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Re: Hibeh Papyrus
Posted:
Sep 12, 2011 6:36 PM
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The Hibeh Papyrus was written around 270 BCE by a Greek scribe and transliterated by David Fowler and Eric Turner in Historia Mathematics, 10 (1983), 344-359. Time units of a day were recorded in 1/45 parts and presented in a n/45 table by Fowler and others. The n/45 information was recorded in unit fractions. Unit fraction denominators ciphered Greek counting numbers 1:1 onto Greek letters. Greek letter denominators followed by (') denoted unit fractions were assumed.
For example: 13/45 should have recorded delta' lambda' rho-pi'
and ciphered 13/45 into 1/4 1/30 (1/(80 +100) and translated as: 1/4 + 1/30 + 1/180
The scribe's narrative offered alternate calculations. It is clear by referencing unit fraction texts dated back 1,500 years that LCM 4 scaled 13/43 such that:
52/180 = (45 + 6 + 1)/180 = 1/4 + 1/30 + 1/180
and that three LCMs 1, 2 and 4 were used in the n/45 unit fraction discussions.
The n/45 table was not offered by the scribe. The n/45 table has been added back, along with nine missing n/45 members (10/45, 12/45, 14/45, 15/45, 18/45, 27/45, 31/45, 33/45 and 39/45), as well as intermediate steps that calculated each n/45 conversion. These added-back n/45 facts assist the reader to understand the 27 festival date and time narrative in historical context by first considering:
1. 1/45 = 1/45 used LCM 1
2. 2/45 = 1/30 + 1/90 meant 4/90 = (3 + 1)/90 used LCM 2
3. 3/45 = 1/15 used LCM 1
4. 4/45 = 1/15 + 1/45 meant 4/45= (3 + 1)/45 used LCM 1 (four times) and
* 4/45 = 1/12 + 1/180 + meant 16/90 = (15 + 1)/180 used LCM 4 (once)
5. 5/45 = 1/9 used LCM 1
6. 6/45 = 1/10 + 1/30 meant 12/90 = (9 + 3)/90 used LCM 2
7. 7/45 = 1/9 + 1/30 + 1/90 meant 14/90 = (10 + 3 + 1)/90 used LCM 2
8. 8/45 = 1/9 + 1/90 meant 16/90 = (15 + 1)/90 used LCM 2
9. 9/45 = 1/5 used LCM 1
10. 10/45 = 1/5 + 1/45 meant (9 + 1)/45 used LCM 1
11. 11/45 = 1/9 + 1/10 + 1/30 meant 22/90 = (10 + 9 + 3)/90 used LCM 2
12. 12/45 = 1/9 + 1/10 + 1/18 meant 24/90 = (10 + 9 + 5)/90 used LCM 2?
13. 13/45 = 1/4 + 1/30 + 1/180 meant 52/180 = (45 + 6 + 1)/180 used LCM 4
14. 14/45 = 1/4 + meant 56/180 = (45 + 9 + 2)/90 used LCM 4
15. 15/45 = 1/3 meant the scribe used LCM 1
16. 16/45 1/3 + 1/45 meant 16/45 = (15 + 1)/45 used LCM 1
17. 17/45 = 1/3 + 1/30 + 1/90 meant 34/90 =(30 + 3 + 1)/90 used LCM 2
18. 18/45 = 1/3 + 1/15 meant 18/45 = (15 + 3)/45 used LCM 1
19. 19/45 = 1/3 + 1/15 + 1/45 meant 19/45 = (15 + 3 + 1)/45 used LCM 1
20. 20/45 = 1/3 + 1/9 meant 20/45= (30 + 10)/90 = 40/90 used LCM 2
21. 21/45 1/3 + 1/10 + 1/30 meant 42/90 = (30 + 9 + 3)/90 = 42/9 used LCM 2
22. 22/45 = 1/3 + 1/10 + 1/18 meant 44/90 = (30 + 9 + 5)/90 used LCM 2?
23. 23/45 = 1/2 + 1/90 meant 46/90=(45 + 1)/90 = 46/90 used LCM 2
24. 24/45 = 1/2 + 1/30 meant 48/90 = (45 + 3)/90 used LCM 2
25. 25/45 = 1/2 + 1/30 + 1/45 meant 50/90 = (45 + 3 + 2)/90 used LCM 2
26. 26/45 = 1/2 + 15 + 1/90 meant 52/90 = (45 + 6 + 1)/90 used LCM 2
27. 27/44 = 1/2 + 1/15 + 1/30 meant 54/90 = (45 + 6 + 3)/90 used LCM 2
28. 28/45 = 1/2 + 1/10 + 1/45 meant 56/90 = (45 + 9 + 2)/90 used LCM 2
29. 29/45 = 1/2 + 1/9 + 1/30 meant 58/90 = (45 + 10 + 3)/90 used LCM 2
*29/58 = 1/2 + 1/10 + 1/30 + 1/90 meant 58/90 = (45 + 9 + 3 + 1)/90 used LCM 2 two other times
30. 30/45 = 2/3 meant 60/90 used LCM 2
31. 31/45 = 2/3 + 1/45 meant 62/90 = (60 + 2)/90 used LCM 2
32. 32/45 = 2/3 + 1/30 + 1/90 meant 64/90 = (60 + 3 + 1)/90 used LCM 2
33. 33/45 = 2/3 + 1/15 meant 66/90 = (60 + 6)/90 used LCM 2
34. 34/45 = 2/3 + 1/15 + 1/45 meant 68/70 = (60 + 6 + 3)/90 used LCM 2
35. 35/45 = 2/3 + 1/10 + 1/90 meant 70/90 = (60 + 9 + 1)/90 used LCM 2
36. 36/45 = 2/3 + 1/10 + 1/30 meant 72/90 = (60 + 9 + 3)/90 used LCM 2
37. 37/45 = 2/3 + 1/10 + 1/30 + 1/45 meant 74/90 = (60 + 9 + 3 + 2) used LCM 2
38. 38/45 = 2/3 + 1/6 + 1/90 meant 76/90 = (60 + 15 + 1) used LCM 2
39. 39/45 = 2/3 + 1/6 + 1/30 meant 78/90 = (60 + 15 + 3) used LCM 2
40. 40/45 = 2/3 + 1/6 + 1/18 meant 80/90 = (60 + 15 + 5)/90 used LCM 2
41. 41/45 = 2/3 + 1/5 + 1/30 + 1/90 meant 82/90 = (60 + 18 + 3 + 1)/90 used LCM 2
42. 42/45 = 2/3 + 1/4 + 1/60 meant 168/180 = (120 + 45 + 3)/180 used LCM 4
43. 43/45 = 2/3 + 1/4 + 1/30 + 1/180 meant 172/180 = (120 + 45 + 6 + 1)/180 used LCM 4
44. 44/45 = 2/3 + 1/4 + 1/20 + 1/90 meant 176/180 = (120 + 45 + 9 + 2)/180 used LCM 4
LCM 1 meant the divisors of 45: 15, 9, 5, 3, and 1 were summed to eight numerators of 1/45, 3/45, 4/45, 5/45, 9/45, 10/45, 15/45, and 18/45 that allowed ciphered unit fraction series to be recorded.
LCM 2 meant the divisors of 90: 45, 30, 18, 15, 10, 9, 6, 5, 4, 3, 2, 1 were summed to 31 n/45 numerator as intermediate steps, and other n/45 conversions as unit fraction series alternatives.
LCM 4 meant the divisors of 180: 90, 60, 60, 20, 18, 15, 12, 10, 9, 6, 5, 4, 3, 2, 1 appeared in five numerators 13/45, 14/45, 42/45, 43/45, and 44/45, and other n/45 conversions as unit fraction series alternatives.
One purpose of the papyrus listed 27 Greek festivals within Egyptian civil calendar dates and times were recorded in unit fraction series. Raw data was restated to local month names that followed the classical Greek local calendar practice. Ciphered Greek unit fraction series considered input provided by David Fowler and Eric Turner: Hibeh Papyrus Historia Mathematics, 10 (1983), 344-359 and other data reported by Bruce Friedman, and by LCM conversion methods provided by earlier Egyptian MK texts. The entire Hibeh Papyrus included a wide range of subjects.
with activated links available within:
http://planetmath.org/encyclopedia/HibehPapyrus.html
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