Hekat (volume)
Encyclopedia
The hekat or heqat 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.
Hana Vymazalova evaluated the hekat unit in 2002 within the Akhmim Wooden Tablet
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
numbers. The remainder scaled Egyptian fractions to 1/320 units named ro. Quotients and unscaed remainders were obtained for the dja, ro and other units when divisor n was great 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, 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 π to around 3.16. The approximation of π 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 = 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
and other scribes. The ancient Egyptian weights and measures discussion further shows that the hekat was 1/30th 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. 1 royal cubit of the Ancient Egyptian weights and measures = 523.5 millimeters. ((523.5 mm)3) / 30 = 4.78221176 liters.
Hana Vymazalova evaluated the hekat unit in 2002 within the Akhmim Wooden Tablet
Akhmim wooden tablet
The Akhmim wooden tablets or Cairo wooden tablets are two ancient Egyptian wooden writing tablets. They each measure about 18 by 10 inches and are covered with plaster. The tablets are inscribed on both sides. The inscriptions on the first tablet includes a list of servants, which is followed...
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
Eye of Horus
The Eye of Horus is an ancient Egyptian symbol of protection, royal power and good health. The eye is personified in the goddess Wadjet...
numbers. The remainder scaled Egyptian fractions to 1/320 units named ro. Quotients and unscaed remainders were obtained for the dja, ro and other units when divisor n was great 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, was found beyond the Rhind Mathematical Papyrus
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum. It dates to around 1650 BC...
. Another text was the Ebers Papyrus
Ebers papyrus
The Ebers Papyrus, also known as Papyrus Ebers, is an Egyptian medical papyrus dating to circa 1550 BC. Among the oldest and most important medical papyri of ancient Egypt, it was purchased at Luxor, in the winter of 1873–74 by Georg Ebers...
, the best known medical text. The hekat unit was defined, in terms of its volume size, in the Moscow Mathematical Papyrus
Moscow Mathematical Papyrus
The Moscow Mathematical Papyrus is an ancient Egyptian mathematical papyrus, also called the Golenishchev Mathematical Papyrus, after its first owner, Egyptologist Vladimir Golenishchev. Golenishchev bought the papyrus in 1892 or 1893 in Thebes...
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 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 = 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
Ahmes was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty . He wrote the Rhind Mathematical Papyrus, a work of Ancient Egyptian mathematics that dates to approximately 1650 BC; he is the earliest contributor to mathematics...
and other scribes. The ancient Egyptian weights and measures discussion further shows that the hekat was 1/30th 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. 1 royal cubit of the Ancient Egyptian weights and measures = 523.5 millimeters. ((523.5 mm)3) / 30 = 4.78221176 liters.