Akhmim Wooden Tablet
From Wikipedia, the free encyclopedia
The Akhmim Wooden Tablet, is an ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. It is currently housed in Cairo's Museum of Egyptian Antiquities. Its text was reported by Georges Émile Jules Daressy in 1901 and analyzed and published by Daressy in 1906. The first half of the tablet details five divisions of a hekat, partitioned from its unity (64/64), by 3, 7, 10, 11 and 13. The answers were written in Eye of Horus quotients, and Egyptian fraction remainders, scaled to a 1/320th 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 divisor (3, 7, 10, 11 and 13). In 2002, Hana Vymazalova, gained a fresh copy of the text from the Cairo Museum, and confirmed that all five two-part answers were correctly returned to the initial (64/64) unity value by the scribe. The proof that all five AWT divisions had been exact was suspected by Daressy, but was not proven in 1906. Typographical errors in Daressy's copy of two problems, the division by 11 and 13 data, were corrected by Vymazalova.
For example, the first problem divided a hekat unity (64/64) by 3, such that a quotient 21 was found by 64/3, with a remainder 1. The remainder meant 1/3 of 1/64 of a hekat, or simply 1/192. The remainder was further scaled to 1/320 units of a hekat. The remainder arithmetic was written in two-parts. The first part restated the quotient 21 to an Eye of Horus binary series by
The second part re-stated the remainder 1/192 to 5/960 such that 1/320 was factored and renamed ρ, meant that the vulgar fraction 5/3 was written as an Egyptian fraction series, and ρ by: (1 + 2 / 3)ρ. The AWT reported the final two-part answer of the division by 3 problem as:
Vymazalova's 2002 paper shows that the AWT scribe was required to prove the answer by multiplying by the initial divisor. Again the division by 3 data meant that
for the first part; and
![\left[\left(1 + \frac{2}{3}\right)\frac{1}{320}\right]\times 3 = \frac{5}{960}\times 3 = \frac{15}{960} = \frac{1}{64}](../../../math/1/6/e/16e5a16821c1b3260afd7541bda552c1.png)
for the second part. That is , adding the first and second parts: 63/64 + 1/64 = 64/64, the initial hekat unity (64/64) was found, as was the case for all five problems.
The Rhind Mathematical Papyrus contains 30 examples of this type of hekat division, with 29 examples being listed in one problem. In RMP #80, Gillings cites the raw data by small fraction divisors, down to 1/64, and large divisors, up to 64, in an appendix. It should be noted that the hekat two-part answers, quotient and remainders, were also re-written into one-part, a remainder only context, 1/10th of a hekat unit. The 1/10th of a hekat, a hin, is well known as the hinu system. The 30th example, RMP #34, followed AWT rules that divided by 70, beyond the AWT's implied 64 (divisor) limit, by increasing the hekat unity numerator to 100 times its size or (6400/64). The division took place in the same manner as set down in the AWT, dividing 6400 by 70, finding a quotient of 91, and a remainder of 30, and so forth, except that no double check was mentioned by Ahmes.
The Ebers Papyrus is a famous late Middle Kingdom medical text. Its raw data was written in hekat one-parts suggested by the AWT, handling divisors greater than 64. Tanja Pemmerening and others accurately read information from this text, for the first time, in 2002, by understanding the one-part divisions of a hekat.
[edit] References
Daressy, G. "Cairo Museum des Antiquities Egyptiennes." Catalogue General Ostraca, Volume No. 25001-25385, 1901.
Daressy, Georges, “Calculs Egyptiens du Moyan Empire”, Recueil de Travaux Relatifs De La Phioogie et al Archaelogie Egyptiennes Et Assyriennes XXVIII, 1906, 62–72.
Gillings, R. Mathematics in the Time of the Pharaohs. Boston, MA: MIT Press, pp. 202-205, 1972. ISBN 0-262-07045-6. (Out of print)
Peet, T. E. "Arithmetic in the Middle Kingdom." J. Egyptian Arch. 9, 91-95, 1923.
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.
