Akhmim Wooden Tablet
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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 Plaster is a building material used for the protective or decorative coating of walls and ceilings and for Molding (decorative), moulding and casting decorative elements. In English, "plaster" usually means a material used for the interiors of ...
. The tablets are inscribed on both sides. The
hieroglyphic Egyptian hieroglyphs (, ) were the formal writing system used in Ancient Egypt, used for writing the Egyptian language. Hieroglyphs combined logographic, syllabic and alphabetic elements, with some 1,000 distinct characters.There were about 1,00 ...
inscriptions on the first tablet include a list of servants, which is followed by a mathematical text. T. Eric Peet, ''
The Journal of Egyptian Archaeology The ''Journal of Egyptian Archaeology (JEA)'' is a bi-annual peer-reviewed international academic journal published by the Egypt Exploration Society. Covering Egyptological research, the JEA publishes scholarly articles, fieldwork reports, and re ...
'', Vol. 9, No. 1/2 (April 1923), pp. 91–95, Egypt Exploration Society
The text is dated to year 38 (it was at first thought to be from year 28) of an otherwise unnamed king's reign. The general dating to the early
Egyptian Middle Kingdom The Middle Kingdom of Egypt (also known as The Period of Reunification) is the period in the history of ancient Egypt following a period of political division known as the First Intermediate Period. The Middle Kingdom lasted from approximatel ...
combined with the high regnal year suggests that the tablets may date to the reign of the
12th Dynasty The Twelfth Dynasty of ancient Egypt (Dynasty XII) is considered to be the apex of the Middle Kingdom by Egyptologists. It often is combined with the Eleventh, Thirteenth, and Fourteenth dynasties under the group title, Middle Kingdom. Some s ...
pharaoh
Senusret I Senusret I (Middle Egyptian: z-n-wsrt; /suʀ nij ˈwas.ɾiʔ/) also anglicized as Sesostris I and Senwosret I, was the second pharaoh of the Twelfth Dynasty of Egypt. He ruled from 1971 BC to 1926 BC (1920 BC to 1875 BC), and was one of the most ...
, c. 1950 BC. The second tablet also lists several servants and contains further mathematical texts. The tablets are currently housed at the
Museum of Egyptian Antiquities The Museum of Egyptian Antiquities, known commonly as the Egyptian Museum or the Cairo Museum, in Cairo, Egypt, is home to an extensive collection of ancient Egyptian antiquities. It has 120,000 items, with a representative amount on display a ...
in
Cairo Cairo ( ; ar, القاهرة, al-Qāhirah, ) is the capital of Egypt and its largest city, home to 10 million people. It is also part of the largest urban agglomeration in Africa, the Arab world and the Middle East: The Greater Cairo metro ...
. The text was reported by Daressy in 1901 and later analyzed and published in 1906. The first half of the tablet details five multiplications of a ''
hekat The hekat or heqat (transcribed ''HqA.t'') was an ancient Egyptian volume unit used to measure grain, bread, and beer. It equals 4.8 litres, or about 1.056 imperial gallons, in today's measurements. retrieved March 22, 2020 at about 7:00 ...
'', a unit of volume made up of 64 ''dja'', by 1/3, 1/7, 1/10, 1/11 and 1/13. The answers were written in binary
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 ...
quotients and exact Egyptian fraction remainders, scaled to a 1/320 factor named ''ro''. The second half of the document proved the correctness of the five division answers by multiplying the two-part quotient and remainder answer by its respective (3, 7, 10, 11 and 13) dividend that returned the ''
ab initio ''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative singular of ''initium'' ("beginning"). Etymology Circa 1600, from Latin, literally "from the beginning", from ab ...
'' hekat unity, 64/64. In 2002,
Hana Vymazalová Hana Vymazalová (born 1978), is a Czech Egyptologist. She graduated in Egyptology and Logic at the Faculty of Arts, Charles University in Prague. Her dissertation focused on accounting texts from the archives of pharaoh Raneferef. Her interests ...
obtained a fresh copy of the text from the Cairo Museum, and confirmed that all five two-part answers were correctly checked for accuracy by the scribe that returned a 64/64 hekat unity. Minor typographical errors in Daressy's copy of two problems, the division by 11 and 13 data, were corrected at this time.Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archive Orientallai, Charles U., Prague, pp. 27–42, 2002. The proof that all five divisions had been exact was suspected by Daressy but was not proven until 1906.


Mathematical content


1/3 case

The first problem divides 1 ''hekat'' by writing it as 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (5 ''ro'') (which equals 1) and dividing that expression by 3. * The scribe first divides the remainder of 5 ''ro'' by 3, and determines that it is equal to (1 + 2/3) ''ro''. * Next, the scribe finds 1/3 of the rest of the equation and determines it is equal to 1/4 + 1/16 + 1/64. * The final step in the problem consists of checking that the answer is correct. The scribe multiplies 1/4 + 1/16 + 1/64 + (1 + 2/3) ro by 3 and shows that the answer is (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64) + (5 ''ro''), which he knows is equal to 1. In modern mathematical notation, one might say that the scribe showed that 3 times the ''hekat'' fraction (1/4 + 1/16 + 1/64) is equal to 63/64, and that 3 times the remainder part, (1 + 2/3) ''ro'', is equal to 5 ''ro'', which is equal to 1/64 of a ''hekat'', which sums to the initial hekat unity (64/64).


Other fractions

The other problems on the tablets were computed by the same technique. The scribe used the identity 1 ''hekat'' = 320 ''ro'' and divided 64 by 7, 10, 11 and 13. For instance, in the 1/11 computation, the division of 64 by 11 gave 5 with a remainder 45/11 ''ro''. This was equivalent to (1/16 + 1/64) ''hekat'' + (4 + 1/11) ''ro''. Checking the work required the scribe to multiply the two-part number by 11 and showed the result 63/64 + 1/64 = 64/64, as all five proofs reported.


Accuracy

The computations show several minor mistakes. For instance, in the 1/7 computations, 2 \times 7 was said to be 12 and the double of that 24 in all of the copies of the problem. The mistake takes place in exactly the same place in each of the versions of this problem, but the scribe manages to find the correct answer in spite of this error since the 64/64 hekat unity guided his thinking. The fourth copy of the 1/7 division contains an extra minor error in one of the lines. The 1/11 computation occurs four times and the problems appear right next to one another, leaving the impression that the scribe was practicing the computation procedure. The 1/13 computation appears once in its complete form and twice more with only partial computations. There are errors in the computations, but the scribe does find the correct answer. 1/10 is the only fraction computed only once. There are no mistakes in the computations for this problem.


Hekat problems in other texts

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 ...
(RMP) contained over 60 examples of ''hekat'' multiplication and division in RMP 35, 36, 37, 38, 47, 80, 81, 82, 83 and 84. The problems were different since the hekat unity was changed from the 64/64 binary hekat and ro remainder standard as needed to a second 320/320 standard recorded in 320 ro statements. Some examples include: * Problems 35–38 find fractions of the ''hekat.'' Problem 38 scaled one hekat to 320 ro and multiplied by 7/22. The answer 101 9/11 ro was proven by multiplying by 22/7, facts not mentioned by Claggett and scholars prior to Vymazalova.Clagett, Marshall ''Ancient Egyptian Science, A Source Book''. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 * Problem 47 scaled 100 ''hekat'' to (6400/64) and multiplied (6400/64) by 1/10, 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90 and 1/100 fractions to binary quotient and 1/1320 (ro) remainder unit fraction series. * Problem 80 gave 5 Horus eye fractions of the ''hekat'' and equivalent fractions as expressions of another unit called the ''hinu''. These were left unclear prior to Vymazalova. Problem 81 generally converted hekat unity binary quotient and ro remainder statements to equivalent 1/10 hinu units making it clear the meaning of the RMP 80 data. The '' Ebers Papyrus'' is a famous late Middle Kingdom medical text. Its raw data were written in ''hekat'' one-parts suggested by the Akhim wooden tablets, handling divisors greater than 64.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


References

Other: *Gardener, Milo, "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, MD Publications, New Delhi, pp 157–173. https://independent.academia.edu/MiloGardner/Papers/163573/The_Arithmetic_used_to_Solve_an_Ancient_Horus-Eye_Problem *Gillings, R. ''Mathematics in the Time of the Pharaohs''. Boston, MA: MIT Press, pp. 202–205, 1972. . (Out of print)


External links

* Scaled AWT Remainders {{DEFAULTSORT:Akhmim Wooden Tablets Ancient Egyptian society Egyptian fractions Ancient Egyptian texts Units of volume Egyptian Museum