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Kahun Papyrus, its math contents
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Message 1.Â
Posted by Milo Gardner (U14346264) on Thursday, 25th February 2010
The Kahun Papyrus, an 1850 BCE Middle Kingdom texts begins with a 2/n table that contains several 2/n conversions that differ with the RMP 2/n table.
Planetmath and Wikipedia offer summaries of the text per:
1.
2.
Beyond the use of different scaling LCMs the KP scribe offered the same arithmetic progression formulas that was later used in two RMP problems.
John Legon attempted to write a definitive 1992 paper on the arithmetic progression issue, leaving open one question. Since 1992, John Legon and I have clarified the issue, hence, the topic is nearly complete on a micro level.
However, on a meta level, several issues remain open. For example, did the KP scribe use a simpler form of mental arithmetic than the shorthand math notes reveal?
More on this topic, if anyone is interested.
In summary, from my point of view, the KP and RMP arithmetic progression formulas were the same class of mathematics that Carl F. Gauss developed as a young child to solve the addition of 1 to 100, taking one addition at a time by matching
1 + 100 = 101,
2 + 99 = 101
3 + 98 = 101
...
48 +53 = 101
49 + 52 = 101
50 + 51 = 101
50 times such that 50 x 101 = 5050
Added scribal details can be reported, if anyone is interested.
Hopefully brevity is appreciated.
Best Regards,
Milo Gardner -
Message 2
Posted by Milo Gardner (U14346264) on Friday, 26th February 2010
Beyond RMP 40 and RMP 64, and Gauss' youthful re-discovery of an ancient arithmetic progression, Scott Williams, U. of Buffalo reports RMP 79 as using a geometric series per:
"There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced seven hekat. What is the sum of all the enumerated things
houses 7
cats 49
mice 343
heads of barley 2401
hekats of barley 16807
total 19607
Two columns of data leads to the sum (in the bottom row) of the five terms of the geometric sequence with a ratio 7 beginning with 7: 7+72+73+74+75.
While the second column offers the usual method for multiplying 7*2801, it is observed that the former sum equals the latter product.
To an archeologist the table above and the relationship between the two columns may be meaningless, and several have said this. However, to an arithmetician, the relationship between the two columns is clear since (we know) the formula for the geometric series of the first n terms of a geometric series
{1, r, r2, ..., rn}
of ratio r and beginning with
1 is 1+r+r2+...+rn = (rn-1)/(r-1).
We have 7 times this value with r=7 and n=4.
Thus, 7*(74-1)/(7-1) = 7*(16807-1)/6 = 7*16806/6 = 7*2801 = 19607.
Thus, problem 79 is a table exhibiting the formula for the sum of a geometric series!! "
As important as the number aspect of this 1650 BCE problem, its word level is equally important.
Ahmes' 1650 BCE poem or rhyme was passed down to medieval times, in the context of ancient number theory, in which the Old English rhythm:
"As I was going to St. Ives,
I met a man with seven wives
Every wife had seven sacks
Every sack had seven cats
Every cat had seven kits
Kits, cats, sacks, and wives
How many were going to St. Ives?"
maintained life.
Oystein Ore, "Number Theory its History" reported RMP 79 in 1948, as well as readable aspects of fragments of ancient Egyptian number theory that were available in 1948.
In the last 60 years additional fragments have been decoded, returning 2/n table methods and other big chunks of Middle Kingdom proto-number theory.
The history of mathematics often offers a meta view of life in ways that the written word only achieves from time to time.
Best Regards,
Milo Gardner -
Message 3
Posted by Milo Gardner (U14346264) on Saturday, 27th February 2010
For those Ö÷²¥´óÐã message board members that enjoy geometry, a KP volume of a cyclinder, with radius 12 cubits, and height of 8 cubits , reported as 1365 1/3 KHAR will be posted here in the next week or so.
At this point, for those wishing to participate, or comment on some level, please read the problem, per Clagett
or some another quality transliteration.
In addition, understand a few ancient meta details of Egyptian geometry reported on Planetmath per:
by working RMP 53, RMP 54 and/or RMP 55 cited by Clagett or another quality transliterated source, as Ahmes might have done.
Best Regards,
Milo Gardner -
Message 4
Posted by Milo Gardner (U14346264) on Saturday, 27th February 2010
For anyone wishing to participate in this Kahun Papyrus Egyptian geometry discussion, at some point would you cite your qualifications by providing a link.
My qualifications are found on Wikipedia: per
Best Regards to all,
Milo Gardner -
Message 5
Posted by Milo Gardner (U14346264) on Monday, 1st March 2010
A four level review of the Kahun Papyrus calculation reported 1365 1/3 khar as the volume of a diameter of 12 cubit and height of 8 cubit cylinder.
1. Level one reports a modern calculation that approximated pi to the tradition 256/81 valuer, per:
a. radius 6 cubits, squared = 36 cubits^2
b. pi = 2581/81 x 36 cubits^2 = 113 7/9 cubits^2
c. 113 7/9 cubits^2 x 8 cubits = 910 2/9 cubits^3
d. 910 2/9 cubits^2 x 3/2 = 1365 1/3 khar
that validates the KP scribe's ancient calculation.
2. Level 2
The KP scribe reported the value of 12 fowls in terms of set-ducks paid to him in the following problem by:
a. 3 re-geese unit value 8 set-ducks = 24
b. 3 terp-geese unit value 4 set-ducks = 12
c. 3 Dj. Cranes unit value 2 set-ducks = 6
d. 3 set-duck unit value 1 set-duck = 3
total value 45 set-ducks. Not calculated, but included in the valuation of 12 - 1 = 11
with 100 - 45 = 55, cited 55/11 as the total value as 5 times the value of one set-duck.
The RMP includes a more complex Egyptian bird-feeding rate problem:
that links MK bird valuations to hekat measurements in a manner that shows Egyptian fraction arithmetic was primarily focused upon economic issues a point discussed on Planetmath per:
3. Level 3 infers that Ahmes, recorded in RMP 38, would have known that the KP scribe over estimated the grain in the cylinder by over 9 khar, since:
a. 36 cubit^2 x 22/7 = 113 1/7 cubit^2
b. 113 1/3 cubit^2 x 8 = 905 1/7 cubit^3
c. 905 cubit^3 x 3/2 = 1356 3/14 khar.
differences (1365 1/3 - 1356 3/14) = 9 5/42 khar
4. Level four jumps forward to Greek times
reporting: Plato who spoke of mathematics by:
"How do you mean?
I mean, as I was saying, that arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument. You know how steadily the masters of the art repel and ridicule any one who attempts to divide absolute unity when he is calculating, and if you divide, they multiply, taking care that one shall continue one and not become lost in fractions.
That is very true.
Now, suppose a person were to say to them: O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there is a unity such as you demand, and each unit is equal, invariable, indivisible, -what would they answer? "
from Chapter 7. "The Republic" (Jowell translation)."
and the work of Archimedes per:
that includes Archimedes showing that pi was an irrational number with well-defined rational number limits far smaller than Egyptians had recorded by 256/81 and 22/7 approximations.
In summary, the Kahun Papyrus calculation of 1365 1/2 khar would likely have been corrected by Ahmes with pi set at 22/7, 175 years later, and by improved by Archimedes by mathematics that includes calculus, with Egyptian and Greek businessmen freed from the astract number of Egyptians.
Comments, from anyone in the Ö÷²¥´óÐã message board community?
Best Regards to all,
Milo Gardner -
Message 6
Posted by Wyldeboar (U11225571) on Monday, 1st March 2010
Did they have Tumbleweeds in Ancient Egypt too????
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Message 7
Posted by Milo Gardner (U14346264) on Wednesday, 3rd March 2010
Concerning the KP 2/n table, and the RMP 2/n table, two differences were reported.
First, the KP 2/3 conversion did not follow the RMP 1/3 + 1/3 method, nor the EMLR's 1/2 = 1/6 + 1/6 method. Only the proof 2/3 of 3 = 2 was cited.
The KP listed conversions of 2/5, 2/7, 2/9, 2/11, 2/13, 2/16, 2/17, 1/19, 2/21, reporting the same unit fraction series as Ahmes wrote down 175 years later.
The KP showed no calculations, the same situation that was reported in the RMP.
Second, the KP provided a proof for each of its conversions, a set of facts that were not reported in the RMP. For example the KP reported:
2/7 = 1/4 + 1/28
was proven by:
a. 1/4 of 7 = 1 + 1/2 + 1/4
b. 1/28 of 6 = 1/4
c. 1 + 1/2 + 1/4 + 1/4 = 2
and so forth. -
Message 8
Posted by Mike Alexander (U1706714) on Tuesday, 9th March 2010
Interesting that they seem to have had a grasp of GP methods. APs are far more intuitively obvious - you can virtually see the n/2(2a + (n-1)d) formula just by laying out measuring rods in interlocking staircases (painting the ends of each rod to length "a" helps). But the GP formula is much less obvious, so it's very clever to have got there without modern algebra.
With modern algebra, it's fairly easy to prove, once you know the trick of multiplying the series (1 + r + r^2 + r^3 .... + r^(n-1)) by (1-r). This gives 1 -r +r -r^2 + r^2 ... -r^(n-1) + r^(n-1) - r^n. Clearly all the terms in the middle cancel, giving 1-r^n. So series sum * (1-r) = 1-r^n, ie sum = (1-r^n)/(1-r).
In cases where the first term is not 1 but some arbitrary value a, you just scale, so the sum = a(1-r^n)/(1-r).
If r < 1, the series is converging, so for an infinite series r^x tends to 0, thus the infinite series can be summed as a/(1-r).
Ah, it's all coming back to me, 20+ years after school! -
Message 9
Posted by Milo Gardner (U14346264) on Tuesday, 9th March 2010
Mike,
Thank you for the comments. Ancient Egypt did substitute in their formulas much as modern algebra does today. Victor Katz, editor, and Annette Imhausen, Egyptologist, recently published a book on ancient mathematics:
that slowly opened doors to ancient Egyptian math, oddly missing the Akhmim Wooden Tablet's (64/64) hekat unity door, used 29 times in RMP 82, and 320 ro, another hekat unity, used in RMP 35-38, and RMP 66 per a Planetmath entry:
One ancient to modern algebra difference is that Ahmes and the KP scribe wrote 2/n tables and focused upon on unities, and optimizing 'red auxiliary numbers (explained in RMP 36), returning problem calculations to one, a common starting ancient point, in an array of methods.
Modern algebraic proofs do not return answers to unity. That is, unities are not common starting points, a valid point that you indirectly cited by your modern analysis of the ancient KP 2/n table.
Thank you again for re-vitalizing your algebraic skills. Commenting on ancient Egyptian math tends to do that. The same thing happened to me a little over 20 years ago.
Milo Gardner -
Message 10
Posted by Milo Gardner (U14346264) on Friday, 12th March 2010
Another interesting aspect of ancient Egyptian math story not covered by Katz and Imhausen was the over million size of integers and quotients. The largest Egyptian rational numbers that I have seen appear in the Kahun Papyrus text quoting a Planetmath encyclopedia entry:
"G. The Kahun Papyrus contains other numerical information. One data set, eight lines of large quotient and remainder rational numbers was preceded with several lines of missing data. The fragments of data may be related to a calculation ending with 1/12. The historical context of the data, cited below is unclear.
1. 925157 + 1/3
2. 708453 + 1/3
3. 709533 + 1/3
4. 508098 + 2/3 + 1/8 + 1/16
5. 407042 + 2/3
6. 440003 + 1/6
7. 209200
8. 1/12
As a wild guess, a prime number analysis may offer a few hints to decoding aspects of the ancient data:
1. 2775460 divided by 3, factors (2, 2, 5, 73, 1901) divided by 3
2. 21283600 divided by 3, factors 2, 2, 2 ,5 ,13, 4093) divided by 3
3. 2128600 divided by 3, factors (2, 2, 2, 5, 5, 29, 367) divided by 3
4. 508098 + 41/48 = 243887050 divided by 48, or
121943525 divided by 24, factors (5, 5, 11, 443431) divided by 24
5. 1221128 divided by 3, factors (2, 2, 2, 152641) divided by 3
6. 2640019 divided by 6, factors (61, 113, 383) divided by 6
7. factors (2, 2, 2, 2, 5, 5, 523)
8. 209200 times 12 = 25700900
The data has been converted to rational numbers and factored to consider astronomical cycles as a possible ancient context. More on this data when, or ever, reliable data becomes available."
Anyone know of bigger numbers used by ancient Egyptians? Better yet, can anyone suggest alternative decoding paths that do no include astronomy? Archimedes counted the sands on the seashore. Did Egyptians make that type of effort?
Best Regards,
Milo Gardner -
Message 11
Posted by Milo Gardner (U14346264) on Saturday, 13th March 2010
To read an entire Kahun Papyrus transliteration, Clagett's 1999 summarized several hieratic texts:
7qKr36D8glkPCy8pUO7Y&hl=en&ei=Dri8SoPfMI7uswOQgZndBQ&sa=X&oi=book_result&ct=result&resnum=3#v=onepage&q=&f=false
To focus on Kahun Papyrus issues click on the Planetmath encyclopedia entry:
updated to cite 14 lines of text were missing in the large number problem. The KP scribe labeled the extant lines 15, 16, 17, 18, 19, 20, 21 and 22. -
Message 12
Posted by Milo Gardner (U14346264) on Saturday, 13th March 2010
OOOPs, Clagett's 1999 Ancient Egyptian Science, VOL III found by a google search:
offers transliterations of the RMP, KP, and other hieratic texts discussed within outdated 20th century 'snap shots of the past'. Clagett offered traditional incomplete and distracting comments often unfairly parsing ancient scribal meta mathematics, as the ancient scribes thought and wrote about their era's mathematics.
Reconstructing the Middle Kingdom era's math requires scribal shorthand styles, that omitted initial and intermediate steps to be replaced. Tanja Pemmerening, a Charles U., Prague, graduate student began to replace missing steps in 2001 by showing that the Akhmim Wooden Tablet, oddly not discussed by Clagett, returned five divisions of a hekat unity, 64/64, to unity by multiplying two-part answers by the initial divisor, following arithmetic steps that indirectly parallel modern arithmetic methodologies. -
Message 13
Posted by Milo Gardner (U14346264) on Saturday, 13th March 2010
OOOPs, one more time. Hana Vymazalova broke the AKHMIM WOODEN TABLET hekat unity code, (64/64) in 2002, the year that Tanja Pemmerening broke the code of the dja, 1/64 of a hekat, recorded in many of 1,000 medical prescriptions.
Sadly. my view is that Vymazalova and Pemmerening's voices are faintly heard in academia, a situation that may be slowly improving.
During the early part of the 21st century, Babylonian scholars tend to be loudly heard. For example, Joran Friberg reported:
that during the Demotic era, roughly from 1500 BCE to 300 BCE, Babylonian and Egyptian mathematics came in close contact. Friberg, Jens Hoyrup, Eleanor Robson, Ö÷²¥´óÐã's RMP expert, and others report in their own styles, after over 1,000 years of contact, using the P. Cairo as a guide, Babylonian mathematics dominated Egyptian mathematics. But are Friberg, et al, correct?
Outside of academia Babylonian voices tend to be muted, with little beyond the P. Cairo mentioned in sufficient detail. Algorithms tend to be the motif of Robson and Imhausen, offering little of substance to refute the contents of the RMP reported on Wikipedia and elsewhere.
Ancient Egyptian scribal voices of Ahmes and his contemporaries tend to dominate the internet via Wikipedia, Planetmath, and blogs per:
by concluding, for example, that "Egyptian unit fractions were consistent from 2,050 BCE to 1454 AD, and last used by Arabs in 1637 AD. Unit fraction arithmetic required finite quotients and remainders. Missing initial and intermediate ancient data was incorrectly read by 20th century scholars. The missing data was validated as scaling (n/p)*(m/m) to mn/mp data by divisors of mp that best summed to mn. Arabs and medieval scribes (Fibonacci) scaled (n/p - 1/m) =(mn - p)/mp to unit fraction series by a related ancient method", points not considered in any meaningful way by Friberg, et al.
In conclusion, an open debate on the contents of the hieratic texts, and later cultural influences accepted in the Greek, Hellene, Arab and medieval worlds may be nearing. I look forward to that day. -
Message 14
Posted by Milo Gardner (U14346264) on Saturday, 13th March 2010
Looking forward to the day when Babylonian scholars are challenged on current minimalist views of Egyptian mathematics, Harvard recently made a move in that direction by appointing its first Egyptology in 68 years:
Harvard To Acquire First Egyptology Professor in Decades | The Harvard Crimson
www.thecrimson.com
After years dedicated to shedding light on the work of the late Harvard Egyptology Professor George A. Reisner, Class of 1889, Peter D. Manuelian �81 will become the first egyptology professor at Harvard since his predecessor�s death 68 years ago.
The next move could offer Egyptian mathematics at Harvard. Currently, Egyptology courses in European and US universities offer reading and writing of hieroglyphic and hieratic script but virtually no hieratic arithmetic (taught by Ahmes or the hieratic texts).
Reading and writing hieratic 2/n tables, the beginning paragraphs of the Kahun and RMP, the EMLR and its 26 lines of text need to be parsed by least common multiples (LCM)s. Understanding the ancient LCM scaling method, as introduced by Ahmes in RMP 14-19, a modern student could easily advance to RMP 36 'red auxiliary number' methods per:
as ancient students followed its well worn study path.
At this point the students of Babylonian and Egyptian math history could be asked to answer the question, Why Study Egyptian fractions:
Any suggested debate between the current array of Babylonian and Egyptian math professors, that think little of Egyptian mathematics, would be brief, with no one required to show up. Future College course outlines of Egyptian math would be handed out, ending the debate.
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