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Egyptian Mathematical Leather Roll
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Message 1.Â
Posted by Milo Gardner (U14346264) on Friday, 19th February 2010
This text, deeded to the British Museum in 1863, and not unrolled until 1927, was not fairly read until 2002 per:
Times are changing in Egyptology circles, slowly for sure, but any change for the better is greatly appreciated.
Has anyone seen this report before? A summary of the 2002 paper was also published in 2005 by a non-Western history of medicine and mathematics encyclopedia. -
Message 2
Posted by somewhatsilly (U14315357) on Friday, 19th February 2010
Hi Milo,
For those amongst us whose mathematical education ended a very long time ago, perhaps you could either give a simple summary or direct us to a website for the mathematically illiterate but interested.
thanks.
Ferval -
Message 3
Posted by somewhatsilly (U14315357) on Friday, 19th February 2010
I've realised I've just demonstrated my failings in this discipline. I did of course mean innumerate!
Ferval -
Message 4
Posted by Milo Gardner (U14346264) on Saturday, 20th February 2010
Beyond the Planetmath link, Wikipedia covers the EMLRs' contents well, per:
My math education was obtained in the early 1960s, and laid dormant for 30 years. Hey, with a little work and a few fresh ideas, all the best ideas come back, dropping the silly ones. -
Message 5
Posted by Milo Gardner (U14346264) on Saturday, 20th February 2010
Hi Ferval,
Innumeracy can be solved by taking meta points of view. Myopia is the illness that fosters innumeracy
and its many siblings. -
Message 6
Posted by TwinProbe (U4077936) on Saturday, 20th February 2010
Hi Milo
I mean no discourtesy but with a new poster it is sometimes extraordinarily difficult to distinguish between a fresh and innovative thinker with something important to say, and an individual who having built a castle in the air is now trying to live in one of the attic bedrooms.
As well as a liking for history and archaeology many of us bring other skills to this message board like geology, computer science, linguistics and indeed maths. None of us, so far as I am aware, is a retired military cryptographer so please be patient.
May I join Ferval in requesting a simple introductory account of your work and what you are setting out to prove. Please try not to be gnomic and I will do my best to follow your reasoning.
With thanks, TP -
Message 7
Posted by Mutatis_Mutandis (U8620894) on Sunday, 21st February 2010
The text is confusing, but the gist of the problem is not that complex. Egyptian scribes had to find a mathematical notation for the result of a non-integer division, in other words a method to represent numbers smaller than 1.
Our own convention is to use the decimal point, which actually is a shorthand for a series of fractions, with the rule that the denominators are a series of powers of 10:
0.325 = 3/10 + 2/100 + 5/1000
Egyptian scribes used a notation that is still known as "Egyptian Fraction". It also is a series of fractions, but the rule is that the numerator of every fraction must be 1. Resulting in a notation for the same number of:
1/5 + 1/8
Unfortunately for them, while the Egyptian notation is neat, mathematically it is rather inconvenient. It can represent any fraction, but it is not that easy to calculate for an arbitrary division, and several different representations are possible for the same number. So scribes used tables and a set of rules, of which we know only a few, to execute a division.
Some of the tables are a given: The Rhind papyrus includes a table for divisions 2/p where p is a prime number; and the leather roll has a table for 1/p. The difficulty is in reconstructing the rules the Egyptians used when they employed these tables to perform divisions. -
Message 8
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
Dear TP,
Possibly this post will provide a little clarity:
This month's Â鶹ԼÅÄ review of the RMP unfairly excluded non-geometry texts, such as the Egyptian Mathematical Leather Roll (EMLR):
a well known scribal students' introduction to ancient 2/n tables, and ancient number theory.
It is highly recommended Â鶹ԼÅÄ interviewers request that British Museum staff fully research and fairly report the contents of the EMLR along with the contents of the RMP and its 2/n table which reveals ancient number theory.
The Kahun Papyrus and the RMP 2/n tables report the first known form of the fundamental theorem of arithmetic (FTA). The FTA factored rational numbers into prime numbers, as needed, defining the first rigorous finite arithmetic.
In the Old Kingdom a binary infinite series truncated conversions of rational numbers to 6-terms. For example, one (1) was written in the Horus-Eye system by:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64
with 1/64 thrown away in an awkward round-off system.
To always correct for Horus-Eye rational number round-off errors the EMLR, 2/n tables, and Middle Kingdom problems and proofs (i.e. 87 RMP problems) selected an LCM (m/m) that scaled n/p such that:
n/p = n/p(m/m)= mn/mp
allowed the best divisors of denominators mp to be summed to numerators mn, in optimized, but not optimal ways.
In volume weights and measures systems, the Old Kingdom hekat round off errors were corrected two ways:
a. The dja, a 1/64 unit, written as 64/n, was reported by Tanja Pemmerening in 2002 and 2005 as a 'healing' unit, exactly eliminating the Old Kingdom round-off error.
b. Since the Old Kingdom hekat definition used pi as 256/81, Ahmes in RMP 38 seems to report that 22/7 was used to reduce granary inventory losses that were caused by the 256/81 approximation.
In the case of the EMLR conversions of 1/p and 1/2n to non-optimized unit fraction series were practiced from several points of view. For example, 1/8 was converted by three scaling factors, the largest being (25/25) a LCM that was also used to convert 1/16. -
Message 9
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
Hi TP,
Thank you for citing the interdisciplinary aspect of this forum. To generally decode and translate any ancient text, as written by an ancient scribe, into modern languages, several disciples must be consulted.
First is the philologist, that transliterates the ancient script. In the ancient Egyptian case, facts concerning hieroglyphic and hieratic scripts must fairly report the raw data.
Concerning mathematical texts, the ciphered numbers must be decoded by professional cryptanalysts and mathematicians. Rarely have cryptanalysts, a skill I learned over 50 years ago, been asked to assist in translating transliterated texts to ancient forms of arithmetic, algebra, arithmetic progressions, geometry and other ancient mathematical patterns.
Of course, mathematicians skilled in the history of mathematics must assist cryptanalysts. Trained as a high math teacher, taking as an elective, the History of Number Theory, authored by Oystein Ore, greatly assisted my thinking of the EMLR arithmetic patterns. The EMLR translation was the first ancient text that I published in 2002.
Wikipedia offers a few other details of my qualifications to discuss the EMLR and ancient Egyptian mathematical texts in my Wiki bio:
Thanks again for taking a deeper look at ancient Egyptian number theory, a form of mathematics that was continuously used for 3,600 years, reduced in historical importance by the rise of the 400 year old base 10 decimal system, built on an algorithm. -
Message 10
Posted by TwinProbe (U4077936) on Sunday, 21st February 2010
Hi Mutatis_Mutandis
Many thanks, that is quite clear. In fact better than clear since I have never appreciated that decimals really represent a series of fractions. I think I see the system on which the 'Egyptian Fraction' is based.
When you say 'arbitrary division' you mean, I take it, an actual example of an unprepared division I might give an ancient Egyptian mathematician eg divide 55 by 7?
I understand what you mean by 1/p and 2/p. I have never asked myself which civilisation first developed the concept of prime numbers, although I did know that the Egyptians appreciated that a triangle with sides 3,4,5 units was right-angled.
Thanks again,
TP -
Message 11
Posted by TwinProbe (U4077936) on Sunday, 21st February 2010
Hi Milo
Thanks for getting back to me. I think I understand now. You explain very clearly the skills needed to comprehend an ancient scientific or mathematical text.
Are you interested in ancient history in general? If you are we don't seem to have discussed ancient science recently.
Best wishes,
TP -
Message 12
Posted by somewhatsilly (U14315357) on Sunday, 21st February 2010
Hi Milo,
Thanks for your additional posts. Along with M M you have helped me to at least a glimmer of understanding. I must add this to the ever lengthening list of things I'd like to find out more about and I'll explore the web site more fully.
I'm interested in your comments about the BM. Was it the '100 objects' programme you felt was so inadequate or do you have a wider objection to their treatment of these texts? As a specialist and enthusiast I'm sure you would find a brief summary such as that frustrating but I've found that other broadcasts in the series, when dealing with topics I know more about, both fair and a reasonable reflection of current understanding. Perhaps you could expand on your point of view and
where, and why, you feel they're going wrong?
Regards
Ferval -
Message 13
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
TP,
Egyptians did not compute with unit fraction series. Vulgar fractions were used for intermediate steps. For example, RMP 38 multiplied 320 ro, a scaled unity, by 7/22 and found 101 9/11, written out as a unit fraction series.
Ahmes proved that 101 9/11 was correct by multiplying by 22/7, obtaining 320 ro. QED
That is, scribes defined multiplication and division as inverse operation as we do today, a surprise to philologists who long had mis-reported Egyptian division as following 'false position', an 800 AD Arabic method of finding roots. -
Message 14
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
TP,
To divide 55 by 7, a quotient 7 and remainder 6/7 were written out in unit fraction series. The RMP 2/n table shows one method to convert 2/7 by writing
2/7*(4/4) = 8/28 = (7 + 1)/28 = 1/4 + 1/28
meaning that:
6/7*(4/4) = 24/28 = (14 + 7 + 2 + 1)/28 =
1/2 + 1/4 + 1/14 + 1/28
In summary,
55/7 = 7 + 1/2 + 1/4 + 1/14 + 1/28
was written from right to left, omitting the + sign and using Greek notation for a unit fraction
n = n' (the Egyptian number symbols drew a line of the ciphered letter sound, an awkward computer notation)
28' 14' 4' 2' 7 -
Message 15
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
To TP and Ferval,
I am interested in ancient astronomy and many things ancient. The history of science and the history of economic thought are two major threads.
Of course, Egyptian astronomy has long been considered an unsolved topic on several levels. A friend and I have read over Clagett and his 1999 references, over a three month period, and we been unable to parse an ancient Egyptian lunar calendar.
We were hoping to find a 135 lunar month calendar during the 1,000 BCE period, one that connected to a Canary Island 270 lunar month calendar (placed there by exiled Libyan/Phonecian Pharaohs) that left mummies from 900 BCE to 400 AD.
The larger story of ancient trips to the New World, ending 1,000 years before Columbus used the Canary Island route, following the same latitude, is of interest. The route was partially documented by an Acano cycle,
a red and black colored lunar calendar painted in Canary Island caves, decoded by Dr. Barrios 15 years ago.
The Mayan 405 lunar month calendar also used red and black colors but not in a clear way that added 135 months to improve its calendars, connected to the Canary Islands. Hopefully someone will find a Mayan cave painted as Canary Island caves have been decoded.
The history of economic thread is also found on my Planetmath site under the name "Economic context of Egyptian fractions",
This is a topic that shows that the Egyptian economy was decentralized by 2050 BCE by creating a commodity based monetary system. The monetary system used a double entry accounting system, scaling many items as our coins,and larger bills are considered today, with zero accounts being denoted by the same hieratic word.
Absentee landlords paid wages in hekat units, paying lesser amounts during heavy flood years, ranging from four to eight hekats per worker.
Let me stop at this point, having reported more than I had expected.
Best Regards,
Milo -
Message 16
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
Ferval,
Commenting on:
I'm interested in your comments about the BM. Was it the '100 objects' programme you felt was so inadequate or do you have a wider objection to their treatment of these texts?
***My complaint is with the Rhind Mathematical Papyrus program. It tended to continue the 1920's supposition that the Middle Kingdom went into intellectual decline, compared to Old Kingdom mathematics.
Reading the Middle Kingdom mathematical texts, beginning with number theory, skipping over the 'strawman' Greek geometry point, the Middle Kingdom was mathematically advanced compared to the Old Kingdom.
That is, there was no intellectual decline, despite yelling at the roof tops by Biblical and other main stream scholars.
For example, Otto Neuegebaur, writing in "Exact Sciences of Antiquity" expressed a personal love for Babylonian culture by disrespecting Egyptian fractions, especially Ahmes' 2/n table. Neugebauer's misleading "Egyptian Middle Kingdom intellectual decline" point was oddly continued by a Â鶹ԼÅÄ contributor, Eleanor Robson, commenting in her own style. Robson is an excellent Babylonian scholar. She loves Babylonian and Egyptian cursive algorithms, but she does not understand or consider Egyptian finite arithmetic that was built upon the fundamental theorem of arithmetic in a manner that solved a serious set of Old Kingdom round-off problems.
***
As a specialist and enthusiast I'm sure you would find a brief summary such as that frustrating but I've found that other broadcasts in the series, when dealing with topics I know more about, both fair and a reasonable reflection of current understanding. Perhaps you could expand on your point of view and where, and why, you feel they're going wrong?
*** I am not prepared to comment on the remaining 99 broadcasts. I have only glanced at the 'flood text', a biblical issue that goes beyond my reading lists.
Hence, please consider my specialty to begin with military cryptanalytics, used by Ventris and Chadwick to decode Linear B, and by others to prove that Linear A can not be broken with available texts. My specialty is tempered by a life-long love for ancient astronomy, ancient science and the history of economic thought.
In conclusion, whatever caused the end of the Old Kingdom, the Middle Kingdom worked within a decentralized economy that was structure by a well defined finite arithmetic to scale commodity payments. Commodities, based on hekat grain units, allowed absentee landlords earnings, workers pay and a 10% profits tax to be paid to Pharaoh. Pharaoh paid an administrative body that worked to resolve local economic disputes, and so forth. Double entry accounting was also a well known Middle Kingdom innovation that often is pushed aside as an anomaly (it was not).
Best Regards,
Milo -
Message 17
Posted by somewhatsilly (U14315357) on Sunday, 21st February 2010
Milo
Thanks for that, It's given me a few things to think about and I'll listen to the programme again with more care.
I see you are a ancient astronomy buff so perhaps you can help me with one of the points on the link you posted. It refers to Göbekli Tepe as being a possible astronomical calendar, do you have a source for that as I've never come across that hypothesis from Schmidt or Hodder? I could well be behind the times there, it's a couple of years since I heard Hodder talking about it and I haven't read the most recent excavation reports from the site, only fairly brief outlines.
Regards
Ferval -
Message 18
Posted by Milo Gardner (U14346264) on Sunday, 21st February 2010
Ferval,
The Göbekli Tepe point was made by a Swiss friend of mine. His analysis has not been confirmed, so please disregard it if you wish.
My interest is with the ancient 99-moon calendar and the sidereal double checking work that improved it after 1100 BC to 135-moons (non-confirmed either), and in Greek/Roman times to 235-moons (validated over and over again), the later point confusing other ancient and medieval lunar calendar improvements such as the Hebrew-Islamic 270-moon calendar used on the Canary Islands, and the Mayan 405-moon lunar calendar used a short boat ride away from the Canary Islands -- as Comalcalco adobe temples bricks record 10% of the brick authors being non-Mesoamerican.
Comalcalco, a site in northern Tobasco state, near the Veracruz state line, was visited in 1991. An on-site museum adobe brick displayed a Phoencian calendar date (I forgot the date, but it was before Carthage was over run by Rome). -
Message 19
Posted by Milo Gardner (U14346264) on Tuesday, 23rd February 2010
This month's Â鶹ԼÅÄ review of the RMP was myopic, taking a small slice of a large loaf, suggesting that a whole loaf of inforrmation had been digested.
The fairly translate the RMP its sibling British Museum EMLR loaf of bread:
must be digested first. The Wikipedia EMLR summary cites a half-dozen critical slices that must be parsed before the RMP 2/n table can be rigorously and fairly placed on a plate for dining per:
"The Egyptian Mathematical Leather Roll (also referred to as EMLR) was a 10" x 17" leather roll purchased by Alexander Henry Rhind in 1858. It was sent to the British Museum in 1864, along with the Rhind Mathematical Papyrus but the former was not chemically softened and unrolled until 1927 (Scott, Hall 1927).
The writing consists of Middle Kingdom hieratic characters written right to left. There are 26 rational numbers listed. Each rational number is followed by its equivalent Egyptian fraction series. There were ten Eye of Horus numbers: 1/2, 1/4 (twice), 1/8 (thrice), 1/16 (twice), 1/32, 1/64 converted to Egyptian fractions. There were seven other even rational numbers converted to Egyptian fractions: 1/6 (twice–but wrong once), 1/10, 1/12, 1/14, 1/20 and 1/30. Finally, there were nine odd rational numbers converted to Egyptian fractions: 2/3, 1/3 (twice), 1/5, 1/7, 1/9, 1/11, 1/13 and 1/15, training patterns for scribal students to learn the RMP 2/n table method.
The British Museum examiners found no introduction or description to how or why the equivalent unit fraction series were computed (Gillings 1981: 456-457). Equivalent unit fraction series are associated with fractions 1/3, 1/4, 1/8 and 1/16. There was a trivial error associated with the final 1/15 unit fraction series. The 1/15 series was listed as equal to 1/6. Another serious error was associated with 1/13, an issue that the 1927 examiners did not attempt to resolve.
The British Museum Quarterly (1927) naively reported the chemical analysis to be more interesting than the document's additive contents. One minimalist reported that the Horus-Eye binary fraction system was superior to the Egyptian fraction notation.
One review includes the Middle Kingdom Egyptian fraction conversions of binary fractions corrected an Eye of Horus numeration error. The Old Kingdom Horus-Eye arithmetic was rounded-off to 6-term binary fraction series, throwing away 1/64 units in an infinite series numeration system. Horus-Eye fractions are related to modern decimal algorithms, with both systems rounding off, (Ore 1944: 331-325). Note that the Horus-Eye definition of one (1): 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + … dropped off the last term 1/64th, (Gillings 1972: 210). Modern decimals' round-off rules are closely related the Old Kingdom's round-off methodologies. The Middle Kingdom correction converted rational numbers to optimized finite series that generally eliminated the traditional Eye of Horus round-off errors.
Preceding the RMP 2/n table by 200 years the EMLR used red auxiliary numbers scaled by least common multiples (LCMs) scaled 26 1/p and 1/pq unit fractions to non-optimal Egyptian fraction series using a red auxiliary method that Ahmes described in RMP 36. The EMLR LCM scaled 1/p and 1/pq by Egyptian multiplication and division methods that allowed additive red auxiliary numerators to define final unit fraction answers. In total 22 unique unit fractions were converted by eight multiples (2, 3, 4, 5, 6, 7, 10, and 25), written as 2/2, 3/3, 4/4, 5/5, 6/6, 10/10 and 25/25, Egyptian fractions represented a solution to the Eye of Horus round-off problem by converting any rational number to an exact unit fraction series by selecting LCMs. The RMP 2/n table converted 51 rational numbers by selecting 14 optimized LCMs.
Summary: Middle Kingdom Egyptian arithmetic was written in non-optimal and optimal unit fraction series in a finite numeration. The Middle Kingdom finite system corrected Old Kingdom infinite series round-off errors. The Old Kingdom Eye of Horus numeration system had rounded-off a 1/64 unit. Early 1900s researchers minimized the EMLR’s significance. The EMLR, the Kahun Papyrus (KP) 2/n table, Rhind Mathematical Papyrus, and the RMP 2/n table demonstrated that LCMs scaled rational numbers to solvable levels by red auxiliary numbers. The EMLR and 2/n tables used the same LCM method, the EMLR in a non-optimal manner, and in 2/n tables, and in other mathematical texts, in optimized ways. The EMLR used 8 non-optimal LCMs that introduced student scribes to higher uses of Egyptian fraction mathematics reported in all other Egyptian fraction mathematical texts."
To understand each EMLR slice, from LCMs, scaled rational numbers, RMP 36 and optimized red auxiliary numbers, to beginning scribal student's 26 non-optimized EMLR answers, a meta position is required to be taken.
The Â鶹ԼÅÄ and British Museum consultants attempted to read the RMP inside-out, following linguistic practices, rather than taking mathematician outside-in meta points of view, that go beyond each problem being discussed, to include the ancient number theory that structured all ancient Egyptian fraction documents.
Best Regards to all,
Milo Gardner
adding military cryptanaytics as a necessary meta aspect of fairly reading ancient Egyptian texts.
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