Hekat (volume)

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 sub-divided 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 Pommerening 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 Horus-Eye imagery. It has been suggested by Pommerening 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 in the Akhmim Wooden Tablet, dividing a hekat unity by small numbers, greatly improving its readability and accuracy. Hekat units were written in two-part and one-part quotients and remainders. The two-part quotients were written as binary fractions, Eye of Horus numbers. The two-part remainders were written as Egyptian fractions, scaled to 1/320 units named ro, when a hekat unity (64/64) was partitioned by divisors n that were smaller than 64. When divisors n were equal to or greater than 64, such as n = 64, written in terms of units of a hin, one-part quotients and remainders were obtained by solving 10/64 hin, or generally 10/n hin. One-part quotients and remainders were also obtained for the dja, ro and other units. For example, one other one-part ro unit was obtained by solving 320/n ro. Gillings cites 29 examples of two-part statements being converted into one-part statements. Ahmes wrote his data, such as by the n = 3 case, by showing (64/64)/3 = 21/64 + 1/192 (a modern statement) as written as(16 + 4 + 1)/64 + 5/3 x 1/320 = 1/4 + 1/16 + 1/64 + 1 2/3ro (two-part ancient statement). Two-part statement was converted by Ahmes to hin units by writing 10/3 hin, or 3 1/3 hin (the equivalent ancient one-part statement).

The hekat measurement unit, and its double entry accounting system, was found beyond the Rhind Mathematical Papyrus. Another text was the Ebers Papyrus, the best known medical text. The hekat unit was defined, in terms of its volume size, 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 found the surface area of a basket equal to: (8d/9)^2 = 64d^2/81, within a cylinder relationship to the hekat. MMP 10 data meant that d = 2 defined pi for use in hekat volumes as 256/81. The 256/81 approximation was also used by Ahmes and other scribes. 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. 1 royal cubit of the Ancient Egyptian weights and measures = 523.5 millimeters. ((523.5 mm)^3) / 30 = 4.78221176 liters.

References

* Gillings, Richard. "Mathematics in the Time of the Pharaohs" Dover, reprint from, Cambridge, Mass, MIT Press 1972, ISBN 0-486-24315-X.

*Pommerening, Tanja, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass" in studien zur Altagyptischen Kulture, Beiheft, 10, Hamburg, Buske-Verlag, 2005

* Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archiv Orientalai, Charles U., Prague, pp. 27-42, 2002.

External links

* [http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_geometry.html Egyptian geometry: Determining the value of pi]
* [http://planetmath.org/encyclopedia/EgyptianFractions3.html Remainder Arithmetic]
* [http://egyptianmath.blogspot.com: History of Egyptian fractions]