cc) in the MK, and 1/64th of an oipe (1/16 of a hekat, or 300 cc) in the NK, meaning that the dja was denoted by HorusEye imagery. It has been suggested by Pemmerening that the NK change came about related to the oipe replacing the hekat as the Pharaonic volume control unit in official lists.
In addition, Hana Vymazalova evaluated the hekat in 2002 from Akhmim Wooden Tablet, dividing it by small numbers, again, greatly improving its readability. Hekat units were written in quotients and remainders. The quotients were written as binary fractions, Eye of Horus numbers. The remainders were written as Egyptian fractions, scaled to 1/320 units named ro.
The hekat was also found is the Rhind Mathematical Papyrus, and many other texts, including the Ebers Papyrus, the best known medical text. This volume unit was defined in the Moscow Mathematical Papyrus by MMP #10, by approximating pi to around 3.16. The approximation of pi was achieved by squaring a circle, increasingly (i.e. for the denominator in terms of setats: 9, 18, 36, 72, and 81, Gillings, page 141) until the vulgar fraction 256/81 was reached, the only relationship that was used in the Egyptian Middle Kingdom. The MMP scribe then found the surface area of a basket = (8d/9)^2 = 64d^2/81 as a cylinder relationship to the hekat, meaning that d = 2 may have defined a hekat or 256/81, the number that was used by Ahmes to approximate pi. The ancient Egyptian weights and measures discussion further shows that the hekat was 1/30th of a royal cubit^3, an analysis that needs to double checked, against the d = 2 suggestion, which means that r = 1, a suggestion that does make sense.
[edit] References
* Gillings, Richard. "Mathematics in the Time of the Pharaohs" Dover, reprint from, Cambridge, Mass, MIT Press 1972, ISBN 048624315X.
* Pemmerening, Tanja, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, PhillipsUniverstat, Marburg, 8112004, taken from "Die Altagyptschen Hohlmass" in studien zur Altagyptischen Kulture, Beiheft, 10, Hamburg, BuskeVerlag, 2005
* Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archiv Orientalai, Charles U., Prague, pp. 2742, 2002.
Best Regards,
Milo Gardner

