Talk:Egyptian mathematics

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Some dates would be useful in the first paragraph. In the section on geometry, there are some examples that I think would be useful additions. (1) the use of a 3-4-5 right triangle in surveying to construct a right angle. This is depicted in some surviving drawing, but I don't remember where I saw it reproduced. (2)The formula for the volume of a truncated square pyramid (needed to know how much stone to import to build a pyramidal tomb). It was equivalent to Volume = (1/6)(H)(B + T + 4M), where H is the height, B is the area of the base, T is the area of the top, and M is the area of the cross section halfway up the pyramid. I find it astounding that this formula could have been discovered without the use of calculus. (3)They used a formula for the area of a quadrilateral which was incorrect except when the quadrilateral was a rectangle. But it gave a good approximation if the quadrilateral was not too far from a rectangle. I remember that "A History of Mathematics" by Howard Eves has a very good chapter on Egyptian and Babylonian mathematics. Aftermath 00:45, 20 Jun 2005 (UTC)

• • • Howard Eves is an excellent resource. Read his book "Number Theory and Its History", McGraw-Hill , 1948, Dover reprints available. To my recollection, Sylvia Couchoud's review of the MMP, and its set of slices of the truncated pyramid. The slices were exactly summed is my best memory. If Sylvia says approximations were used I'll accept the suggested correction. Milo

Contents 1 Remainder arithmetic 2 Geometry 3 False position 4 Milo's contributions 5 Request for citations 6 Request for attention from an expert on the subject 7 Britannica citation 8 Abbreviations 9 Merge

[edit] Remainder arithmetic

I reverted the edit introducing remainder arithmetic because of verifiability issues: it does not give any references, I asked at User talk:172.194.232.169 for references but did not receive an answer, a Google search suggests that this is a theory of Milo Gardner only published in his blog, and it does not agree with my (admittedly very small) knowledge of the usual theories. -- Jitse Niesen (talk) 00:03, 21 January 2006 (UTC) (references have been added, so why is remainder arithmetic still ignored/censored by Wikipedia? Milo)

I'd be happy to send you, or any Wikipedia reviewer - that is willing to put pencil to paper as ancient scribes wrote their math - in an email or two - by attaching a copy of Daressy's 1906 paper, in French or English, as well as Peet's misleading 1923 paper that attacks Daressy in an unprofessional manner. In addition I'll again cite Hana Vymazalova's 2002 paper published in a Prague journal, that shows that a hekat unity was used by several scribes, one being Ahmes, setting 1 = 64/64. This abstract definition of number was almost reported by Daressy in 1906 (the reason for Peet's personal attack on Daressy) - and finally was confirmed by Vymamazolva in 2002. Yes, there is much more to the Egyptian math and the Egyptian fraction story, much more than Wikipedia's overly brief, misleading and 80 year old outdated citations have dared to review or discuss in any serious way.

Considering pertinent primary and secondary references supporting remainder arithmetic, and directly disproving the minimalist additive view of Egyptian fractions strongly suggested by Peet or the 1920's classical math historians, read: http://historyofegyptianfractions.blogspot.com .

The EMLR paper was written in 2002. It is particularly important. It details a method and references that shows 'how and why' Ahmes stressed the 2/nth table, taking up 1/3 of the space in his review of 1650 BCE mathematics. Ahmes had used abstract definitions of nummber, beginning with unity (like 64/64), and progressed to units of measure.

In between Ahmes' abstract defintions of number, contained in his 2/th table, as also used in the EMLR, a two-part method of exactly partitioning a hekat was presented - from which the one-part additive units of measure were created.

Best Regards,

Milo Gardner

Several raw historical data points clearly shown below. Find me any set of references that refutes this approach discussed above? I propose that none exist. If true, Peet, 1923, only discussed the additive 1/320 = ro aspect of teh Akhmim Wooden Tablet, a fact also found in the RMP, totally missing the remainder arithmetic. Peet and other 1920's classical math historians had already pedagogically determined that Egyptian division only consisted of a process inversely related Egyptian multiplication (a false conclusion if I ever saw one!) Milo ***

Egyptian division consisted of the use of abstract rational numbers, often beginning with a unity, and partitioning that unity into two parts. As shorthand, a one-part statement was developed for the average granary worker, they using 10/n = hin units and 320/n = ro units for large and small applications - with several other one-part m/n units being used for other purposes.

To understand a little more of the subject, gegin with

http://akhmimwoodentablet.blogspot.com http://egyptianmath.blogspot.com http://reisner.blogspot.com

Oh yes, who is Wikipedia s 'in-house expert?

Until a qualified in-house 'expert' responds, please post my changes - since the information is factual in every respect! Mathematics is like that --- nothing but the facts.

Glyphdoctors.com is filled with this topic, over the last year. No one from that discussion group that has refuted the remainder arithmetic facts, with over one year of opportunities. Since there are no formal debates showing that remainder arithmetic was not the dominate from of scribal arithmetic,please repost all of my excised and censored posts - and let's get on with life.

Who is Jitse Nissen? Where is his work posted, and debated?

The texts confirm the fact that remainder arithmetic was used by all hieratic scribes. Look at the long 'false position' algebra solution posted for your stated problem:

x + 1/2x = 16

Look how easy it is solved by remainder arithmetic, quotient = 10 and remainder = 2/3 the method that Ahmes did use, per Ahmes's own shorthand, per:

3/2 x = 16

x = 32/3

 = 10 + 2/3

and (97/42)x = 33, 37 (two other RMP problems)

0r,

x = 33*(42)/97 and 37*(42)/97

as I have already submitted to Wikipedia, showing a direct and simple way to solve all of Ahmes' algebra problems - avoiding the false use of 'false position'!

Occam's Razor alone should point out that 'false position' was not used!

That is, please provide an example, or bettter yet, a set of examples of fraction series that were not created by remainder arithmetic? Scholars have injected their personal (additive minimalist) views far too long! Let the hieratic scribal thinking be reported, as the scribes first thought of the subject, using the simpliest terms - using Occam's Razor, and the actual texts, thereby avoiding the often awkward and confusing notions of many 'published experts'!

Best Regards,

Milo Gardner

Wikipedia does not publish original research. Everything has to be verifiably cited to reliable sources. A weblog is not a reliable source. Tom Harrison ^Talk 17:20, 25 May 2006 (UTC)

Again, why does Wikipedia continue to publish secondary sources that have not been validated by the primary historical texts? For example, there is no known text that reports 'false position'. RMP #31, 47/97X = 33 has not been reported by false position. St. Andrews U. in Scotland's web site shows that modern arithmetic best solves this problem, a proposal that I can expand to every RMP alegebra, and vulgar fraction problem in every hieratic text. There are many other examples of 'struggling' scholars pulling scribal methods out of thin air - attached to nothing - getting published, as if their work had been confirmed by one or more ancient texts! The ancient scribes are the only experts that I 'tip my hat' to!

Mr. Harrison, Wikipedia's Egyptian algebra references are outdated - by at least four years. Read and reference the original texts, such as:

http://reisnerpapyri.blogspot.com

and the so forth, and Wikipedia readers will benefit - greatly.

Rehashing the 1920's, as reviewed on:

http://historyofegyptianfractions.blogspot.com

without considering discoveries made since that time is odd - to say the least. The 1920's was a time of flowering 'minimalism', as the reading of their 'published' views of the RMP, EMLR and other texts (the 1920s was defined by love of Greece, with the love of things Egyptian taking a backseat. That is, Peet, Neugebauer, Chace et al (of that era) misdefined scribal subtraction and division in horrible ways. They all missed the use of quotients and remainders that are now easily read by remainder arithmetic - the primary method of scribal division, and adding in the Hultsch-Bruins method, scribal subtraction is also clearly desribed for the remainder portion of scribal division).

BEST REGARDS

MILO

In brief, it has to be published somewhere else first. Tom Harrison ^Talk 20:23, 25 May 2006 (UTC)

Hana Vymazalova, Charles U., Prague in 2002 published in Archiv Orientalni, pages 27-42, showing that five Akhmim Wooden tablet division problems (by 3, 7, 10, 11 and 13) all bean and ended with 64/64, as she appropriately titled a hekat unity.

That is, the AWT scribe used the following relationship:

(64/64)/n = Q/64 + R/(n*64)

with a couple of additional modification to complete his/her two-part answer, namely by n = 3,

(64/64)/3 = 21/64 + 1/(3*64), and coverting the

quotient to a binary series and the remainder term to an Egyptian fraction series, after factoring 1/320 from the remainder. This was done by allowing the remainder 1/54 term to be replaced hy 5/320 such that:

          = (16 + 4 + 1)/64 + 5/3 *1/320

          = 1/4 + 1/16 + 1/64 + (1 + 2/3) ro

since 1/320 was replaced by the word ro.

Hana Vymazalova showed that all five division 2-part answers were multiplied by their original divisor, again finding 64/64, the starting point.

To complete the division by 3 example,

(1/4 + 1/16 + 1/64 + (1 + 2/3)*1/320)* 3 =

63/64 + 5/320 = 64/64

Q.E.D.

Milo Gardner

[edit] Geometry

I have added further detail to the geometry section to give a fuller picture of the situation regarding squaring the circle. I have tried to leave you paragraph unaffected in the process.

--Michael saunders 22:16, 18 March 2006 (UTC)

[edit] False position

I'm not sure that the solution to problem 25 in the Rhind Papyrus uses the False position method. Wouldn't it more correctly be described as "linear extrapolation?" Tom Harrison ^Talk 20:39, 6 May 2006 (UTC)

I strongly agree with Tom's point, with one exception. False position in a linear sense was not found in ancient Egypt. About 2,000 years later a version close to Tom's view was found in India, as noted on Wikipedia. The direct way to solve 3/2 x = 16, was by cross multiplying (mentally as Ahmes did), such that x = 32/3, as also easily divided (again mentally as anyone can do) leaving a quotient = 10 and remainder = 2/3 answer, as all of Ahmes' algebra problems followed.

That is, Occam's Razor, if for no other reason points to the shortest method as the historical method, which is remainder arithmetic, as found in the reisner papyrus:

http://reisnerpapyri.blogspot.com

and many other hieratic texts.

The clearest proof that 'false position' was not used is found in RMP 31, 97/42x = 33, or,

x = 1386/97 = 14 + 28/97 = 14 + 2/97 + 26/97.

Hultsch-Bruins' method of solving 2/97 and 26/97 per

2/97 = 1/56 + (8 +7)/(56*97) = 1/56 + 1/679 + 1/776 (per the 2/nth table)

and,

26/97 = 1/4 + (4 + 2 + 1)/(4*97) = 1/4 + 1/97 + 1/194 + 1/388

such that:

28/97 = 14 1/56 1/97 1/194 1/388 1/679 1/776

leaving out the plus sign, as Ahmes wrote his info from right to left is the only info contained in Ahmes' shorthand, and not 'false position' as 'suggested' by scholars that can not complete the solution to RMP 31, any many other problems.

That is, modern scholars have refused to accepted F. Hultsch's 1895 proposal of reading the 2/nth table, a method that Bruins independently found in 1944. And to further cloud the issue, and their personal inabilities of reading the 2/nth table and its many concise applications in the RMP and elsewhere, these same scholars - beginning in the 1920's, otherwise excellent scholars have intentionally or unintentionally have tried to write over Ahmes' historical 'aliquot part' method of converting 2/p, 2/pq, n/p, and n/pq (as noted above for 2/97 and 26/97) to short and concise Egyptian fraction series - with a FALSE version of 'false position'! False position as described on Wikipedia can not be found in the RMP or any text --- hence Wikipedia should allow its editing of its web site, in this respect - after a few months of waiting for counter-arguments.

Of course, anyone is free to submit counter-arguments, with hard data, of course. But try to be speedy. We have already waited 80 years + for these 1920's errors to be corrected.

Best Regards to all,

Milo Gardner, Sacramento, CA

[edit] Milo's contributions

Milo's edits raise a number of issues.

1. Primary sources. A long-running guideline is "In general, Wikipedia articles should not depend on primary sources but rather on reliable secondary sources who have made careful use of the primary-source material." (see Wikipedia:Reliable sources). The reason for this is clear: it is not easy to interpret primary sources. This is particularly true in this case, where only few sources are available and they are written in a dead language. Of course, it would be great if all these sources were available to the general public. This does not agree with the mission of Wikipedia, but a sister project called Wikisource collects primary sources. I assume they would be very happy to accommodate for instance the Rhind Mathematical Papyrus, if its text is available.

Wikipedia's list of authors excludes Archimedes and many in the mathematical community. Adding Ahmes and other mathematicians of note as a Scotland University has done online, as I have noted on:

http://historyofegyptianfractions.blogspot.com

may be a major inconvenience to Wikipedia's current point of view, and practice, of stressing geometry based texts. Oddly Wikipedia had thereby skipped over the scribal arithmetic that empowered Ahmes' geometry! The history of mathematics, and its many texts from around the world seem best left - in detailed descriptions - math problem per math problem per each important mathematician - to other web pages. Milo

1. Secondary sources. Milo counters that secondary sources are not reliable either. Indeed, the difficulty in interpreting primary sources suggests that secondary sources may get it wrong sometimes. This may well be the case here. Even if it this is the case, Wikipedia still prefers secondary sources. However, what will often happen in these situations is that secondary sources contradict each other. In the current context, they may give different explanation of how the 2/n tables were constructed.

Concerning the RMP and its 2/nth table, the number of conflicting views are too numerous to mention. Scondary sources therefore continue to be not helpful to clear up the reading of the RMP and its 2/nth table. Again, Wikipedia's version of Egyptian math had began and ended with geometry, only a few weeks ago. oddly skipping over large aspects of scribal arithmetic that empowered scribal geometry. Gillings is a good starting point to open the door to the 'secondarty source' controversies found in the RMP, as cited on Simeon Fraser U. Canada's web page, per

http://www.math.sfu.ca/histmath/Egypt/Ahmes/frametop.html.

I thank Tom, or whomever, for expanding the Egyptian math discussion to cover numeration, arithmetic (both sides), algebra, geometry and weights and measures. Adding in all the scribal math as one body of knowledge allows the ancient scribes to point out their methods, leaving modern secondary sources and their wide ranging opinions aside.

Many scholars, and secondary sources, have tended to choose one subject, or one point of view (i.e additive), as your primary editor keeps returning to 'false position' (using an unproved assumption) that it was used in every algebra problem. I can not find 'false posiition' in any of the scribal shorthand notes, either in MAD and other web pages as cited by Wikipedia, suggest that it was used. - Again, I challenge anyone to look at RMP 31 - and submit a 'false supposition' version of it. We may have a lively debate, written in numbers. Finally, to my point of view arithmetic empowered the geometry that scribes used to build pyramids, a supposition on my part. Wikipedia continues to define Egyptian mathematics, as written in the RMP, and for sure the 2/nth table, without citing any of the 80 + old basic secondary source controversial threads on the subject (ie false position). The current de facto position of Wikipedia only shows the slowly changing scope and slowing improving intellectual and arithmetic integrity of Wikipedia. Wikipedia can continue to provide samples from four to five alternative views? I have named five of them, for your review: 1. Egyptian math derived from Babylonian sources, a pet notion of Otto Neugebauer and others (minimalist), 2. Algorithms from Babylonian numeration caused Horus-Eye numeration, and that Ahmes used algorithms to create his 2/nth table (far out), 3. Old Kingdom Egyptian math was based in geometry, since the Old pyramid builders' knowledge could not have been based in arithmetic. Oddly the geometric knowledge was lost by the time of the Middle Kingdom (the singular view of Wikipedia, and one of the positions of Otto Neugebauer and his citation that the 2/nth table marked an intellectual decline in Egyptian mathematic, an interesting supposition - but where are the ancient texts that prove it)?, 4. Reading only the hieratic texts, and finding contextual solutions to the Horus-Eye problem, as surely the 2/nth table and Egyptian arithmetic reports, a scribal method that eliminated the rounding off of vulgar fractions - when expanded into Egyptian fractions, by subtraction or division - was required anytime 1/p, 2/p or n/p calculations were made. Middle Kingdom changes in arithmetic definitions of number, and methods associated with scribal division, opened the door to abstract definitions of number (my primary position), and 5. Scribes employed 'false position' 2000 years before it was used in Greek and Arabic texts (an odd add-on to the Wikipedia position) - pedagogical positions that birthed many controversies 80 + years ago. Minimalists tried to close off those debates -and have failed badly! So, please, ANYONE post RMP #31 examples of 'false position' - off line, or on Wikipedia, and actually cite the hieratic raw data from all the algebra problens in the RMP. If Wikipedia reviewers are willing or able, from: problems #24 to 32 added disussions can follow. Or just cite the scribal step by step solution to RMP # 31: x + (2/3 + 1/2 + 1/7)x =33, since someone forgot to finish it on Wikipeida (I suggest that this problem can not be done by applying 'false position'!). Show me the beef, as all that I ask! All that has been published on Wikipedia and elswhere in the area of false position is conjecture. Cite the scribal division method, step by step, taken directly from the 1650 BC RMP, used by Ahmes, in these cases and I will show you modern algebra, written in Egyptian fractions, methods that did not use the awkward 100 AD method of 'false position'.

Another of the critically omitted threads is: why did Ahmes introduce statements and solutions to 80 + problems, by first writing out an optimized 2/nth table of Egyptian fractions? Why not include this thread, along with the five mentioned above? The implications attached to trying to answer the basic question - opens the arithmetic door to several aspects of these math controversies. Note that potential abstract solutions are easily seen in scribal shorthand, introducing three forms of remainder arithmetic, none of which come close to the 'false suppositions' cited by Wikipedia. Stated in other words, whenever an ancient scribe calculated a remainder, by subtraction or division, an exact Egyptian fraction series was written down as part of the final answer, nothing more, nothing less. Milo

1. Neutral point of view. If we have secondary sources contradicting each other, we try to describe all sources from a neutral point of view. This means that the interpretations from all sources are mentioned, with their arguments, while giving the greatest weight to the interpretation that is accepted by most of the professionals (Egyptologists and maths historians, in this case).

I wish this was the case. Giving the greatest weight to the oldest set of professional Egyptologists and math historians, using the additive point of view, going back 80 years or more, should not be continued, without citing a few of the alternatives. Concerning Egyptian mathematics, NPOV is not being followed. Current policy, the stressing of minimalist views from outdated authors like Otto Neugebauer (Exact Sciences in Antiquity - where Egyptian fraction was said to be a 'sign of intellectual decline') should not continue since there is no abstract side of number possible within Neugebauer and his group's view. By only citing the additive views of Peet, Chace, Neugebauer, et al, a de facto policy of excluding 100% of the alternative views, especially the recently well documented analyses published after 2002 (Vymazalova, et al) grossly distorts the subject to young and old readers alike. Remainder arithmetic definitions of scribal division, one of the most reasonable of the altenatives only adds to our understanding of the ancient texts. Why not consider Occam's Razor, that the simplest method is most likely the historical method, per Sarton, whenever the secondary source was published? Following this history of science policy, remainder arithmetic would clearly jump to the top of the the simplest method list, and so forth! Get a real debate started and see your readership rise. Best Regards, Milo Gardner

1. References. This aspect puzzles me a bit. Using references is standard in academic texts, and the article that Milo wrote for the Non-Western Encyclopedia of the History of Science, Medicine and Tenchology shows that he knows some of the references in the field and how to cite them. I don't understand why he doesn't do the same for his contributions to this article.

I'd have begun to edit your current list of Wikipedia outdated references, many going back 80 years. The list cries out to be brought upto date to include the discoveries made after 2002, a critical year in the history of decoding Egyptian fractions (as I have sent a separate email to Jitse, on this topic). A small number of references going back to F. Hultsch 1895 (per Science Awakening), Daressy in 1901, and so forth have been added. From this broader context of the history of the 2/nth table and weights and measures a few added changes will be suggested to the narrative itself, leading to the 2002 era. Milo

1. False position. When Tom said above "I'm not sure that the solution to problem 25 in the Rhind Papyrus uses the False position method," he means that it does not use the iterative method for nonlinear equations. Instead, in the standard interpretation, it uses a method called sometimes the "rule of false position", akin to linear interpolation. It seems that Milo has misunderstood what Tom wanted to say.

I am saying the Tom does not understand the primary ancient Egyptian text that he is discussing. Ahmes, the RMP scribe includes shorthand footnotes to define and prove his problems, using a form of writing that we commonly call proofs. Ahmes' problem was that his shorthand excludes many logical steps, something mathematicians often do today. Reading fragmented 'proofs' from any era causes confusion for everyone that reads bits and pieces of data. In the case of Ahmes, scholars had not correctly guessed the mental side of Ahmes' thinking that had been omitted. Oddly, scholars, rather then mentioning their translation problems, have commonly filled in several different classes of logical gaps, often taken from thin air, with one overly simplified additive versions scribal math. One overly complex version of scribal math is 'false position' while - at the same time, poorly discribing scribal division as being based in an inverse relationship to scribal multiplication. Scholarly 'assumptions' therefore have commonly been based on assumptions, several wild ones as teh RMP 2/nth table discussions report, rather than rigorously applying the scientific method of hypothesis, testing and proof.

To resolve Ahmes' logical step omissions, other hieratic texts need to be cited, a step that Wikipedia listed scholars secondary sources often avoid. In the case of Egyptian ' False Position' the method could NOT have been used by Ahmes. The methodology did not appear until 100 BC, or as documented 200 AD+ in Arabic texts.

So why is Tom and others pushing for 'false position' to have been used by Ahmes in 1650 BCE? The issue is, there is not one confirmed definition or use of Egyptian division in the RMP. So why cite it? There were only logical gaps created by Ahmes shorthand discussions that are very difficult to decode. Yes, false suppositions abound when all the scribal versions of proposed scribal division are placed in the context of the 2/nth table, Egyptian algebra or Egyptian weights and measures decoding efforts by modern scholars. Differing definitions appear in each of these three scribal areas of mathematics. Yet, to avoid discussing these three different number based realities, several 'false' definitions continue to be offered one simple (logical to Wikpedia), and minimalist gap filling idea, by otherwise professional Egyptologists and math historians. One irony is that the current 'minimalist' term stressed by Wikipedia's current list of references is named 'false position', a term that is more logically reported as 'false supposition'. Milo

Concluding, Milo obviously knows a lot about Egyptian mathematics. I'm sure he can make a valuable contribution to Wikipedia. However, Wikipedia has policies, which parallel the policies in academic encyclopaedia. Be careful to add references to a reliable source (most importantly, not a blog) and give a fair summary of all points of view, giving due prominence to the dominant point of view.

I'd be happy to do that, slowly at first, if reviewers do not continue over-reacting by totally removing my references and comments, prior to their complete entry. Milo

Perhaps it is better to start with describing the sources themselves. Our article on the Moscow and Rhind Mathematical Papyri is inadequate and could easily be expanded.

I'd be happy to add an Akhmim Wooden Tablet article, one of the ancient texts that is needed within everyones' references - whenever Egyptian mathematics is discussed. For those that would like a peek at the subject, log onto http://glyphdoctors.com and enter through the public door, and query "Akhmim Wooden Tablet", and 53 articles will be offered for your review. Milo

I expect that there are less issues with a subject ties to primary sources. -- Jitse Niesen (talk) 11:56, 4 June 2006 (UTC)

I agree with Milo that the current article concentrates too much on one explanation. Other views should be mentioned. -- Jitse Niesen (talk) 05:23, 7 June 2006 (UTC)

Jitse, thank you for your comment. I wonder why reviewers had not seen the geometric myopia that had previously dominated Wikipedia's Egyptian math discussion? My only concern is that all of hieratic texts may not be fairly considered (as one body of knowlege), and that major aspects of numeration, arithmetic (especially subtraction per Hultsch-Bruins, and division by remainders), algebra and weights and measures will continue to be slighted in favor or geometry, or some other unproved pedagogical position. Best Regards, Milo 6/17/2006.

[edit] Request for citations

I'd like to see verifiable citations to a reliable source for "ciphered one-to-one numeral system", and for the translation of the quote beginning,

(Boyer, as cited in the reference section, Milo)

Please be more precise and include a page or section number. -- Jitse Niesen (talk) 14:12, 13 June 2006 (UTC)

"If you are told: A truncated pyramid of 6 for the vertical height by 4 on the base by 2 on the top." Did our quote incorrectly omit by adding up all the slices or segments, or was it added in error? Tom Harrison ^Talk 13:39, 9 June 2006 (UTC)

(all the exact slices or segments of the truncated pyramid were correctly summed, hence the total was exact, per Couchoud or anyone's reading of the text. I have never seen a controversy on the MMP in this respect. Milo)

You did not answer the question. Does the phrase "by adding up all the slices or segments" appear in the manuscript? Does it appear in Couchoud? A quote should be exact.

Having read Couchoud in French, I can not attest to her exact wording on this topic. I have read the MMP and used pencil and paper several times directly related to the topic at hand. The truncated pyramid was broken down, partitioned or segmented, into 'bite sized' areas, and then the area of each 'piece' was found. Then all the separate areas were added, thereby exactly finding the area of the truncated pyramid. How simple can be an ancient problem be? Milo 6/23/06.

I don't know what you mean with "exact slices or segments", but there does not seem to be agreement that the Egyptians subdivided the frustum and added the volumes of the parts; see Talk:Moscow and Rhind Mathematical Papyri for some quotes. -- Jitse Niesen (talk) 14:12, 13 June 2006 (UTC)

I do not know who you have been reading, but the MMP is very easy to read. There is no controversy on this topic. All of the separate areas of the truncated pyramid were added together, much as modern calculus takes slices of a shape and sums the slices to find the formula and area, volume of a shape. The MMP data does not take an infinite number of slices akin to modern calculus. But the MMP methodology of taking big bites out of the truncated pyramid's area, and summing all the bites was discussed by Couchoud (though, not reading French, I can not attest to her word-by-word phrasing).

From a recent search of the text on Amazon, Boyer nowhere refers to "a ciphered one-to-one numeral system". Tom Harrison ^Talk 17:55, 9 June 2006 (UTC)

(I have tried: ciphered numerals, Egypt,obtaining the answer: The best independent reference is covered on Britannica.com. The other googled links tend to link back to my discussions on history of math listservers connecting to Boyer. Milo).

[edit] Request for attention from an expert on the subject

I'd appreciate it if someone more knowledgable than I could review Mister Gardner's additions for consistency with the current mainstream understanding of the subject. Tom Harrison ^Talk 18:02, 9 June 2006 (UTC)

Agreed. The article is in bad shape. I'm not sure I qualify as an "expert" but there's a lot of significant overstatements in this one. --M a s 13:44, 15 June 2006 (UTC)

The body of knowledge contained in the hieratic texts covers scribal numeration, arithmetic (both sides, practical and abstract),algebra, geometry and weights and measures. Prior to 2006 updates Wikipedia had grossly understated numeration, arithmetic, algebra and weights and measures - while grossly overstating scribal geometry. So what specifically is user M a s talking about (hopefully geometry)? Citing a generalization unconnected to a set of specific situations is an odd comment. Milo

Just to clarify, the overstatements that M a s rightly complains about are not due to Milo, but to a previous editor with the name of Roylee. -- Jitse Niesen (talk) 04:39, 18 June 2006 (UTC)

Thanks Milo, Tom, Jitse. I apologize for any missatribution. I took issue with and deleted the following:

Three geometric elements contained in the Rhind papyrus are basic to simplistic analytical geometry: (1) first and foremost, how to obtain an approximation of π accurate to within less than one per cent within a squaring of the circle technique, rather than our modern definition of pi within A = pi* radius^2; (2) an approximation of the square root of 2, and (3) third, the earliest known use of a kind of cotangent.

Analytical geometry is the study of the relationship between algebra and geometry, specifically based off of the works of Descartes, Fermat, and other 15th - 16th century European mathematicians. Analytic geometry has nothing to do with the geometric methods that were used in the Ahmes papyrus. It is not a scholar's or Wikipedia's job to rename a 4600 year old geometric geometric approach as "analytic geometry."

Likewise it's impossible to know what's meant by pi's definition in Ahmes being different than "our modern definition." The argument that it was accurate to less than 1% again is specious. Ahmes did not claim that this pi was constant, he wasn't using pi as the ratio of circumference to diameter, but instead area to diameter. It is only with our "modern" definition of pi that we can see that his approximation is accurate to within 1%. Again, although Ahmes did have an approach to the approximation of pi for use with measuring area of a circle, he made no claim that this approximation was exact, or only approximate.

Another item that I took issue with was the claim that Ahmes was writing a textbook. No context is given as to the meaning of a "textbook." Most modern scholars and historians picture Ahmes' work as a primer for use by scribes on how to apply basic arithmetic to solve geometric problems. But, the claim that it's a textbook anachronistic. Ahmes did have a preface, where he had his stated goal of "Accurate reckoning;" the body, wherein he has his problems; and an appendix, with the 2/n table. But this does not qualify as a textbook.

There's more, but another claim that I had an issue with is

Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned

I really can't see the point. In this case, this claim seems a little unfair to Ahmes, ancient Egyptians, and others. Experiments with infants and subitizing seem to imply that number sense is innate; and I'm confused as to how "8" should be understood abstractly.

Let me show the EMLR version of an abstract 8, written as

1/8 = 1/25 *25/8 = 1/5*(25/40) = 1/5*(24/40 + 1/40)

    = 1/5*(3/5 + 1/40) = 1/5* (1/5 + 2/5 + 1/40)
    = 1/5*(1/5 + 1/3 + 1/15 + 1/40)
    = 1/25 + 1/15 + 1/75 + 1/200

as cited twice, the second time writing 1/16 leading to a generalized rule.

That is, any number 1/p or 1/pq could be written as an Egyptian fraction using the abstract partitioning rule:

1/pq = 1/A *(A/(pq)), with A = p + 1 (the generalized rule)

as used 14 times in the EMLR, and

2/pq = 2/A x (A/pq), with A = (p + 1) (the generalized rule)

used over 21 times in the RMP 2/nth table.

Seen as a primer to proto-number theory, the EMLR and RMP both were pretty good textbooks. Milo 6/29/06

Thaniks, --M a s 18:49, 18 June 2006 (UTC)

[edit] Britannica citation

What point was it that was to be cited to Britannica? Tom Harrison ^Talk 23:28, 14 June 2006 (UTC)

Britannica.com's citation of Egyptian numeration strongly stresses the first use of ciphered numerals in the world, a reference that Tom had requested - from other than myself and Boyer. I now see that St. Andrews review of ciphered numerals has been added at the end -- though reading it, hieroglyphic numerals continue to be over stressed. Hieratic totally replaced hieroglyphic numerals in all but the most religious texts. Everyday practical and abstract versions of number used hieratic - for a very good reason - exactness was required whenever rational numbers were written out - as hieroglyphic numeration could NEVER achieve - except in association with Egyptian fraction remaidners (as the Akhmim Wooden Tablet clearly details per http://akhmimwoodentablet.blogspot.com.) Milo

Thanks; based on Britannica Online, I think we can say that the first ciphered system was the Egyptian hieratic. I would rather cite something other than Britannica, if some other work is available. I'll look in Menninger's Number Words and Number Symbols next time I get to the library. Tom Harrison ^Talk 17:42, 17 June 2006 (UTC)

Menninger's work looked at higher levels of numeration, and rarely points that Boyer, Carl B., A History of Mathematics, John Wiley, 1968, cited with respect of hieratic ciphered numeration. Good luck with Menninger. But rely on Boyer.

[edit] Abbreviations

Here's a general style tip: write the word or phrase first, then in parentheses write the abbreviation. Egyptian Mathematical Leather Roll (EMLR) etc. —The preceding unsigned comment was added by 209.128.81.201 (talk)

[edit] Merge

There's been a suggestion to merge Ancient Egyptian multiplication here. Did anyone have an opinion on whether it should be merged or not? Tocharianne 02:58, 9 January 2007 (UTC)

I would suggest that the wider, and in some places still current, use of Peasant Multiplication merits its own entry which should, of course, be linked to the Ancient Egyptian Mathematics page to show its oldest known description. Peter R Hastings 08:25, 9 February 2007 (UTC)