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The hekat or heqat (transcribed ''HqA.t'') was an ancient Egyptian volume unit used to measure grain, bread, and beer. It equals 4.8 
litre The litre (international spelling) or liter (American English spelling) (SI symbols L and l, other symbol used: ℓ) is a metric unit of volume. It is equal to 1 cubic decimetre (dm3), 1000 cubic centimetres (cm3) or 0.001 cubic metre (m3). ...
s, or about 1.056
imperial gallon The gallon is a unit of volume in imperial units and United States customary units. Three different versions are in current use: *the imperial gallon (imp gal), defined as , which is or was used in the United Kingdom, Ireland, Canada, Austral ...
s, in today's measurements. retrieved March 22, 2020 at about 7:00 AM EST.


Overview

Until the
New Kingdom New is an adjective referring to something recently made, discovered, or created. New or NEW may refer to: Music * New, singer of K-pop group The Boyz Albums and EPs * ''New'' (album), by Paul McCartney, 2013 * ''New'' (EP), by Regurgitator, ...
the hekat was one tenth of a khar, later one sixteenth; while the New Kingdom (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 1/64 of a hekat (75 cc) in the MK, and 1/64 of an (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 replacing the hekat as the Pharaonic volume control unit in official lists. Hana Vymazalova evaluated the hekat unit in 2002 within the
Akhmim Wooden Tablet The Akhmim wooden tablets, also known as the Cairo wooden tablets (Cairo Cat. 25367 and 25368), are two wooden writing tablets from ancient Egypt, solving arithmetical problems. They each measure around and are covered with plaster. The tablets ar ...
by showing that five answers were returned to (64/64) when multiplied by the divisors 3, 7, 10, 11 and 13. The RMP also divided a hekat unity (64/64) by prime and composite numbers ''n'' when 1/64 < ''n'' < 64. The binary quotient used
Eye of Horus The Eye of Horus, ''wedjat'' eye or ''udjat'' eye is a concept and symbol in ancient Egyptian religion that represents well-being, healing, and protection. It derives from the mythical conflict between the god Horus with his rival Set, in wh ...
numbers. The remainder scaled
Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each ...
s to 1/320 units named ro. Quotients and unscaled remainders were obtained for the dja, ro and other units when the divisor ''n'' was greater than 64. For example, one the 1/320 ro unit was written by Ahmes by solving 320/n ro. Gillings cites 29 examples of two-part statements converted to one-part statements in RMP 82. Ahmes recorded 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 × 1/320 = 1/4 + 1/16 + 1/64 + 1 2/3ro (two-part ancient statement). Two-part statements were also converted by Ahmes to an unscaled hin unit by writing 3 1/3 hin. The hekat measurement unit, and its
double entry accounting system Double-entry bookkeeping, also known as double-entry accounting, is a method of bookkeeping that relies on a two-sided accounting entry to maintain financial information. Every entry to an account requires a corresponding and opposite entry t ...
, was found beyond the
Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
. 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 The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...
by MMP #10, by approximating '' π'' to around 3.16. The approximation of π was achieved by squaring a circle, increasingly (i.e. for the denominator in terms of : 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 = 64d2/81, within a cylinder relationship to the hekat. MMP 10 data meant that ''d'' = 2 defined π for use in hekat volumes as 256/81. The 256/81 approximation was also used by
Ahmes Ahmes ( egy, jꜥḥ-ms “, a common Egyptian name also transliterated Ahmose) was an ancient Egyptian scribe who lived towards the end of the Fifteenth Dynasty (and of the Second Intermediate Period) and the beginning of the Eighteenth Dynas ...
and other scribes. The
ancient Egyptian units of measurement Ancient history is a time period from the beginning of writing and recorded human history to as far as late antiquity. The span of recorded history is roughly 5,000 years, beginning with the Sumerian cuneiform script. Ancient history cove ...
discussion further shows that the hekat was 1/30 of a royal cubit3, an analysis that needs to double checked, against the ''d'' = 2 suggestion, which means that ''r'' = 1, a suggestion that does make sense. One royal cubit of the ancient Egyptian weights and measures = 523.5 millimeters. ((523.5 mm)3) / 30 = 4.78221176 liters. However that may be at least a sphere that has a circumference 523.5 millimeters will actually have a metric volume about 2.42269 liters or roughly half of a hekat or about one sixtieth of a royal cubic cubit to two parts in a hundred. A modern schoolbook formula has volume=4/3 pi r3 for example. In the case of a land where pi=256/81 or about 3.1604938 a similar result can be obtained with the different formula that has been suggested by Zapassky and others where over there the volume of a sphere is given by the quotient of the cube of the circumference divided by six pi2 (V=c3/6π2)Zapassky E, Gadot Y, Finkelstein I, Benenson I (2012) An Ancient Relation between Units of Length and Volume Based on a Sphere. PLoS ONE 7(3): e33895. https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0033895 retrieved March 22, 2020 at about 7:38 AM EST and in that case the ancient Egyptian volume should come to about 2.386954 liters or about 98.5% of its true volume.


References


External links

* http://www.eurekalert.org/pub_releases/2012-06/afot-ajh060412.php * 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 pharmaceutical and medical knowledge, an abstract, Philipps-Universität, 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.
eurekalert.org




* Benenson, Itzahk. Some short paper in Hebrew about the noted 2012 pottery volumes studies that looks like it could not possibly have much particulars in it at https://tauweb.tau.ac.il/news/new-method-discovery-ancient-world retrieved March 21, 2020 at about 11:20 PM EST History of mathematics Obsolete units of measurement Units of volume