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Hello,
I'm not sure if this is the correct place for this comment - apologies if it isn't. I've just listened (via i-Player) to yesterday's broadcast on the Rhind Papyrus. I'm a bit of a geek, so I tried to solve the puzzle that was given at the start of the broadcast. Sadly my answer doesn't agree with the answer given by Mr MacGregor (which was 19,607).
I make it 16,807, being 7 (cats) times 7 (houses) times 7 (mice) times 7 (ears of corn) times 7 (gallons of wheat).
Please can you explain?

Thanks & Regards,

Jim Cleary.

An update (I said I was a geek!). The answer to the question asked by Mr MacGregor is 16,807 - that is the number of gallons of corn which we will have thanks to the cats killing the mice. It's 7 to the power 5.

The question in the papyrus is different - it is 'what's the total of cats & mice & houses & ears & gallons?' - and the answer to that is 19,607 : 7 houses + (7 x 7 cats) + (7 x 7 x 7 mice) + (7x7x7x7 ears of corn) + (7x7x7x7x7 gallons of grain).

More info here [Unsuitable/Broken URL removed by Moderator]

Re the flood tablet.
Q/ Why do you claim that the Flood tablet [6/ 700bc] pre dates the Bible flood account. You state that the flood story's start to be told about 4000 yrs ago, 2000 bc.
The bible flood account can be dated at 2370 bc, 1367yrs after the creation account ?.You listen to the people who have a vested interest in belittle the bible accounts without questioning there reasonings. Those that attack the Bible, when challenged, usually resort to character assassination of the one who making the challenge.
Or are you afraid to be on the receiving end of there venom.
Pudsey keith.

Anna, a ³ÉÈË¿ìÊÖ radio moderator, suggested that discussions of the Rhind Mathematical Papyrus, such as:

/dna/mbhistory/F2766774?thread=7347367

should be placed here.

One reason for placing the RMP discussion on the ³ÉÈË¿ìÊÖ radio message board is that the problem took place there. ³ÉÈË¿ìÊÖ grossly misreported by contents and historical content of the Rhind Mathematical Papyrus.

Two complaints were submitted, 21 days ago and 7 days, hoping that a ³ÉÈË¿ìÊÖ staff member would be assigned to research the Feb. 2010 ³ÉÈË¿ìÊÖ RMP broadcast's errors. To date, no response has been received.

Indirectly, a recommendation to discuss the RMP here, was given.

Hence, a set of references to earlier ³ÉÈË¿ìÊÖ message board discussion will be provided, calmly awaiting a formal response from upper ³ÉÈË¿ìÊÖ radio management that controlled the contents of the "History of the World in 100 Objects" to offer appropriate guidance. Will a corrected RMP program be produced and aired on ³ÉÈË¿ìÊÖ radio?

If so, when?

If not, why not?

Best Regards,

Milo Gardner

Let's comment on RMP 79 in the economic context of Egyptian fractions, cited on Planetmath, an on-line math encyclopedia per:



To begin a discussio, I'd like to cite the British medieval rhyme,

"As I was walking to St. Ives
I met a man with seven wives
Every wife had seven sacks
Every sack had seven cats
Every cat had seven kits
Kits, cats, sacks, and wives
How many were going to St. Ives"

This rhyme is vivid restatement of RMP 79, transmitted from 1650 BCE to the modern era. Several, but not all, math aspects of the rhyme were discussed by Oystein Ore, page 118, "Number Theory and Its History", Dover reprint is available.

I read Ore's book in 1964. Several unresolved aspects of Egyptian mathematical arithmetic foundations have been resolved in the last 10 years, facts not reported by Eleanor Robson. Two resolved facts are decentralized economic system and proto-number theory context of ancient Egyptian games, business accounting, and mathematical methods.

Ancient Egyptian scribes worked within theoretical and practical calculations in balanced ways, facts not reported by ³ÉÈË¿ìÊÖ. Ahmes reported the solution of the Old Kingdom Eye of Horus problem. Ahmes exactly converted rational numbers to concise unit fraction series, in weights and measures problems (such as RMP 82 and 29 examples).

The ³ÉÈË¿ìÊÖ #17 RMP program offered Eleanor Robson as an Egyptian math scholar. She is an excellent Babylonian math scholar who knows a little of Egyptian math, but seems to know nothing of the Akhmim Wooden Tablet, and its 1900 BCE solution to the Eye of Horus problem that scaled a hekat to (64/64). Reported in modern remainder arithmetic the AWT and RMP method says:

(64/64)/n = Q/64 + (5R/n)ro

with Q = quotient, R = remainder scaled by 5/5, and ro = 1/320 of a hekat

The method was used by Ahmes over 40 times, 29 times in RMP 82.

The point is that ancient Egypt in 2050 BCE began to use hieratic script that recorded decentralized economy with workers transactions paid in hekats, and/or equivalent commodities. Double entry records were kept for control purposes, much as dollars are recorded today to report assets and liabilities.

Eleanor Robson, nor ³ÉÈË¿ìÊÖ radio 'experts' mentioned the decentralized economic status of the Middle Kiungdom, nor the rich theoretical and practical arithmetic that empowered Rhind Mathematical Papyrus records.

The well intentioned Feb. 2010 program can be easily updated.

I look forward to commenting before or after the release of an updated RMP program. Ahmes would be pleased to know that his work received a second ³ÉÈË¿ìÊÖ airing.

Best Regards,

Milo Gardner

May a comment on RMP 79 be discussed in the context of the Berlin Papyrus? Robson may have read of Schack-Schackenbury's proposal, as Clagett properly footnoted it. My review of the ³ÉÈË¿ìÊÖ RMP program Robson, like Clagett, improperly discussed Egyptian division as 'single false position', within other algorithms that were not present in the RMP. In RMP 38 and RMP 79 Ahmes' division operation was inverse to a version of our modern multiplication operation, as discussed by



in one short paragraph:

"Weights and measures units were scaled as the 2/n tables were scaled. For example, RMP 69 scaled 80 loaves of bread made from 3 + 1/2 hekat of grain to a pesu unit. The pesu unit divided 80 loaves of bread by 3 1/2 hekat and created a rational number distribution method. In RMP 69 the pesu method tracked one loaf made from 14 ro (14/320 of a hekat), or 7/160 of a hekat. The proportional pesu method was used in two Berlin Papyrus problems that solved x^2 + y^2 = 100 and x^2 + y^2 = 400, by applying a geometric analogy. Variables x and y were stated as proportional to one another in the ratio of 1: 3/4 and 2: 3/2, seemingly equal proportions. The scribe solved for 2x in the second problem. Schack-Schackenburg reported RMP 69's use of a proportional method in the context of the Berlin Papyrus proportional method years ago. Clagett footnoted the Schack-Schackenbery suggestion, but misreported Ahmes' pesu and Berlin Papyrus inverse division operation by concluding 'single false position' was present (which surely it was not)."

Dear Ciaran McConnell:

Thank you for your comments on two formal complaints. I hoped that one or more of the numerical complaints would have been cited, at least indirectly. For example, an informal complaint was posted to ³ÉÈË¿ìÊÖ per:

/blogs/ahistoryoftheworld/2010/02/weekly-theme-the-beginning-of.shtml

RMP 79 was mentioned rather than the correct RMP 69 . A discussion of RMP 69 and the Berlin Papyrus is available in this email by an APPENDIX.

My central point is that hieratic Egyptian math was literally built upon finite properties of rational numbers. Modern number theory mathematicians call modern versions of Egyptian fractions finite math. Note that modern and ancient infinite series math is openly adored by Robson, Ritter, Imhausen, Hoyrup, and the pro-Babylonian lobby. If you need references
I'll gladly send them. Sadly no where were the supporters of Egyptian math mentioned by ³ÉÈË¿ìÊÖ radio in its Feb. 2010 RMP program.

Rational numbers, at any time, are infinite, between 0 and 1, the 2/n table domain of Egyptian fractions, and between any two points in a line segment. Egyptian scribes solved this irony. Babylonian scribes never did. Babylonians always rounded off their rations numbers.

Modern ironies of finite math expose modern and ancient principles within the infinite counting of numbers on the number line, and in weights and measures, These principles were understood and implemented by 2050 BCE, not metaphorically, but literally. Egyptian scribes applied two classes of solutions, theoretical and practical statements that stressed rigorous double-checking of answers. That is why Ahmes doubled checked most of his answers. One implication seems to have been, making an arithmetic mistake would have been a major thing in 1650 BCE. Ahmes may few errors!

The first method was included in ancient proto-number theory.

The second ancient method always returned round-off Old Kingdom rational numbers to exact Middle Kingdom weights and measures units. Rational numbers were not allowed to be rounded off in the Middle Kingdom. Only higher order numbers like pi were rounded off.

Taken together the resolution of Old Kingdom "Eye of Horus" problem, was not sensed or reported by Robson or ³ÉÈË¿ìÊÖ in any way. Robson's love of Babylonian algorithms misguided ³ÉÈË¿ìÊÖ to continue the British Museum 'status quo' suggestion that began of over 80 years ago, that the RMP contained only additive information. A great deal of the RMP included non-additive information.

As you may know, Egyptian Old Kingdom scribes wrote

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...

as a 6-term infinite series. The "Eye of Horus" numeration
system rounded off by throwing away up to 1/64 of a unit.

Babylonian scribes in all their numeration history wrote
integers in round-off ways in base 60, throwing away very
small units, thereby
being superior, as a culture, compared
to the Egyptian Old Kingdom.

Archimedes, before 200 BCE re-wrote 1,800 years of infinite
series mathematics to solve an area of a parabola problem via:

4A/3 = A + A/4 + A/16 + A/64 + ...

obviously, one-half phase of the "Eye of Horus" Series.

Archimedes solution to the infinite series problem was a finite
series as, as Heiberg reported in 1906 per:

4A/3 = A + A/4 + A/12

Middle Kingdom Egyptians lived within a decentralized political
system by 2050 BCE and an exact weights and measures, ... scaled
commodity monetary system ... that never rounded off rational numbers.

Of course, Egyptians all of eras rounded off irrational and
higher order numbers like pi. Every culture in every era must
do that. Hopefully you, or a ³ÉÈË¿ìÊÖ staff member may be interested
in the finite and the solution to infinite series problems, per
Archimedes, or
another context.

Best Regards,

Milo Gardner
Sacramento, CA


--- On Sun, 3/28/10

Subject: ³ÉÈË¿ìÊÖ Complaints [T20100316034MS010Z7809683]
Date: Sunday, March 28, 2010, 8:48 AM

Dear Mr Gardner

Thanks for your e-mail regarding 'Around the World in 100 Objects' broadcast on 8 March.

I understand you felt the programme misrepresented Rhind Mathematical Papyrus and inappropriately presented Eleanor Robson as an expert on Egyptian math when she isn't. I also apologise for any frustration you may've been caused by any delays in responding to your complaint.

I assure you that accuracy is the cornerstone of all our programming and we strive to ensure that everything we broadcast is as accurate and relevant as possible. We'd never knowingly include information in a programme we know to be incorrect, and I regret you felt some of the information in this programme was inaccurate.

We appoint all our programme contributors, including Eleanor, on the basis of their talent and experience in their field, and their ability to undertake to roe we ask of them

Nevertheless we know our audience have a wide range of opinions and we'd never expect everyone to agree with every contributor we engage.

It's with this subjectivity in mind that I'd like to take this opportunity to assure you that I've recorded your comments onto our audience log. This is an internal daily report of audience feedback which is circulated to many ³ÉÈË¿ìÊÖ staff including senior management, producers and channel controllers.

The audience logs are seen as important documents that can help shape decisions about future programming and content.

Thanks once again for contacting us.

Regards

Ciaran McConnell
³ÉÈË¿ìÊÖ Complaints

APPENDIX:

"May a comment on RMP 79 (should have read RMP 69) be discussed in the context of the Berlin Papyrus? (Eleanor) Robson, the ³ÉÈË¿ìÊÖ Egyptian expert, may have read of Schack-Schackenbury's proposal, as Clagett properly footnoted it. My review of the ³ÉÈË¿ìÊÖ RMP program Robson, like Clagett, improperly discussed Egyptian division as 'single false position', within other algorithms that were not present in the RMP. In RMP 38 and RMP 79 Ahmes' division operation was inverse to a version of our modern multiplication operation, as discussed by



in one short paragraph:

Weights and measures units were scaled as the 2/n tables were scaled. For example, RMP 69 scaled 80 loaves of bread made from 3 + 1/2 hekat of grain to a pesu unit. The pesu unit divided 80 loaves of bread by 3 1/2 hekat and created a rational number distribution method. In RMP 69 the pesu method tracked one loaf made from 14 ro (14/320 of a hekat), or 7/160 of a hekat. The proportional pesu method was used in two Berlin Papyrus problems that solved x^2 + y^2 = 100 and x^2 + y^2 = 400, by applying a geometric analogy. Variables x and y were stated as proportional to one another in the ratio of 1: 3/4 and 2: 3/2, seemingly equal proportions. The scribe solved for 2x in the second problem. Schack-Schackenburg reported RMP 69's use of a proportional method in the context of the Berlin Papyrus proportional method years ago. Clagett footnoted the Schack-Schackenbery suggestion, but misreported Ahmes' pesu and Berlin Papyrus inverse division operation by concluding 'single false position' was present (which surely it was not)"

Please consider not to skip over valid numerical specifics. Your email prematurely jumped to a generalized summary of RMP issues to be placed on one or more ³ÉÈË¿ìÊÖ distribution lists.

In 2002, I published a paper on the Egyptian Mathematical Leather Roll (EMLR), mentioned on Wikipedia per:



A copy of the 2002 paper was submitted to the British Museum, as RMP complaints on your well intended ³ÉÈË¿ìÊÖ "History of the World ... #17 Rhind Mathematical Papyrus program ... were submitted. Shortly thereafter the 2002 submission of the EMLR paper to the BM an email was received from a BM staff member.

I received your email today that read much as the BM email was drafted. A warm statement of concern that outlined no meaningful path to resolved any Egyptian math issue. Please add a note to the ³ÉÈË¿ìÊÖ distribution lists to consider looking for a path to resolve one or more small set of ancient facts, fairly translated into the present.

Finding one error in Robson's thinking leads to hundreds of corrects. Take your finger out of the Dike, and allow the BM owned document to be owned by the world.

That is, holding onto the BM 'status quo', as your email implies, grossly under reports the contents of the updated 1927 views of the EMLR and updated views of the RMP, that do not affirm 1927 views.. F. Hultsch in 1895, H. Schack-Schackenberg in 1900 and G. Daressy in 1906 published wider view of hieratic texts that the BM ignored in 1927, and today per the out-dated and mis-leading ³ÉÈË¿ìÊÖ view of the RMP.

As you may know, the EMLR was companion document to the RMP. Both texts were owned by Henry Rhind, and on his untimely death, both were deeded to the British Museum around 1862. Both document were allowed to gather dust.

Luckily, German scholars that had been reading the Berlin Papyrus since 1862 gained a copy of the RMP and published it 1879, opening a debate that continues to 2010. Eleanor Robson added nothing to the debate! Her view of algorithmic scribal thinking does not parse the wide array of Egyptian math texts.

The EMLR was not unrolled until 1927, 75 years after its delivery to the BM. The EMLR and its 26 lines of texts report the scribal LCM scaled 2/n table method -- a simple method that Robson does not understand in any meaningful historical way.

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