Ancient and ArchaeologyÌý permalink
Rhind Mathematical Papyrus
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Message 1.Ìý
Posted by Milo Gardner (U14346264) on Tuesday, 25th May 2010
The Ö÷²¥´óÐã radio program ... the history of the world in 100 objects ... MUDDLED a review of the RMP ... in program #17 ... a Babylonian historian ... that knew nothing of the RMP's deeper contents ... was Ö÷²¥´óÐã's chosen math expert.
Had RMP 69 and Berlin Papyrus:
been discussed on Ö÷²¥´óÐã radio as the Math HistoryList parsed the two topics Ö÷²¥´óÐã would have corrected over 90 years of oversights in misreporting 1650 BCE arithmetic. Peet in 1923 and other British Museum funded scholar suggested that the RMP contained 'single false position', a medieval root finding method, as its primary division method/operation.
I look forward to Ö÷²¥´óÐã radio updating its Feb. 2010 program #17 in the series " A history of the world in 100 objects", and correctly reporting the RMP's two finite math division operations to include
1. hekat, a volume unit's, division method
(64/64)/n = Q/64 + (5R/n)*1/320
Q =quotient and R = remainder, 1/320 = ro
2. A hekat unity stated as 320 ro divided by 7/22 in RMP 38 and 10 hekat 3200 ro by 365. Ahmes inverted the divisors 7/22 and 1/365, to 22/7 and 365 to prove the correctness of the unit fraction answers by a multiplication operation,
thereby correction the 1920's 'single false position' scholar error.
Best Regards,
Milo Gardner -
Message 2
Posted by Milo Gardner (U14346264) on Wednesday, 26th May 2010
The Ö÷²¥´óÐã radio RMP program of Feb. 2010 and British Museum (BM) radio panel should consider discussing a second text owned and misread by the BM for 83 years the Egyptian Mathematical Leather Roll (EMLR):
The 26 line EMLR was a student scribe's introduction to the RMP 2/n Table and Middle Kingdom arithmetic. Ö÷²¥´óÐã's Feb. 2010 Egyptian math expert ... Eleanor Robson ... reported nothing of the red number aspects of scaling 1/p and 1/p to non-optimal unit fraction series, as RMP 36 and RMP 37 clearly defined Ahmes' 1650 BCE 2/n table construction method.
Again, I am appreciative of Ö÷²¥´óÐã radio taking on the deeper aspects of "World History in 100 Objects" and its program #17 ... the RMP ...
Your Ö÷²¥´óÐã radio staff has selected a significant ancient text. I look forward to Ö÷²¥´óÐã staff completing the project as Ahmes, the RM scribe, reported the 2/n table, as decoded by the EMLR, RMP 36 and RMP 37.
Best Regards,
Milo Gardner -
Message 3
Posted by Stoggler (U14387762) on Wednesday, 26th May 2010
By the way, it's spelt
Regards -
Message 4
Posted by Andrew Host (U1683626) on Wednesday, 26th May 2010
Hi Milo,
If you wish to comment directly on the accuracy of this programme it would be best to use the 'Contact Us' link on this page:
These boards are primarily for discussion and debate. If you're finding that other memberes aren't really picking up on this subject you might want to try engaging in the existing debates on these boards and getting to know the regulars a bit.
Cheers
Andrew -
Message 5
Posted by Milo Gardner (U14346264) on Wednesday, 26th May 2010
Hi Andrew,
Thank you for the link. It has been tested. Only my USA ZIP code and out of the UK residence status was accepted. A hyperlink was posted on the page that asked for another ancient item that can be considered as "A History of the World in 100 Objects" topic. When clicked-on the screen returned to the ZIP code out residence questions.
The EMLR link was not posted to the site.
Can you assist? As a Ö÷²¥´óÐã Host, your involvement to achieve this minor positing step would be greatly appreciated.
Best Regards,
Milo Gardner
Sacramento, California -
Message 6
Posted by Andrew Host (U1683626) on Thursday, 27th May 2010
Hi Milo,
I'm no expert on how that site is run I'm afraid. I suggest you you email them via the Contact Us link on that page.
Cheers
Andrew -
Message 7
Posted by Stoggler (U14387762) on Thursday, 27th May 2010
By the way, it's spelt programme
¸é±ð²µ²¹°ù»å²õÌý
Oops, sorry Milo. Hadn't realised you were American. In which case, there's nowt wrong with your spelling of programme/program then.
Apologies
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Message 8
Posted by Milo Gardner (U14346264) on Thursday, 27th May 2010
Thanks for the suggestions. Hopefully this email will bring appropriate responses:
Dear Richard Parkinson, and Ö÷²¥´óÐã radio staff
The RMP began with a 2/n table that took up 1/3 of the text. To decode the 2/n table's construction method the Egyptian Mathematical Leather Roll:
and RMP 36 and RMP 37 must be parsed as recorded by Middle Kingdom scribes.
The Ö÷²¥´óÐã radio broadcast of Feb 2010 muddled the 2/n table and the RMP 87 problems that applied 2/n table unit fraction arithmetic operations. Ö÷²¥´óÐã panel members oddly accepted the 1927 additive version of the EMLR rather than the EMLR's 2002 and 2008 updated contents.
Thank you very much for considering to update your web page including a future radio broadcas the RMP to correct Feb. 2020 Ö÷²¥´óÐã program errors.
Best Regards,
Milo Gardner
Sacramento, CA, USA
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Message 9
Posted by Milo Gardner (U14346264) on Wednesday, 30th June 2010
Hi all:
As an alternative to Ö÷²¥´óÐã's misleading 15 minute RMP broadcast ... aired on Feb. 9, 2010 ... a one hour Math 2.0 Webinar is being prepared:
I'll let you folks know how the event is received.
In summary, Fibonacci and Ahmes' rational number method that converted 4/13 to unit fraction series will be discussed. Both scribes began with 1/4. Fibonacci selected 1/18 for a second partition. The two calculations follow:
A. Fibonacci:
1. (4/13 - 1/4) = (16 - 13)/52
2. (3/52 - 1/18)= (54 - 52)/936 = 1/468
3. 4/13 = 1/4 + 1/18 + 1/468
recorded left to right, following Ahmes' direction.
B. Ahmes
1. 4/13 x (4/4) = 16/52
2. finding the divisors of 52, 13, 4, 2 and 1
3. (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52
recorded from right to left, absent the + sign.
Best Regards,
Milo Gardner -
Message 10
Posted by Milo Gardner (U14346264) on Monday, 5th July 2010
Dear forum members, the Math forum discusses RMP 38 in a manner that may be of interest per:
the details follow:
Thank you for the comment. RMP problems must be read considering all scribal details. Truncating RMP 38 data, for example, omitting a discussion of Ahmes' inverse aspect of scribal division and multiplication operations, throws a beautiful baby out with the bath water.
An Ahmes blog:
reports a vivid RMP 38 context: "In RMP 38 two rational numbers, (35/11)/10 = 35/110 = 7/22, were multiplied 320, by a doubling ' method citing:
1. Initial calculation
(320 ro)*(35/11) = (320 ro)*(2/3 + 1/3 + 1/6 + 1/11 + 1/22 + 1/66)/10 = 101 + 9/11 ro
2. Proof
(101 9/11 ro) was multiplied by 22/7, and returned one hekat , 320 ro.
This class of hekat calculation infers that the traditional Old Kingdom pi value of 256/81 was corrected by considering : " ... that (7/22) and (22/7) were shown and proved to be inverses, and that the AE scribes were skilled and aware of the natural inverse operations of multiplication and division. In effect, the AE were adept at finding reciprocals" (Bruce Friedman)!"
On a deeper level, modern scholars must discuss 320 times 7/22 = 101 9/11, at some point, by showing how and why 9/11 was converted to a unit fraction series. Omitting fundamental arithmetic discussions of Ahmes' proof that 101 + 9/11 was converted to a unit fraction series in the context of a 2/n table method throws a much larger baby out with RMP 38 bath water.
RMP 38 is a wonderful example of scribal arithmetic. Reading RMP 36 in terms of Ahmes' 2/n table method shows that Middle Kingdom scribes scaled n/p by an LCM m to mn/mp. When possible mn/mp was written as a unit fraction series by selecting the divisors of denominator (GCD) mp that summed to numerator mn.
In the 101 + 9/11 case
101 remained as a quotient, and
9/11 was scaled to LCM 6 such that
54/66 selected divisors of 66: 44, 33, 22, 11, 6, 3, 2, 1 that summed to numerator 54 by
(44 + 6 + 3 + 1)/66 = 2/3 + 1/11 + 1/22 + 1/66
(101 + 2/3 + 1/11 + 1/22 + 1/66)ro = (101 + 9/11)ro
To ponder additional Ahmes arithmetic details a Math 2.0 Webinar will focus on Ahmes and Fibonacci's common proto-number theory rational number conversion methods. The one hour open forum discussion will be available for viewing after July 21, 2010 on:
Best Regards,
Milo Gardner -
Message 11
Posted by Prof Muster (U14387921) on Tuesday, 6th July 2010
DEAR..........
I got Alzheimers disease phase-1 so I can'nt recall specifics, but never the less I hoped to have contributed a generally overlooked aspect problem of ancient numerology,
ABOUT the matter of Obsolete Ancient NUMERALS,
On two occasions in a Conference on Atlantis egyptian measurements, by a Publishinghouse Heliotopos from Greece,
it was perceived that an egyptian Hekat were a half Kilometre
And a Greek Stade were one kilometre.
So when Plato reported in his Atlantis -Dialogue,
about the circumference of the Disk-City Atlantis/Acropolis,
He mentioned a length of 3.000 Stades but a width of 2.000 Stades
Additionally a number of inhabitants as 2.000.000
Obviously, there must be something wrong here,
ASome amateur historians suggested that the Hieroglyph-Sign for 1.000 was reminicent of the Hieroglyph-Sign for 100. Whatever.
But the only-Solution to this problem is much simpeler.
The Crux lays in the matter of misinterpreting Greek obsolete Numerals,
Plato wrote in a Greek dialect the Attic-Greek which was abolished in 331 bc because Macedonia occupied Athens and up til then the Greek Attic Dialect was considdered the Language of an Arch enemy.
In the New Macedonian -Greek Dialect named KOINE,
Corynthian Numerals were employed rather than Athenian, which were slichtly differend enough to confuse later compilers who were NOT familiare with Pre 331 Greek Dialects.
Plato wrote his Treatise/Dialogues around 366 bc.
The Result is that Late greek/Roman compilers were left to be gueaasing whether obsolete Greek numerals were used or modern-Koine ones.
Resulting in grave/great errors in conveying Numerals of Trade distances and in faulty geography-coordinates, not realized by then and later compilers as faulty.
SINCE Plato was supposed to have translated then egyptian measurements into then Greek/Attic ones
later cmpilers forgetting the Alexandrian- Greek macedonian language-transformations
naturally assumed the by a tenfold eggagerated numerals to be correct.
Anyway the Atlantean -numerals and obviously year-accounts too, as Plato rendered them from egyptian originals were sandrdly over-exaggerated by 10.
Now I have not got the acomparitive list of pre and after 331 bc Greek numerals list at hand but
The Attic-Greek sign for TEN was a small sized X
The Macedonians however already used a small sized X for the Pronounciation sign of a soft G
Thus the Koine-Sign for TEN was enlarged from a small sized x to a Capital sized X but still meaning 10 not Hundred
Somewhere along the road the Larger Capital size X was regarded in the Middleages as meaning not the NUMERAL Ten but the Multiplication sign of Ten
and so the Sizes and antiquity of the Atlantian civilisation was cannonized as 10.000 bc whereas
for most mathematicians it was obvious that Plato's account had a decimal-fault but beiing mathemathicians and not historians they could not readily pinpoint why this was the Case.
Atlantis-Canon dictates that it conserns a sunken Oceanic Isle inbetween Two large Continents thus Cuba and Mexico. By Cayce / Donnelly and oceanographer Dr.Greg Little.
The Fake Atlantis sunk in 10.000 bc
The real Ad-Land was only immersed by a Tsunami in 1055 bc in the region of Aden now a desolate deserted place, non reminding anyone of a veritable Eden/ Paradise in its haydays
So the Greek/Roman translators/ mathemathisians, left this matter an unnecessarily confusingly open question
For deviating from the official-fake-Atlantis Canon,
as newly discoverer of the real Atlantis being, Ad-land in Aden, I was banned from 4 Ancient-history Websites in 4 languages, by socalled
Atlantologits who, as reading English only, had no idea of translation faults inbetween Greek and Latin compilers.
Immagine it has become common view that the greek translations of Plato into Latin were faultless resulting in garbled Philosophy overlooking -obsoleted- obsolete numerals !
The more confusing is/was that for sizes or liquid contence differnd numerals were rendered not knowing wether this was for marble/metal coins or replica's or earthen-vases.
The other day Dr. Jan Best and Dr Woodhuizen, ancient-greek, languages professors of Leiden and Amsterdam-university have found out that the TEN known PHAISTOS- Disks,
represented NOT an older variant of linear-B-Greek but LUWISH/ Punician language, and stating tax-returns from Cretan Kings NESTOR and Neleus to their Assyrian overlordship King Salmanasser-3
Since the obsoleted Luwish Language-Studies are an underdog amongst the Antique Asian laguages courses, their Phaistos-Discovery was swept under the carpet as was my Discovery of the real Ad-Lant/Atlantis in Aden.
With the Mysterywriter Agatha Cristy I would have said, Why didn't they ask Evans?
Sincerely,
Prof Muster [Personal details removed by Moderator]
dd. July 2010 -
Message 12
Posted by Milo Gardner (U14346264) on Tuesday, 6th July 2010
Prof. Munster's rambling post can be refuted/clarification by Plato's "The Republic" comments. mentioned from a Planetmath post:
"Plato's Mathematics (Platonism) from Wikipedia
Platonism is a form of realism suggesting that mathematical entities are abstract with no spatiotemporal or causal properties, and eternal and unchanging. This view of numbers is claimed by certain classes of mathematicians in the modern time period. The term Platonism is used because such a view is seen to parallel Plato's belief in a World of Ideas (typified by Allegory of the cave): the everyday world can only imperfectly approximate of an unchanging, ultimate reality, a topic that requests modern scholars to read the ancient math texts as the texts were written in historical time period. Plato's cave and Platonism have meaningful implications, not just superficial connections, because Plato's ideas were preceded and influenced by Pythagoreans of ancient Greece, and Egyptians that used the same form of unit fraction arithmetic, the scaling of rational numbers into exact unit fraction series. Ancient Near East may have believed that the world was, quite literally, generated by numbers, but the numeration systems defined by exact rational numbers provided the desired clarity.
A modern problem of mathematical platonism may precisely suggest where and how does the mathematical entities exist, and how do we know about them? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? One answer might be ultimate ensemble, which is a theory that postulates all structures that exist mathematically also exist physically in their own universe. Yet, to see the world of numbers through ancient eyes, read Plato's Republic:
Plato spoke of the ancient mathematical world in his life-time by asking in the "Republic", 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).
In context, chapter 8, H.D.P. Lee translation, reports the education of a philosopher containing five mathematical disciplines:
1. arithmetic, written in unit fraction 'parts' using theoretical unities and abstract numbers;
2. plane geometry, and,
3. solid geometry consider the line to be segmented into rational and irrational unit 'parts';
4. astronomy;
5. harmonics, that include music.
Translators of the works of Plato rebelled against practical versions of his culture's practical mathematics. However, Plato himself, and Greeks generally copied 1,500 older Egyptian fraction abstract unities, a hekat unity scaled to (64/64) in the Akhmim Wooden Tablet, 320/320 in RMP 35-38, and 53/53 = 2/53 + 3/53 + 5/53 + 15/53 + 28/53 in RMP 36, thereby not getting lost in unit fraction series.
Godel's platonism postulates a mathematical intuition that allows perceptions of mathematical objects, but not the precise mathematical language that describes the object. This view resemblances things Husserl said about mathematics, and supports Kant's proposed idea that mathematics can be analytic-synthetic distinction: conceptual containment (synthetic), A priori, and a posteriori (philosophy). Philip J. Davis and Reuben Hersh suggest in 'The Mathematical Experience' that most mathematicians act as Platonists, even though, if pressed to defend the position carefully, they may retreat from this formalism taken from the philosophy of mathematics.
Mathematicians may infer opinions that amount to nuanced versions of Platonism. These ideas are best described as neo-platonism.
Modern neo-platonic points of view do not provide clear templates to read or decode ancient Greek and Egyptian finite arithmetic texts. To directly decode Greek finite arithmetic, algebra, and geometry in a historical context, multitude must be understood as ancient scribes reported the idea. To Plato, and 1,500 year older Egyptians, multitude referenced rational number n/p scaled by LCM m to mn/pm and/or by (n/p - 1/m) = (mn - p)/mp. Egyptians commonly used the first method that scaled 4/13 to 1/4 + 1/26 + 1/52. Greeks, Arabs, and Fibonacci's Liber Abaci scaled 4/13 to 1/4 + 1/18 + 1/468 by the second method. -
Message 13
Posted by Milo Gardner (U14346264) on Wednesday, 7th July 2010
Dear Professor Muster, and forum members:
A Planetmath "Plato's Mathematics" entry contains links that clarify several Egyptian, Greek, Arab and medieval arithmetic issues per:
Best Regards to all,
Milo Gardner
Sacramento, CA -
Message 14
Posted by Prof Muster (U14387921) on Thursday, 8th July 2010
Dear Sir, Esq,
WARNING for OBSOLETE NUMERICALS reused without knowing whether they are still valid.( more 'Rambling-Notes"?)
Despite your above sermon, I still maintain that while a poetic greek story, translated into plain Latin prose-story like Plato's Atlantis Dialogue, can be readilly wrong translated blurring the original ( no more existing-)Greek text,
Statistically, the same should be true for Plato's philosophy in The Republic
Without hairsplitting, it should be noted that none of Plato's original texts in the Attic Dialect survived due to the fact that after Plato's death The Attic Dialect becanme obsolete,
this fact was unknown to Latin (Litterature-)compilators( also of mathematical charts.)
If the obsoleted numericals ( and sentences)did not fit the new Koine-Greek language the result was edited/adapted in context but thwe true meaning lost without the context
I am not a scholar in ancient greek, but this I do know that 5 Latin translators interpreted Plato's Dialogues and came up with 5 different -aproximate-translations non of them valid,
Example, Phrases like: Atlantis was an Isle positioned in the'Stomach' of the Atlantic Ocean and facing the Mediaterranean-Sea.
Other translations speak of either ' At the Head of the Continent facing the Straits, or even just plain,' inbetween two continents'
Whilst this conserns only a geographic positioning coordinate-indicative,
How wrongly would an itricate philosophical or even mathematical Theme be delivered from 5 translations !
Even in the early Jewish-Bible rendering non of the 4 Vulgate Gospel-Texts nor of the Hebrew-Tora, is original, this could be easily checked in the 19-th century by Bible critics since they had access to the 'original'Greek Septuagint Texts
The Hebrew Tora was only'canonized' in: 600 ad,
Whilst the Vulgate Latin translation of Hieronymus & Augustine was' canonized' in 510 ad.
The oldest surviving Koine-Greek texts with mathematical charts surviving came from the Egyptian City of Oxxyrinchus alkl from after the abolishment of Attic-Dialect by Alexander the Great as a non-grata Attic-enemy-censored language.
Ho megas theos dedoke, Timeo Danaos et donna ferentes. -
Message 15
Posted by Milo Gardner (U14346264) on Friday, 9th July 2010
Myopia is living in a grove of trees without recognizing the character and meaning of surrounding forest(s).
The same is true concerning the unavailability of Greek language and mathematical texts. The language side may be fully lost as the previous post adequately details.
However, the math side of Plato's "The Republic" has been saved by the availability of Egyptian and medieval texts that fully describe the proto-number theory arithmetic that Greeks like Plato and Archimedes used.
Medieval scribes, like Leonardo de Pisa (Fibonacci) demonstrated 124 pages of the 500 page "Liber Abaci" (Sigler 2002 translation), Latin writing Europe's arithmetic book for 250 years by 7 methods (distinctions).
The 7 methods converted rational numbers n/p to exact unit fraction series .. by finding an Least Common Multiple (LCM) m, recorded in a subtraction context as 1/m. The first six rules demonstrated that (n/p - 1/m) = (mn - p) = 1, recorded in a right to left manner as Egyptians and Greeks recorded unit fraction series.
(Fibonacci used two other notations hint of two other Greek arithmetic notations as well ... for brevity .. only the easy notation will be parsed).
Rule 7 was the most difficult case, when remainder (mn -p) was not equal to unity (one). For example 4/13 was scaled by LCM 4 as Sigler clearly explains by:
(4/13 - 1/4) = (16 - 13)/52 = 3/52
with a second LCM 18 completing the conversion by
(3/52 - 1/18) = (54 - 52)/936 = 1/468
such that
4/13 = 1/4 + 1/18 + 1/468
Additional Arab texts that wrote in Islamic/Hindu numerals after 800 AD also followed this system .. since these were Fibonacci's mentors .. defined an 800 AD to 1454 AD abstract arithmetic notation .. consistent for a 654 year period.
The Egyptian Middle Kingdom period, 2050 BCE to 1550 BCE period, a 500 year period defined rational numbers n/p in a manner that was slightly different from Fibonacci's subtraction context. Ahmes, a 1650 BCE scribe, wrote up a 2/n table, and described in RMP 36 and RMP 37. Ahmes' demonstrate method contained two parts, the first may have had six sub-parts (as Gillings discussed), and the second one part (that J.J, Sylvester in 1891 grossly misread .. by misreading Fibonacci's 7th conversion method.
Ahmes' first method converted 4/13 by LCM 4, as Fibonacci used, writing:
4/13 x (4/4) = 16/52
Ahmes' second step inspected the divisors of GCD 52, in this case selecting 13 + 2 + 1, recorded in red such that:
4/13 = (16 + 2 + 1)/52 = 1/4 + 1/26 + 1/52
recorded from right to left, with no (+) sign.
Ahmes described in RMP 31 (28/97) and RMP 36 (30/53) two cases that one LCM would not solve the problem. To convert 28/97 and 30/53 Ahmes applied a 2/n table rule replacing
28/97 = 26/97 + 2/97
(solving 26/97 by LCM 4, and 2/97 by LCM 56)
and
30/53 = 28/53 + 2/53
solving 28/97 by LCM = 4, and 2/53 by LCM 30.
meaning that n/p =(n-2)/p + 2/p was applied as needed!
Summary: the abstract mathematics that Egyptians, Greeks, Arabs an medieval scribes used allows other-wise lost Greek information (connecting 1500 BCE to 800 AD) to be partially recovered ... yes the proto-number theory 'hard drives' of 3,500 + years of Egyptian fraction arithmetic were not erased!
Best Regards,
Milo Gardner -
Message 16
Posted by prof_muster (U14549653) on Monday, 12th July 2010
The Arabs have translated Plato's Atlantis-City numbers/numerology the same 'wrong way' as the Romans did earlier.
Overlooking that they described a geo-metry of a City that is even by modern standards TEN times too large.
On the other hand they calculated Earth's circumperence ( 30.000 miles.)correctly but the distance Earth- Moon by 90.000 miles instead of 400.000. -
Message 17
Posted by Milo Gardner (U14346264) on Monday, 12th July 2010
The facts of both of the previous paragraphs are misleading and/or wrong. For example, the second paragraph reports 400, 000 miles ... actually 400,000 refers to kilometers ...
Orbit and relationship to Earth
Main articles: Orbit of the Moon and Lunar theory
The Earth has a pronounced axial tilt; the Moon's orbit is not perpendicular to Earth's axis, but lies close to the Earth's orbital plane.
The Earth-Moon system, not drawn to scale. The axial arrows give the direction of axial rotation. Note that the axial tilt of the Moon is shown in the same direction as that of the Earth (it appears reduced only due to the similar magnitude of its orbital inclination).
The Moon makes a complete orbit around the Earth with respect to the fixed stars about once every 27.3 days[nb 5] (its sidereal period). However, since the Earth is moving in its orbit about the Sun at the same time, it takes slightly longer for the Moon to show the same phase to Earth, which is about 29.5 days[nb 6] (its synodic period).[12] Unlike most satellites of other planets, the Moon orbits near the ecliptic and not the Earth's equatorial plane. The Moon's orbit is subtly perturbed by the Sun and Earth in many small, complex and interacting ways. For example, the plane of the Moon's orbital motion gradually rotates, which affects other aspects of lunar motion. These follow-on effects are mathematically described by Cassini's laws.[78]
There are several known near-Earth asteroids that have unusual Earth-associated horseshoe orbits: 3753 Cruithne, 54509 YORP, (85770) 1998 UP1 and 2002 AA29.[79] They are co-orbital with the Earth, so that their orbits bring them close to Earth for periods of time but then alter in the long term, and they are not natural satellites of Earth.[80]
Seasons
Although the Moon's minute axial tilt (1.54 degrees) means that seasonal variation is minimal, it is just enough to create a 3-degree variation in the Sun's elevation at the poles, resulting in a very slight "summer" and "winter".[81] From images taken by Clementine in 1994, it appears that four mountainous regions on the rim of Peary crater at the Moon's north pole remain illuminated for the entire lunar day, creating peaks of eternal light. No such regions exist at the south pole. Similarly, there are places that remain in permanent shadow at the bottoms of many polar craters,[59] and these dark craters are extremely cold: Lunar Reconnaissance Orbiter measured the lowest summer temperatures in craters at the southern pole at 35 K (−238 �C),[82] and just 26 K close to the winter solstice in north polar Hermite Crater. -
Message 18
Posted by Milo Gardner (U14346264) on Tuesday, 20th July 2010
For those that are interested in discussing Egyptian math live, log-on:
The time Wed., July 21, at 9:30 EST, USA ... a world clock is available on the above web page.
For those that wish to listen after the fact, recordings will be available on the same site.
Two tours will be made covering one hour. The first tour starts at 1202 AD within one of Fibonacci's three rational number notations. The simplest notation connects to Ahmes's 1650 BCE rational number method reported in the 2/n table.
The second tour begins in the Egyptian Old Kingdom 3,000 BCE and progresses to 2050 BCE. The Old Kingdom arithmetic notation used an algorithm. The 2050 BCE to 1550 BCE Middle Kingdom ciphered numeration system and unit fraction rational number system did not use an algorithm, points that may be of interest to Ö÷²¥´óÐã readers.
Enjoy,
Milo Gardner
Sacramento, CA -
Message 19
Posted by Milo Gardner (U14346264) on Thursday, 22nd July 2010
Last nights's Egyptian math webinar can be accessed by linking to:
The site will request that Elluminate, a JAVA program be downloaded, before the one hour program can be viewed,
Enjoy
Milo Gardner -
Message 20
Posted by Milo Gardner (U14346264) on Friday, 23rd July 2010
(correcting the link)
Last night's Egyptian math webinar can be accessed by linking to:
The site will request that Elluminate, a JAVA program be downloaded, before the one hour program can be viewed,
Enjoy
Milo Gardner -
Message 21
Posted by Milo Gardner (U14346264) on Monday, 9th August 2010
The geometry contained in MMP 10, the RMP and Kahun Papyrus is being discussed on
a Facebook page per:
080310: Skype notes
Review of Leon Cooper: "A new Interpr...etation of Problem 10 of the MMP" [From HM Vol 37]
Leon's definitions/translation first provided in German by Struve and later translated to English by Peet and copied by Leon and further reduced to simplify the chaos by myself.
Now noted exactly in arithmetical agreement with Struve's work and hemispherical determinations.
Line #1. This is a basket calculation
Line #2. The mouth dimension is
Line #3. Four and one half [9/2]
Line #4. Find surface
Line #5. Double 4 1/2 to nine and divide by nine
Line #6. Specifying why doubled = because it's half an egg shaped thing [i. e. a hemisphere is half a sphere]
Line #7. 9 - 1 = 8
Line #8. Find one ninth of 8
Line #9. Result = 2/3 [+] 1/6 [+] 1/18 [vulgar fraction value 8/9; Milo notes scaled by 2/2]
Lines #10 & #11. 8 - 8/9 = 7 [+] 1/9 [vulgar fraction 64/9]
Line #12. 64/9 times 4 [+] 1/2 [vulgar fractions are 64/9 x 9/2]
Line #13. Result 32 is the surface
Line #14. Correct [QED]
Leon as a linguist has offered the following analysis of three key words that have been the source of many previous issues/queries and affect the interpretation of MMP 10:
1. NB.T = NEBET = A shape name. Leon agrees with Struve but not Peet or Neugebauer
2. TP.R = TEPER = All parties agree that this refers to the diameter across the mouth of the object. *See line #3 above.
3. cD = BIG D = Struve refers to this as a noun and as the dimension of the radius of the hemisphere while Peet caled this the barrel length of the semi-cylinder. Leon defines this as an adjective describing the good condition of the specified object.
The result of Leon's input on my perspective is that BIG D is probably NOT a dimension and I am not interested so much in the fact that it has any meaning as long as it is not a number or symbolic of a missing number from a copying / scribal error.
Leon and I [and Struve] agree this is a hemisphere calculation.
In the style of the line #1-14 breakdown of MMP 10, I will prepare similar breakdowns for the RMP and KP [*Khar conversion in granary problem] problems involving obvious arithmetic and calculations of circles. "
All positive ideas are invited. Putting the Egyptian geometry 'humpty dumpty' back together again is fun. Linguists often prematurely stop by identifying puzzle pieces by defining words. Linguists, required team members in any Egyptian math decoding project, often do not discuss the broader context of Egyptian math words reported in complete ancient sentences.
The Facebook page will report a broader geometry and arithmetic context of MMP 10.
Best Regards,
Milo Gardner -
Message 22
Posted by Milo Gardner (U14346264) on Tuesday, 24th August 2010
Egyptian unit fraction arithmetic was finite with quotients and exact remainders. Scribes scaled n/p by LCM m to mn/mp. RMP 36 recorded divisors of mp in red that summed to numerator mn. RMP 37 and Fibonacci recorded (n/p - 1/m) =(mn - p)/mp series in related finite arithmetic methods. Egyptian geometry was algebraic. Diameter D/2 placed radius (R) and 256/81 replaced pi in the area (A) of a circle. In RMP 41(9 - 9/9)^2 = 64 and RMP 42(10-10/9)^2 = 6400/81 used the formula [(8/9)D]^2 = A.
Ahmes Papyrus, New and Old Classifications
ABSTRACT
Old Kingdom Egyptian arithmetic was algorithmic and decimal. Arithmetic was recorded in rounded-off base 10 infinite series. Old Kingdom Egyptian numeration statements were related to the Babylonian base 60 numeration statements. The Egyptian and Babylonian numeration systems rounded off rational numbers in unit fraction series. Old Kingdom Egyptian numerals anticipated awkward Roman-numerals mapping more than one symbol to one number idea.
Egyptian hieratic script used a new numeration system in 2050 BCE. The numeration system mapped one number onto one symbol. The ciphered numeral system was the first in the Western Tradition. The arithmetic used in the numeration system was finite. Scribes scaled 1/p, 2/p, ..., n/p by LCM m to mn/mp before recording quotient and remainder unit fraction series. RMP 36 recorded divisors of mp in red that summed to mn unit fraction series without using algorithms.
Rational numbers recorded exact unit fraction series in non-algorithmic patterns. Ahmes scaled 2/53 by LCM 30 2,850 years earlier in RMP 35-38 and RMP 66. Arabs and Fibonacci scaled n/p by (n/p - 1/m) = (mn - p)/mp,with (mn -p) = 1 whenever possible. 20th century scholars minimally added back missing scribal number theory, not noticing a LCM m unifying role that was active for 3,600 years. Scholars did not decode Ahmes' division operation, nor Ahmes' finite arithmetic and algebraic geometry formulas.
Ahmes geometry was algebraic in area (A) of a circle formulas. In RMP 41 and RMP 42 Ahmes modified pi(R^2) by replacing R with D/2 and pi with 256/81. Ahmes solved for A and D in two steps.
Step one: A = (256/8)1((D/2)^2) = (64/81)D^2 (replaced R with (D/2) and reduced terms).
Step two A =[(8/9) D]^2 cubits^2 (geometry formula 1.0)
RMP 41: d = 9; applied A = [(8//9)D]^2. Step one A^1/2 = (9 - 9/9) = 8; Step two A = 8 x 8 = 64 cubits^2.
RMP 42: D = 10, height (H) = 10; applied formulas A = [(8/9)D]^2 and V =([(8/9)D)^2)H Step one: A = (10 - 10/9) = 80/9; Step 2: V = (80/9) x (80/9) x 10 = 64000/81 = (790 + 10/81)cubit^3 Step 3: V= 1/2 x (790 + 10/81) = (395 + 5/81) + (790 + 10/81) = (1185 + 10/81)khar used the formula
V = (H)[(8/9)D]^2 cubits^3 (geometry formula 1.1)
V = (3/2)(H)[(8/9)]^2 khar (geometry formula 1.2)
One problem in the Moscow Mathematical Papryrus, MMP 10, used the same A = [(8/9)D]^2 cubits^2 (formula 1.0).
One problem in the Kahun Papyrus further modified the area of a circle formula to find khar units per:
V = 2/3(H)[(4/3)D]^2 khar (geometry formula 2.0)
-
Message 23
Posted by Milo Gardner (U14346264) on Sunday, 7th November 2010
Aspects of hekat calculations are reported in RMP 43 and the Kahun Papyrus where the volume formula
V = (2/3)(H)[(4/3)(4/3)(D)(D)] (khar)
divided khar data by 20 to reach 400-hekat and 100-hekat units. RMP 41, 42, 43, 44, 45, 46, and 47 mixed hieratic symbols for 400-hekat and 100-hekat. Scribal two-part quotient and remainder answers are incorrectly scaled and therefore muddled on Wikipedia:
by:
"Problem 47 gives a table with equivalent fractions for fractions of 100 quadruple hekat of grain. The quotients are expressed in terms of Horus eye fractions...
1/10 gives 10 quadruple hekat
1/20 gives 5 quadruple hekat
1/30 gives 3 1/4 1/16 1/64 (quadruple) hekat and (1 2/3 ro) (error)
1/40 gives 2 1/2 (quadruple) hekat
1/50 gives 2 (quadruple) hekat
1/60 gives 1 1/2 1/8 1/32 (quadruple) hekat (3 1/3) ro (error)
1/70 gives 1 1/4 1/8 1/32 1/64 (quadruple) hekat (2 1/14 1/21) ro (error)*
1/80 gives 1 1/4 (quadruple) hekat
1/90 gives 1 1/16 1/32 1/64 (quadruple) hekat 1/2 1/18 ro (error)
1/100 gives 1 (quadruple) hekat (error)
*Ahmes actually reported one (5R/n)ro remainder as 2 1/14 1/21 1/42 since (150/70)ro = (2 + 1/7)ro with 1/7 = 1/7(6/6) = 6/42 = (3 + 2 + 1)/42 = 1/14 + 1/21 + 1/42 (a 2/n table rule per 2/101, and an EMLR rule per 1/101).
Ahmes mixed initial 400-hekat multiplications by 1/10 and 1/20 reporting correct answers with 100-hekat (scaled to 6400/64) multiplications by 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90 and 1/100 into Q/64 quotient and (5R/n) remainders reporting incorrect answers whenever 4-hekat quotients were added to 1-ro remainders. Correct RMP 47 quotient and remainder answer are reported by two sets of balanced statements:
A. Table A with 4-hekat quotient and 4-ro remainder answers:
following 4 x (6400/64)/n = (Q/64) 4-hekat + (5R/n)4-ro
1/10 gives (10) 4-hekat
1/20 gives (5) 4-hekat
1/30 gives (3 1/4 1/16 1/64) 4-hekat + (1 2/3)4-ro
1/40 gives (2 1/2) 4-hekat
1/50 gives (2) 4-hekat
1/60 gives (1 1/2 1/8 1/32) 4-hekat + (3 1/3)4-ro
1/70 gives (1 1/4 1/8 1/32 1/64) 4-hekat + (2 1/14 1/21 1/42) 4-ro
1/80 gives (1 1/4) 4-hekat
1/90 gives (1 1/16 1/32 1/64) 4-hekat (1/2 1/18)4-ro
1/100 gives (1) 4-hekat
B. And Table B with 1-hekat quotient and 1-ro remainder answers:
following 1 x (6400/64)/n = (Q/64)1-hekat + (5R/n)1-ro
1/10 gives (10) 1-hekat
1/20 gives (5) 1-hekat
1/30 gives (3 1/4 1/16 1/64) 1-hekat + (1 2/3)1-ro
1/40 gives (2 1/2) 1-hekat
1/50 gives (2) 1-hekat
1/60 gives (1 1/2 1/8 1/32) 1-hekat + (3 1/3)1-ro
1/70 gives (1 1/4 1/8 1/32 1/64) 1-hekat + (2 1/14 1/21 1/42)1-ro
1/80 gives (1 1/4) 1-hekat
1/90 gives (1 1/16 1/32 1/64) 1-hekat (1/2 1/18)1-ro
1/100 gives (1) 1- hekat
C. The quadruple (400) hekat case is also made by 4-sack and 4-hekat economic shipping units recorded in Northumberland Papyri 1, 2 and 3 published by Barns and Gunn, JEA, 1948. Quadruple sack and hekat initial scaled values (47/60) 4-hekat (wheat), 2-hekat (flour) scaled to (15/16) 4-hekat (flour), 383 deben (meal) and 31 loaves of 10 3/4 deben/loaf summed to 333 1/4 deben. Balanced 4-hekat, 4-ro or 1-hekat, 1-ro quotients and remainders was practiced by Ahmes. Ahmes reported khar divided by 20 into 400 hekat units by two volume formulas. The 400 hekat and 100-hekat initial divisions byt rational numbers have been translated into one hekat into 4800 ccm, 1/10 of a hekat (hin) into 480 ccm, 4-ro into 60 ccm, and 1-ro = 15 ccm by scholars, often muddling scribal 4-hekat and 4-ro intermediate details.
Scribal formulas can now be used to double check scribal hekat data as well as double checking modern metric translations of hekat and its many sub-units.
Best Regards,
Milo Gardner -
Message 24
Posted by Milo Gardner (U14346264) on Thursday, 9th December 2010
The RMP, EMLR, MMP and the AWT are discussed in a 12/7/10 New York Times "Science" article:
cited the oldest puzzles are found in Egyptian math texts. Solving puzzles brings moment sof certainty to otherwise uncertain world problems.
The oldest puzzle corrected divisions of a hekat (a commodity grain currency) by prime divisors that calculated binary Eye of Horus quotients (Q/64) and exactly scaled 1/320 of a hekat remainders (5R/n)*1/320, with 1/320 replace by the word ro. RMP 47 list the remainder arithmetic relationship as:
(6400/64)/n = Q/64 + (5R/n)ro
with n = 10, and 20 solved in quadruple hekats and 30, 40, 50, 60, 70, 80, 90 and 100 solved in single or quadruple hekat (Ahmes' shorthand is hard to read)
The remainder term (5R/n) considered all prime numbers p and composite numbers pq as n.
The value of a given product was assigned based on the grain units contained, be the product bread ,beer or ducks, geese, quail and dove (fed in pens, and valued less if caught in the wild .. RMP 83).
Best Regards,
Milo Gardner -
Message 25
Posted by Milo Gardner (U14346264) on Tuesday, 14th December 2010
A one hour webinar focuses on math education aspects of the topic per:
Egypt Math Glossary
Egyptian math
Best Regards,
Milo Gardner -
Message 26
Posted by Stoggler (U14387762) on Tuesday, 14th December 2010
To be honest Milo, I don't think anyone is really interested as you are the only person posting on this topic. If people were interested, they would have responded
-
Message 27
Posted by Milo Gardner (U14346264) on Tuesday, 11th January 2011
Dear Stoggler,
Scanning your Ö÷²¥´óÐã posts ... your posts are all over the historical map ... a master of nothing might be an appropriate title for your work.
At least the New York Times documented ancient Egyptian math and economic as I have consistently reported for years, one blog reporting:
Mind mentioning any of your publications? Your silence on this point will be deafening.
Happy New Year to all,
Milo Gardner -
Message 28
Posted by Stoggler (U14387762) on Wednesday, 12th January 2011
Milo
All I was doing was pointing out that you were the only person really posting on this particular topic and it seemed that no one else was really interested in the subject matter, hence the lack of any engagement from others on these boards.
I was not, AND I CANNOT MAKE THIS POINT ANY PLAINER, being offensive or having a dig at you or your subject matter. Merely pointing out that you seemed to be talking to yourself.
So I am mystified why you should feel it necessary to be offensive and start questioning what publications I've had published, which is nothing to do with the point and is not a prerequisite for people to post on the Ö÷²¥´óÐã History messageboards.
As it happens, I am not a professional historian and have not had any papers published, I am merely someone with a keen interest in matters historical and enjoy reading about the subject and debating with like-minded individuals. That is after all one of the reasons for the Ö÷²¥´óÐã History messageboards.
There, I haven't been silent - so it's far from deafening. Your offensiveness however speaks volumes. I will leave you to carry on posting to yourself on this thread then, and bid you good day sir! -
Message 29
Posted by Andrew Host (U1683626) on Wednesday, 12th January 2011
Hi all,
It is the aim and the (hopefully!) the spirit of these boards for members to have a space to engage in lively and informative discussion of a wide range of historical subjects in a friendly and polite manner.
As a Host I want to encourage all members to get engaged in debate and get to know each other as part of an online community.
If this thread hasn't garnered a great deal of involvement from other members and I wonder if there might be more gained from a different approach.
Other members have written long papers on pet subjects and have posted them here:
This is a useful space where longer arguments, theses etc can be set out and commented on.
Alternatively it may help to get more discussion going if perhaps a few questions were posed - inviting comment from other members.
If discussion isn't the aim of this thread then I would recommend using H2G2.
Many thanks
Andrew -
Message 30
Posted by Milo Gardner (U14346264) on Wednesday, 12th January 2011
Dear Stoggler and Andrew,
Stoggler's restated point of view "All I was doing was pointing out that you were the only person really posting on this particular topic and it seemed that no one else was really interested in the subject matter, hence the lack of any engagement from others on these boards." does not fairly cover the RMP turf.
My point discusses attested contents of the RMP, a scholarly effort that I was drawn into by chance, reviewed by:
a scholarly task that Ö÷²¥´óÐã readers are encouraged to independently research.
The British Museum version of the RMP grossly understates the contexts of the scribal mathematics, a point that British universities seem not to address. Why is that?
Sitting in the USA one answer is clear. The British school system is Classical based, over valuing Greeks and under valuing the mentors under which Greeks studied ... the Egyptians.
Is it not true, that Classical Studies report Plato and other Greeks studied in Egyptian schools? So where are the Greek arithmetic operations and styles of writing unit fraction seriess that were learned in Egyptian schools? Only neo-Pythagorean and neo-Platonic math are reported in British schools.
The arithmetic foundations of of Pythagorean and Platonic math are the central topics of the RMP. Today the scribal arithmetic of Ahmes is called number theory.
Thank you very much for not being silent. I hope that you and others actually read the RMP for yourself, and double check the British Museum's out dated 'additive ' and minimalist views of the EMLR and the RMP ... both documennts donated by the family of Henry Rhind in 1864.
As easily determined, the EMLR was not unrolled until 1927. Its actual contents, allowed a beginning scribal student to be introduced to the 2/n table and two other n/p conversion rules. Look for my paper on that topic as well, published in India where non-European scholarly efforts are alive and well.
Best Regards,
Milo Gardner -
Message 31
Posted by Milo Gardner (U14346264) on Wednesday, 12th January 2011
Dear Ö÷²¥´óÐã members,
The under reported issue of Greek arithmetic can be introduced by Plato's Republic, discussed in brief by:
A broad view of Greek arithmetic can be attested by allowing well documented Egyptian mentors of the Greeks and Egyptian fraction arithmetic ... recorded in the RMP ... to be discussed as Egyptian and Greeks scribes recorded abstract math and everyday economic transactions.
Best Regards to all,
Milo Gardner -
Message 32
Posted by Milo Gardner (U14346264) on Saturday, 15th January 2011
Dear Andrew,
Thank you for the suggestion -- " ...If this thread hasn't garnered a great deal of involvement from other members and I wonder if there might be more gained from a different approach.
Other members have written long papers on pet subjects and have posted them here:
www.bbc.co.uk/dna/h2...
This is a useful space where longer arguments, theses etc can be set out and commented on. "
I'll not be entering the suggested space. My point continues to be: The Ö÷²¥´óÐã has not independently commented on the British Museum's 1920s view of the RMP as a New York Times reporter did on Dec. 7, 2010 with respect to the oldest puzzle.
I have no quarrel with average British citizens. British citizens are wonderful puzzle solvers. I enjoy crosswords, cryptoquotes daily, and older unsolved pizzles since my 1990 retirement. During a two year stay in Germany, 1957-59, I hung out with three fun loving Brits that severed on our base. We had a great time.
My intellectual quarrel is with British Universities, the British Musem and the British Press. All three oddly accept Peet and 1920s views of the RMP that prematurely concluded that Ahmes' 1650 BCE 2/n table was built upon an additive premise.
Ahmes used three methods that always converted rational number n/p to concise unit fraction series. The first method was demonstrated in the 2/n table and the RMP 7-20. To convert n/p Ahmes selected a least common multiple (LCM) m within a unity context (one purpose of RMP 7-20) such that:
n/p x m/m = mn/mp
with the best divisors of denominator mp written in red that summed to numerator mn, calculated a concise unit fraction series.
When one LCM m could not be found a two step method discussed in the EMLR (by 1/8 x 25/25 = 25/400, and a second LCM
by a brute force method reported by an out-of-order series solved IN 2001, published in India IN 2002, and a Non-European Encyclopedia in 2006)
Ahmes could not solve 28/97, but did solve 26/97 + 2/97, by LCM 4 and LCM 56, respectively. Ahmes also could not solve 30/53 but solved 28/53 + 2/53, by LCM 2 and LCM 30, respectively).
The first, second and third methods were discussed in RMP 36. For open minded British thinkers work RMP 36 independently as a puzzle and discover Ahmes' actual thinking.
Claggett's 1999 transliteration is available on line:
Use it as a guide, for he too did not add back scribal steps that were omitted by Ahmes' shorthand notation.
Best Regards to all,
Milo Gardner -
Message 33
Posted by Milo Gardner (U14346264) on Monday, 17th January 2011
Dear Ö÷²¥´óÐã members,
For those that have not linked to the above EMLR blog,
a review of three EMLR alternative conversions of 1/8 by 'brute force' to non-optimal unit fraction series included two LCMs: LCM 25 and LCM 6, or a less likely LCM 25 and LCM 3 discussed by:
G. Modified multiple of 25 (2 questions)
Alternative one:
Increased denominator by LCM 25, and use LCM 6 (likely reading of the text):
25. 1/8*(25/25) = 25/200 = (8 + 17)/200 = 1/25 + 17/200
with,
17/200*(6/6) = 102/1200 = (80 + 16 + 6)/1200 = 1/15 + 1/75 + 1/200
hence an out-of-order series indicated a 2-phase method writing:
1/8 = 1/25 + 1/15 + 1/75 + 1/200 (line 8)
26. 1/16*(25/25) = 25/400 = (17 + 8)/400 = 1/50 + 17/400
with,
17/400 *(6/6) = 102/2400 = (80 + 16 + 6)/2400 = 1/50 + 1/30 + 1/150 + 1/400 (line 9)
25a . Alternative two (an unlikely reading of the text)
Decrease the denominator by factoring 1/5 in a second step per:
1/8*(25/25) = 25/200 = (24 + 1)/200 = 24/200 + 1/200
factor 1/5 from 24/200 = 1/5 *(24/40) = 1/5*(3/5)
such that.
3/5*(3/3) = 9/15 = (5 + 3 + 1)/15 = 1/3 + 1/5 + 1/15
meant
1/8 = 1/5*(1/3 + 1/5 + 1/15) + 1/200
re-written as
1/8 = 1/25 + 1/15 + 1/75 + 1/200
to show that an out-of-order series indicated a pair of LCMs, 3 and 25, had been used.
SUMMARY
The EMLR recorded 26 lines of unit fraction information. To decode one or more scribal methods in which 1/p and 1/pq unit fractions were converted to a unit fraction series, non-optimal LCM m values must be determined.
Scribal LCM m values can be seen as scaling factors by adding back initial details. Seen in terms of the RMP 2/n table (concise but not optimal unit fraction series) suggests that the EMLR answer sheet recoded basic test results. The EMLR student data can be seen recording conversions of 1/p and 1/pq to non-optimal unit fraction series as an opportunity to learn RMP 2/n table conversion methods at a later time.
REFERENCES
1. Gillings, Richard J, "Mathematics in the Time of the Pharaohs" Dover books, New York, 1982, ISBN 0-486-24315-X
2. Gardner, Milo R., 'The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term'. in History of the Mathematical Sciences, editors Ivor Grattan-Guiness, B.S. Yadav, Hindustan Book Agency, New Delhi, 2002, ISBN 81-84931-45-3.
3. Gardner, Milo, " Mathematical Roll of Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, Nov. 2005. -
Message 34
Posted by Milo Gardner (U14346264) on Wednesday, 19th January 2011
Dear Stoggler, Andrew and Ö÷²¥´óÐã message board members:
As an aid to jump-stating a two-sided debate, let's consider a proposed British view of the RMP.
A. Several intellectual positions hi-light vital historical contents of the RMP. Stoggle, Andrew and others can agree, or disagree, as they wish with each of the following five references:
1.Marshall Claggett, 1999:
2. Annette Imhasen, editor Victor Katz, 2007
3. Eleanor Robson, 2006 (summarized by Victor Katz, 2007)
4. Joran Friberg, 2005
and,
5. Tanja Pemmerening, 2005
?
B. The above books offer intellectual positions that tend to coincide with British Museum and British Isle universities
1. Open University
that hi-light outdated additive views of Peet, Chace, et al per Clagett,
2. St. Andrews
that hi-lights proposed additive views of Peet, Chace, et al, and algorithmic views of Eppstein that may connect to the RMP.
and,
C. the British press, represented by Ö÷²¥´óÐã, that tend to agree with the majority of the above references.
In summary, a standardized 1920s British additive position on the RMP has morphed into an algorithmic view formalized by Jim Ritter (1991), David Eppstein (1991), Eleanor Robson (2005), Joran Friberg (2005) and Annette Imhausen (2008).
The current British Museum, British Isles academia and British Press position tends to be pro-Babylonian thereby minimizing non-algorithmic methods used by Ahmes in the RMP.
As a formal debate emerges, it may be important to discuss alternative views of the EMLR opening alternative decoding doors to the RMP to honor Henry Rhind. The two-sided debate can conclude by considering the Dec.7, 2010 NY Times oldest puzzle article:
and foundations of the Middle Kingdom economic system built on the hekat outlined by Pemmerening.
D. In summary, Middle Kingdom texts reported unit fraction arithmetic, algebra, geometry and weights and measures within proto-number theory ideas and practical economic transactions built upon the hekat. The hekat created wage payment units in ways that the New York Times reported in Dec. 7, 2010, not necessarily as Clagett, Imhausen, Robson, and Friberg connect Egyptian scribal math to Babylonian-like algorithms. Pemmerening's hekat work falls into a category that must connect to 2/n table methods, at some point, as well as correctly scaling a wide range of hekat units used in the Middle Kingdom.
Stoggler and Andrew can agree or disagree with the above five books, and the New York Times article, one by one, if they desire. The same is true for any reader of this message board that is knowledgeable with this field of study.
In closing comments are requested for the purpose of structuring a formal debate on pointing out the most likely contents of the RMP. This type of two-sided debate is long overdue.
Best Regards to all,
Milo Gardner -
Message 35
Posted by Milo Gardner (U14346264) on Sunday, 23rd January 2011
Dear Ö÷²¥´óÐã message board members:
One side of the RMP debate will be placed in a formal academic paper. Could someone from the other side of this debate do the same thing ... or point out which scholars best represent your position?
I'll post a link to my updated paper in April or May 2011. As a preview:
Yesterday a few words of the abstract were pasted together along with portions of three appendicies summaries of 26 lines of the EMLR, 51 lines of the RMP 2/n table, and 40 lines of the AWT/RMP. Details of RMP 36 will appear in an appendix that discuss solved puzzles consistent with the NY Times writing style and unified ancient scribal thinking.
In the past my formal papers overstated intellectual contents of scribal math by offering specific lines of text reported in a brute force manner that fragmented the scribal thinking.
The 2011 paper will use everyday language and provide scholarly back up in easy-to-read summaries. The summaries will unpack and unify contents of the EMLR, RMP 2/n table, the two-part quotient (Q/64) plus scaled remainder (5R/n)ro and a 4th appendix RMP 36 considering:
1.
2.
The NYT's writer spent 90 minutes with me connecting general background issues related to four Egyptian texts, the RMP, EMLR, AWT, and the MMP. Taken together the four texts demonstrate abstract scribal methods that divided a hekat, and equivalent commodities units, used in decentralized wage payments per:
3. Gardner, 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.
Gaps left by the NYT article and the 2006 Ganita Bharati paperwill be filled by taking abstracts from:
4. Gardner, Milo R., 'The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term'. in History of the Mathematical Sciences, editors Ivor Grattan-Guiness, B.S. Yadav, Hindustan Book Agency, New Delhi, 2002, ISBN 81-84931-45-3
5. ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
and other documents.
Formal debates are fun. I hope more than one of your take up this challenge.
Best Regards to all,
Milo Gardner
Sacramento, California -
Message 36
Posted by The Gosport One (U14343205) on Monday, 24th January 2011
1. hekat, a volume unit's, division method
(64/64)/n = Q/64 + (5R/n)*1/320
Q =quotient and R = remainder, 1/320 = ro
--still stuck at the beginning
64/64 = 1 , so why not put 1/n ? also what is ro ? Quotient and Remainder of what? -
Message 37
Posted by Milo Gardner (U14346264) on Tuesday, 25th January 2011
Dear Gosport One:
The Akhmim Wooden Tablet describes one beginning point to scribal thinking by dividing one hekat by 3, 7, 10, 11 and 13. To achieve these five divisions, one hekat was replaced by (64/64) and multiplied by 1/3, 1/7, 1/10, 1/11 and 1/13.
The simplest problem (64/64) times 1/3 wrote the quotient as
64/3 = 21/64 with a remainder of 1/192
The AWT scribe converted the quotient 21 to binary units by
(16 + 4 + 1)/64 = (1/4 + 1/16 + 1/16) hekat
and converted the remainder 1/192 to 1/320 units named ro by multiplying
1/192 x (5/5) = 5/980 = (5/3)1/320 = (1 + 2/3)ro
such that a complete answer
1/3 of a hekat was written as:
(1/4 + 1/16 + 1/64)hekat + (1 + 2/3)ro
Georges Daressy solved this level of the problem in 1906 for the n = 3, 7 and 10 cases. Daressy failed in the n = 11 and n= 13 cases.
Hana Vymazalova, 2002, completed all five problems on the proof level only showing that the AWT scribe included in his shorthand notes:
1/3 of a hekat = multiplied by 3 = 1/3 x3 = 1 was we do today, but wrote
[1/3 =(64/64)/3 =[(1/4 + 1/16 + 1/64)hekat + (1 + 2/3)ro] x 3 =
[21/64 + (5/3)1/320] x 3 = 63/64 + 1/64 = 64/64
as 1/7, 1/10, 1/11 and 1/13 of a hekat problems were solved by Vymazalova and the AWT scribe.
Thank you for the question. There is more to the methodology. Work both sides of the n = 7, 10, 11 and 13 of (64/64) hekat unity problems and gain a few ancient scribal skills.
Best Regards,
Milo Gardner -
Message 38
Posted by Wyldeboar (U11225571) on Wednesday, 26th January 2011
Oh Gawd! They're multiplying!!!
-
Message 39
Posted by Milo Gardner (U14346264) on Wednesday, 26th January 2011
"Oh my god they are multiplying" ... is another way to trivialize scribal math ... consider this proposed abstract ... a full paper will follow in a few weeks ...
Solving Five Fragmented Egyptian Puzzles by a unified Ancient and Modern Arithmetic Method
ABSTRACT
The mission statement of this paper solves major aspects of scribal arithmetic, an 80 year old problem. For 80 years scholars have discussed possible common arithmetic operations connecting the Berlin Papyrus (BP), the Rhind Mathematical Papyrus ( RMP), the Akhmim Wooden Tablet (AWT), the Egyptian Mathematical Leather Roll (EMLR,) and the Moscow Mathematical Papyrus (MMP) into unified ways. The RMP and MMP report backup wage data recording commodity transactions and other information. In 20101 the New York Times reported puzzle aspects of the commodity system. Loaves of bread, volumes of beer and other commodities became wages by an inverse proportional pesu unit. The pesu inversely measured hekats of grain recorded in four levels of rational number calculations. Level one scaled rational number ( n/p) by least common multiple (LCM) m to (mn/mp) to calculate EMLR and RMP 2/n table unit fraction series. Level two decodes one hekat replaced by (64/64) and 320 ro. Multiplications of (64/64) by 1/n created two-part quotient plus 1/320 of a hekat (ro) remainders, and multiplications of 320 ro by (1/n) created one-part hekat quotient plus hekat remainders. Concerning the five hieratic texts, six hekat divisions were reported in the AWT, 13 divisions in the MMP, and 31 divisions in RMP. Level three decodes back up wage data and arithmetic and geometric progression data. Level four decodes arithmetic methods that double checked scribal calculations. Combining the four levels of the five texts into a unified ancient and modern arithmetic method demonstrates corrected BP, AWT, MMP, EMLR and RMP shorthand information that translates into double checked modern arithmetic statements. -
Message 40
Posted by Milo Gardner (U14346264) on Wednesday, 16th February 2011
"Oh my god they are multiplying" ... is another way to trivialize scribal math ... consider this proposed abstract ... a full paper will follow in a few weeks ...
Solving Five Fragmented Egyptian Puzzles by a unified Ancient and Modern Arithmetic Method
ABSTRACT
The mission statement of this paper solves major aspects of scribal arithmetic, an 80 year old problem. For 80 years scholars have discussed possible common arithmetic operations connecting the Berlin Papyrus (BP), the Rhind Mathematical Papyrus ( RMP), the Akhmim Wooden Tablet (AWT), the Egyptian Mathematical Leather Roll (EMLR,) and the Moscow Mathematical Papyrus (MMP) into unified ways. The RMP and MMP report backup wage data recording commodity transactions and other information. In 20101 the New York Times reported puzzle aspects of the commodity system. Loaves of bread, volumes of beer and other commodities became wages by an inverse proportional pesu unit. The pesu inversely measured hekats of grain recorded in four levels of rational number calculations. Level one scaled rational number ( n/p) by least common multiple (LCM) m to (mn/mp) to calculate EMLR and RMP 2/n table unit fraction series. Level two decodes one hekat replaced by (64/64) and 320 ro. Multiplications of (64/64) by 1/n created two-part quotient plus 1/320 of a hekat (ro) remainders, and multiplications of 320 ro by (1/n) created one-part hekat quotient plus hekat remainders. Concerning the five hieratic texts, six hekat divisions were reported in the AWT, 13 divisions in the MMP, and 31 divisions in RMP. Level three decodes back up wage data and arithmetic and geometric progression data. Level four decodes arithmetic methods that double checked scribal calculations. Combining the four levels of the five texts into a unified ancient and modern arithmetic method demonstrates corrected BP, AWT, MMP, EMLR and RMP shorthand information that translates into double checked modern arithmetic statements.
Ìý A revised abstract will focus on Egypt's wage payment and monetary system ... I hope that the new focus adds clarity to a long neglected topic:
ABSTRACT
The mission statement fills-in math building blocks of 4,000 year old scribal arithmetic. For 80 years scholars reported practical foreground aspects of the Akhmim Wooden Tablet (AWT), the Berlin Papyrus (BP), the Egyptian Mathematical Leather Roll (EMLR), the Kahun Papyrus (KP), the Moscow Mathematical Papyrus (MMP), the Rhind Mathematical Papyrus (RMP), and other texts, minimizing fragmented background aspects. Previously undefined building blocks of commodity based wage payments are newly decoded and offered as examples of corrected scribal arithmetic methods. To publish improved scribal arithmetic translations, additional puzzle aspects of commodity data elements have been solved. Loaves of bread, hekat units of beer, and other commodities were paid as wages and decoded by applying a scribal inverse proportion. Aspects of the inverse proportion, named pesu, measured grain in unit fractions in four classes of calculations, facts that have been generally understood for 110 years (Berlin Papyrus, Schack-Schackenberg, 1900). New translations of class one rational numbers (n/p) scaled by least common multiples (LCM) m to (mn/mp) in the EMLR (Gardner, 2002) and RMP calculated optimized unit fraction series. Fifty problems, six in the AWT (Vymazalovaq, 2002), 13 in the MMP, and 31in the RMP (Gardner, 2006) decode class two a hekat unity replaced by (64/64) and 320 ro. New translations of scribal multiplications of (64/64) by (1/n) created two-part quotient plus 1/320 of a hekat, ro remainders, as well as multiplications of 320 ro by (1/n) created one-part hekat quotient plus hekat remainders. Improved class three scribal calculations expose wages, arithmetic progression and geometric progression data summarized by AWT and KP and RMP links to daily bird feeding rates of grain. Class four calculations encoded/decoded scribal double checking methods. Combining corrected AWT, BP, EMLR, KP, MMP, and RMP data by meta math offers important corrections to the standard translations: Peet (1923), Chace (1927), Gillings (1972), Robins-Shute (1987) and (Clagett, 1999). -
Message 41
Posted by cladking (U6255252) on Sunday, 6th March 2011
I'm interested in your work on this subject though, frankly, it's difficult for me to follow all of it. Your knowledge seems extensive so you may be the right person to ask a related question.
To you believe that this work represented the state of the art in Egyptian mathematics?
I can't shake the feeling that this was more in the nature of a 10th grade math text than it was state of the art or all the Egyptian knowledge of math. It even seems to be organized something along the lines of a texytbook. -
Message 42
Posted by Milo Gardner (U14346264) on Friday, 11th March 2011
Dear Cladking:
Thank you for the question. If I may be so bold as to rephrase your question: does the RMP contain the highest level of Egyptian science? ... the answer would be yes, and no. The yes part reports the highest level of Egyptian mathematics included clues to the earliest economy that used grain and other commodities as money, paying a minimum wage of 1/15 hekat of grain aa day ... an amount that would buy 1/2 loaf of bread and 2 gallons (1/2) jug of beer by exactly converting rational numbers to concise unit fractions in the 2/n table, and elsewhere in the RMP.
The no part says that the basis of Egyptian math, Egyptian astronomy, has not been decoded in serious ways. There are as many astronomical texts as there were mathematical texts ... scholars can not read the astronomical texts well. Hopefully, one day they will.
As you may be able to see, the mathematical texts are beginning to reveal their number based secrets, not as an art (your area of interest), but as the language of science, as history of science scholars report their topic.
Best Regards,
Milo Gardner -
Message 43
Posted by cladking (U6255252) on Monday, 14th March 2011
Thank you very much for the answer. I'll just try to shake that feeling that this is a mid level math book for young students.
I can hardly imagine the difficulties of describing astronomical movements without calculus.
>>"As you may be able to see, the mathematical texts are beginning to reveal >>their number based secrets, not as an art (your area of interest), but as the >>language of science, as history of science scholars report their topic."
It is apparent, even painfully obvious, to me that the great pyramid builders were scientists first and foremost. How their writings have been misinterpreted to be incantation and spells is beyond my grasp. I have little doubt that going forward your work will come to be seen as much more important. -
Message 44
Posted by Milo Gardner (U14346264) on Wednesday, 16th March 2011
Dear Cladking:
Thank you for generally describing your qualifications. For awhile I expected a 10th grade student. You seem to be the instructor.
Welcome to Egyptian math, a topic not well enhanced by ancient astronomical texts,
Calculus may not have been used in 2050 BCE, or in the OK pyramid building days. A variation of the Chinese Remainder theorem may have been worked on ... as Chinese developed teh CRT in 1900 BCE ... and Mayans developed another version 1500 years later.... Until the Egyptian astronomical texts are decoded speculation often passes as fact .
Of course, per your complaint, the spiritual side of the Book of the Dead discussed rounded off balance beam methods that used binary weights to weigh the hearts of the dead ... distorting Egyptology and Egyptian math.
By 2050 BCE prime number weights became available calculated within finite Egyptian fractions. The rational number system eliminated round off errors. The RMP included examples of area and volume calculations per RMP 41-47 ... Stressing volume and RMP 47 read this over quickly:
and place deben weights into another file.
Scholarly transliterations of hekat problems, especially RMP 47, have trunder reported Ahmes' arithmetic, and geometry. The hekat was very near 4800 CC in modern metrology. The hekat measured beer, bread and other commodities well enough in grain content to pay wages to large labor forces ... even decentralized crews working for absentee landlords ... outside the cities.
Thanks again for the chat.
Best Regards,
Milo Gardner -
Message 45
Posted by Milo Gardner (U14346264) on Wednesday, 16th March 2011
Dear Cladking,
Suspecting that you are looking for an introductory Egyptian fraction text ... how about the EMLR
?
Of 26 EMLR unit fraction series two out of order ,1/8 and 1/16, series used LCM 25 and LCM 6 as introductions to the 2nd of three RMP 2/n table construction rules.
The EMLR converted 1/8 by first scaling 1/8 by LCM 25
1/8 x 25/25 = 25/200
and removed 1/25, taking the remainder
(25/200 - 1/25) by writing
(25 - 8)/200 = 1/25 + 17/200;
and second, scaling 17/200 by LCM 6 considering
17/200 x 6/6 = 102/1200 = (80 + 16 + 6 )/1200 = 1/15 + 1/75 + 1/200
meant
1/8 = 1/25 + 1/15 + 1/75 + 1/200
an introductory method that converted
1/16 = 1/50 +( 1/30 + 1/150 + 1/400
Oh yes, the second RMP 2/n table rule was written out in RMP 31 with respect to 28/97 = 26/97 + 2/97 and RMP 36 with respect to 30/53 = 28/53 + 2/53 to converting the rational numbers to concise unit fraction series. The 2/n table rule converted n/p by replacing n/p with (n-2)/p + 2/p and solved
(n -2)/p by one LCM, and 2/p by a second LCM (as introduced by the EMLR).
Without cheating, can you, or one of your students, guess which LCM was used by Ahmes to converted 26/97, and which LCM converted 28/53? Both are very easy to find.
Clue: Ahmes used LCM 56 to convert 2/97 and LCM 30 to convert 2/53 as the 2/n table shorthand used difficult LCMs to find ... with a little practice LCMs are easy to find for any n/p conversion problem ... 4th grade work I'd say ....
Best Regards,
Milo Gardner -
Message 46
Posted by Milo Gardner (U14346264) on Sunday, 20th March 2011
Adding clarity to the RMP considers the 275 year older EMLR (adapted from: )
1. (1/8)(5/5) = 5/40 = (4 + 1)/40 = 1/10 + 1/40
2. (1/4)(5/5)= 5/20 = (4 +1)/20= 1/5 + 1/20
3. (1/3)(3/3)) = 3/9 = (2 + 1)/9 = 1/4 + 1/12
4. (1/5)(2/2)) = 2/10 = 1/10 + 1/10
5. (1/3)(2/2) = 2/6= 1/6 + 1/6
6. (1/2)(3/3) = 3/6= 1/6 + 1/6 + 1/6
7. 2/3 = 1/3 + 1/3
8. (1/8)(25/25)= 25/200 = (8 +17)/200 = 1/25 + (17/200)(6/6)
= 1/25 + (80 + 16 + 6)/1200 =1/8 = 1/25 + 1/15 + 1/75 + 1/200
9. (1/16)(25/25)= 25/400 = (8 +17)/400 = 1/50 + (17/2400)(6/6)
= 1/50 + (80 + 16 + 6)/2400 =1/16 = 1/50 + 1/30 + 1/150 + 1/400
10. (1/15)(10/10) = 10/150 = ( 6 + 3 + 1)/150 = 1/25 + 1/50 + 1/150,
corrects the initial rational number (was cited as 1/6)
11. ( 1/6)(3/3) = 3/18 = (2 + 1)/18 = 1/9 + 1/18
12. (1/4)(7/7) = 7/28 = (4 + 2 + 1)/28 = 1/7 + 1/14 + 1/28
13. (1/8)(3/3) = 3/24 = (2 + 1)/24 = 1/12 + 1/24
14. ( 1/7)(6/6) = 6/42 = (3 + 2 + 1)/42= 1/14 + 1/21 + 1/42
15. (1/9)(6/6) = 6/54 = (3 + 2 + 1)/54 = 1/18 + 1/27 + 1/54
16. (1/11)(6/6) = 6/66 = (3 + 2 + 1)/66= 1/22 + 1/33 + 1/66
17. (1/13)(?) = 1/28 + 1/49 + 1/196 (corrected by?)
(1/13)(6/6) = 6/78 = (3 + 2 + 1)/78 =1/26 + 1/39 + 1/78
18. (1/15)(6/6) = 6/90 = (3 + 2+ 1)/90 = 1/30 + 1/45 + 1/90
19. (1/16)(3/3) = 3/48 = (2 + 1)/48 = 1/24 + 1/48
20. (1/12)(3/3) = 3/36 = ( 2 + 1)/36 = 1/18 + 1/36
21. (1/14)(3/3) = 3/42 = (2 + 1)/42 = 1/21 + 1/42
22. (1/30)(3/3)= 3/90 = (2 + 1)/90 = 1/45 + 1/90
23. (1/20)(3/3)= 3/60 = (2 + 1)/60 = 1/30 + 1/60
24. (1/10)(3/3) = 3/30 = (2 + 1)/30 = 1/15 + 1/30
25. (1/32)(3/3)= 3/96 = (2 + 1)/96= 1/48 + 1/96
26. (1/64)(3/3) = 3/192 = (2 + 1)/192 = 1/96 + 1/192
Also Discussed by:
-
Message 47
Posted by Milo Gardner (U14346264) on Monday, 21st March 2011
After scribal students understood EMLR-like conversions, 2/n table conversions were practiced. The Kahun Papyrus offered an abbreviated version of the 51 member RMP table. Using the same modern format as the 26 line EMLR info, a modern translation of the 2/n table looks like this:
Taken from:
2/3 = 1/3 + 1/3
2/5(3/3) = 6/15= (5+ 1)/15 = 1/3 + 1/15
2/7(4/4) = 8/28= (7 + 1)/28 = 1/4 + 1/28
2/9 (2/2) = 4/18= (3 + 1)/18 = 1/6 + 1/18
2/11(6/6) = 12/66= (11 + 1)/66 = 1/6 + 1/66
2/13(8/8) = 16/104= (13 + 2 + 1)/104 = 1/8 + 1/52 + 1/104
2/15(2/2) = 4/30= (3 + 1)/30 = 1/10 + 1/30
2/17(12/12) = 24/204= (17 + 4 + 3)/204 = 1/12 + 1/51 + 1/68
2/19(12/12) = 24/228= (19 + 3 + 2)/228 = 1/12 + 1/76 + 1/114
2/21((2/2) = 4/42= (3 + 1)/42 = 1/14 + 1/42
2/23(12/12) = 24/276= (23 +1)/276 = 1/12 1/276
2/25(3/3) = 6/75= (5 + 1)/75 = 1/15 + 1/75
2/27(2/2) = 4/54= (3 + 1)/54 = 1/18 + 1/54
2/29(24/24)= 48/696= (29 + 12 + 4 + 3)/696 = 1/24 + 1/58 + 1/174 + 1/232
2/31(20/20) = 40/1620= (31 + 5 + 4)/1620 = 1/20 + 1/124 + 1/155
2/33(2/2) = 4/66 (3 + 1)/66 = 1/22 + 1/66
2/35(30/30) = 60/1050= (35 + 25)/1050 = 1/30 + 1/42
2/37(24/24) = 48/888= ( 37 + 8 + 3 )/888 = 1/24 + 1/111 + 1/296
2/39(2/2)= 4/78= (3 + 1)/78 = 1/26 + 1/78
2/41(24/24)= 48/984= (41 + 4 + 3 )/984 = 1/24 + 1/246 + 1/328
2/43(42/42)= 84/1806= (43 + 21 + 14 + 6)/1806 = 1/42 + 1/86 + 1/129 + 1/301
2/45(2/2)= 4/90= ( 3 + 1)/90 = 1/30 + 1/90
2/47(30/30)= 60/1410= (47 + 10 + 3)/1410 = 1/30 + 1/141 + 1/470
2/49(4/4)= 8/196= (7 + 1)/196 = 1/28 + 1/196
2/51(2/2) = 4/102= (3 + 1)/102 = 1/34 + 1/102
2/53(30/30)= 60/1590= (53 + 5 + 2 )/1590 = 1/30 + 1/318 + 1/795
2/55(6/6) = 12/330= (11 + 1)/330 = 1/30 + 1/330
2/57(2/2) = 4/114= (3 + 1)/114 = 1/38 + 1/114
2/59(36/36) = 72/2124= (59 + 9 + 4) /2124 = 1/36 + 1/236 + 1/531
2/61(40/40) = 80/2440= (61 + 10 + 5 + 4)/2440 = 1/40 + 244 + 1/488 + 1/610
2/63(2/2)= 4/126= (3 + 1)/126 = 1/42 + 1/126
2/65(3/3)= 6/195= (5 + 1)/195 = 1/39 + 1/195
2/67(40/40)= 80/2680= (67 + 8 +5 )/2680 = 1/40 + 1/335 + 1/536
2/69(2/2)= 4/138= (3 + 1)/138 = 1/46 +1/138
2/71(40/40)= 80/2840= (71+ 5 + 4)2840 = 1/40 + 1/568 + 1/710
2/73(60/60)= 120/4380= (73 + 20 + 15 + 12)/4380 =
1/60 + 1/219 + 1/292 + 1/365
2/75(2/2)= 4/150= (3 +1)/150 = 1/50 + 1/150
2/77(4/4)= 8/388= (7 + 1)/388 = 1/44 + 1/308
2/79(60/60)= 120/4740 =(79 + 20 + 15 + 6 )/4740 =
1/60 + 237 + 1/316 + 1/790
2/81(2/2)= 4/162 = (3 + 1)/162 = 1/54 + 1/162
2/83(60/60)= 120/4980= (83+ 15 + 12 +10)/4980 =
1/60 + 1/332 + 1/415 + 1/498
2/85(3/3)= 6/255= (5 + 1)/255 = 1/51 + 1/255
2/87(2/2)= 4/174= (3 + 1)/174 = 1/58 + 1/74
2/89 =(60/60)= 120/5340= (89 + 15 +10 + 6)/5340 =
1/60 + 1/356 + 1/534 + 1/890
2/91(70/70) = 140/6370= (91 + 49)/6370 = 1/70 + 1/130
2/93(2/2)= 4/186= (3 + 1)/186 = 1/62 + 1/186
2/95(60/60) = 120/5700 = (95 + 15 + 10)/5700 = 1/60 + 1/380 + 1/570
2/97(56/56)= 112/5432= (97+ 8 + 7 )/5432 = 1/56 + 1/679 + 1/776
2/99 (2/2) = 4/198= (3 + 1)/198 = 1/66 + 1/198
2/101(6/6)= 12/606= (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606
further discussed by:
It appears that student scribes passed EMLR-like and 2/n table-like courses before graduating.
Any questions? Most new readers of the RMP 2/n table should have many questions. An outline of the EML/R and 2/n table courses can be provided off-line for the motivated readers.
Basic question(s) can be posted here.
Best Regards to all,
Milo Gardner -
Message 48
Posted by Milo Gardner (U14346264) on Wednesday, 23rd March 2011
Detailed EMLR and 2/n table questions can be posted to the math-history-list by following this link:
Both tables were built upon least common multiples (LCM) and red auxiliary numbers. The 26 line EMLR used 7 LCMs and the 51 line 2/n table used 15 LCMs.
A short list of common construction patterns connect the two tables, details that can be discussed per reader questions.
Best Regards,
Milo Gardner
-
Message 49
Posted by Milo Gardner (U14346264) on Saturday, 9th April 2011
A nine year old EMLR link to the RMP has been informally updated on Planetmath:
Comments would be appreciated.
A formal paper on this topic is being prepared as well.
Best Regards,
Milo -
Message 50
Posted by Milo Gardner (U14346264) on Wednesday, 27th April 2011
The formal AWT, EMLR and RMP paper has been submitted updating Egyptian fractions, unit fractions, hekats and wages. A future paper will discuss deben puzzles. The deben was a weight unit used in the Middle and New Kingdoms.
As some of you know rational numbers were trivially written into unit fraction series by medieval scribes. Rational number 2/67 was trivially solved by LCM 34. Fibonacci in the 1202 AD Liber Abaci.
applied the first of three rational number notations within a subtraction context that considered LCM m written as:
(n/p - 1/m) = (mn = p)/mp,
with (mn -p) set to unity (1) as often as possible.
Ahmes's LCM m stairway to the best n/p unit fraction series scaled
n/p by LCM m to mn/mp
and inspected the divisors of mp that best summed to mn in 2/n tables, algebra, geometry, hekat and deben problems. Ahmes avoided trivial solutions. The best LCM m, in the 2/67 case was 40. LCM 40 allowed the best divisors of mp (67 + 8 + 5) to be written in red ink. Ahmes' list of alternatives included 6 entries:
1. 2/67(34/34) = 68/2278 = (67 + 1)/2278 = 1/34 + 1/2278
2. 2/67(36/36) = 72/2412 = (67 + 3 + 2)/2412 = 1/36 + 1/804 + 1/1206
3. 2/67(36/36) = 72/2412 = (67 + 4 + 1)/2412 = 1/36 + 1/603 + 1/2412
4. 2/67(38/38) = 76/2546 is impossible since (78 - 67) = 9 are not additive divisors of 38. Only 2 + 1 are less than 9.
5. 2/67(40/40) = 80/2680 = (67 + 8 + 4 + 1)/2680 = 1/40 + 1/335 + 1/670 + 1/2680
6. 2/67(40/40) = 80/2680 = (67 + 8 + 5)/2680 = 1/40 + 1/335 + 1/536
since 1/536 offered the smallest last term denominator.
The deben paper will hopefully include Greeks, Arabs, and medieval scribal unit fraction notations, potentially correcting a longer Egyptian fraction historical record.
It may be important to note that the Egyptian fraction notation, that began in 2050 BCE, unofficially ended in 1454 AD. The 1202 AD Liber Abaci Fell out of use in Europe after a 250 year run as Europe's arithmetic book. The year was 1454 AD when the Ottoman Empire ended the 1,000 reign of Byzantium. Byzantium had been Europe's economic link to the Silk Road. Re-acquiring Silk Road products required new sea routes, and new weights and measure systems. Arab middlemen that continued to use unit fraction math until 1637 AD were avoided. Portugal opened new routes to India and China around Africa. Spain opened new routes across Mexico and the Pacific to China. Both routes were approved by the Pope in 1498, along with partitioning New World lands to Portugal and Spain related to a specific longitude (a subject for another day).
Formal replacement of Egyptian fraction weights and measures took place in 1585 AD. Two based 10 decimal books, one for business and one for science were written by Simon Stevin. The two books were approved by the Paris Academy.
Most of us have learned the 400 year old base 10 decimal system in grammar school, absent its Egyptian fraction parentage. Is it time to correct that historical oversight as well?
Best Regards,
Milo Gardner
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