The Rhind Mathematical Papyrus

>>8027713
Very cool, I'll be following this thread.

>>8027713

Me too, sounds pretty cool. Pretty, pretty, pretty cool.

Unfortunately the Egyptians' attitude towards math was purely utilitarian. They did not contemplate theoretical stuff, but only sought methods to perform calculations directly connected to real life problems like measuring areas of polygon shapes etc.

The Rhind Papyrus was originally a single roll that was about 18 feet long by 13 inches high, and writing is found on both sides (although a large part of the backside is blank). A few fragments of this are kept in New York, now. This is the text I will drill into shortly, but I want to mention a few others first. You can skip ahead to the wiki if you like:

https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus

Now, when Mr. Rhind got the papyrus, he also got this small, tough, rolled-up piece of leather that didn't get un-rolled until the 1920s (chemical treatment to do that), just in time to be photographed for this Chace edition that I've checked out. This is a small table of more Egyptian fractions, and it is called the Mathematical Leather Roll:

https://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll

Moscow and Berlin each have a so-called (Egyptian) papyrus as well, and wiki lists a few other items along these lines. I wanted to see if any quadratics would be mentioned in the Rhind papyrus (I know ancients were aware of them), but at first glance it doesn't sound like it. Quadratics, however, were treated by Babylonians, Greeks, and very slightly in the above Berlin papyrus, with the Pythagorean babby example in the link:

https://en.wikipedia.org/wiki/Berlin_Papyrus_6619

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So. Egyptian fractions, not very advanced stuff. Historical interest. Having set the stage, let us recapitulate the CONTENTS of the Rhind papyrus.

The AUTHOR of the text is one Egyptian scribe by the name of Ahmose (or Ahmes), as he identifies himself (along with a pharoah or two) in the TITLE PAGE.

This is followed by a table that does the following: take 2, divide it by all the odd numbers from 3-101, and then convert those fractions into some (arbitrary!) Egyptian fractions, showing work in each case. This is known as the "2/N TABLE", and I don't feel like going into detail on it, so here's a simple link of this part of the text instead, with what was done:

https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus_2/n_table

This is followed again by a much shorter "1-9/10 TABLE" which gives particular Egyptian fractions for each of 1/10 ... 9/10. I haven't checked yet but I would imagine that both of these tables set the stage for the

PROBLEMS. According to Chace, I've come up with a total of 91 problems; these will be the focus of the remainder of my treatment. As for Chace's treatment, they are numbered as 1-87 inclusive, along with four additional items noted as 7B, 59B, 61B, and 82B. Note that this may vary slighly from wiki's treatment of the overall text (my library book is literally 90 years old. Conventions may have changed since then.)

And that's the Rhind Papyrus, in a nutshell.

Pic related is a nice look at the Chace treatment of the Egyptian, which isn't really what I"m interested in. He has relatively modern transliterations of the problems, which is where I'll spend the rest of my time. I reserve the right to summarize what's being done in even more modern terms.

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The first 21 problems, problems 1-20, 7B are simple divisions and multiplications, which will probably have been reckoned in terms of the above "1-9/10 TABLE" once I finish going through them. So let me first simply give the modern version of this much smaller table, which comes right before the problems begin. Arbitrarily chosen Egyptian fractions!

[math]

\displaystyle \frac{1}{10} = \frac{1}{10}, \\
\displaystyle \frac{2}{10} = \frac{1}{5}, \\
\displaystyle \frac{3}{10} = \frac{1}{5} + \frac{1}{10}, \\
\displaystyle \frac{4}{10} = \frac{1}{3} + \frac{1}{15}, \\
\displaystyle \frac{5}{10} = \frac{1}{2}, \\
\displaystyle \frac{6}{10} = \frac{1}{2} + \frac{1}{10}, \\
\displaystyle \frac{7}{10} = \frac{2}{3} + \frac{1}{30}, \\
\displaystyle \frac{8}{10} = \frac{2}{3} + \frac{1}{10} + \frac{1}{30}, \\
\displaystyle \frac{9}{10} = \frac{2}{3} + \frac{1}{5} + \frac{1}{30}. \\

[/math]

I should also mention that for whatever reason, the Egyptians/Ahmose allowed an exception to this "get everything into Egyptian fractions" condition: /he allows of the fraction [math] \frac{2}{3} [/math] ./ And it is used above.

For this very simple fragment, I did allow myself to check the original, and I rapidly made sense of the glyphs. Kinda neat.

Now the problems themselves. 1-6 are simple divisions, and so are simple repetitions of the previous table's information, with work shown to prove out the concepts.

Problems 1-6 are, respectively, "Divide 1,2,6,7,8 and 9 loaves among 10 men. Each man receives x [quotient]." In every case, the appropriate Egyptian fraction from the table just posted is given as an answer, being that quotient.

Some multiplications follow, but I want to check these first. This is where I start working through the thing.

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7-20, 7B are prosaic multiplications, and 7B is a simple re-statement of 7 (as well as of 10, just with slightly different work shown), so now I understand what those "B" problems are about, they're just copies/errata. For whatever reason, Ahmose keeps using the same two multiplicands over and over again here, so I'll just tidy it up by starting with what he uses and making my own shorthand. Ahmose's original Egyptian-fractional rendering is assigned by me as two (modern) constants, S as in Seven-fourths, and E as in Eight-fourths, thus:

[math] \displaystyle \frac{1}{1} + \frac{1}{2} + \frac{1}{4} = \frac{7}{4} = S \;\;\; ; \;\;\; \frac{1}{1} + \frac{2}{3} + \frac{1}{3} = 2 = E . [/math]

The problems, then, amount to the below. I think this will start getting less dreary soon, but in any event I'm committed now. No, there's no typos below (shouldn't be), he did write (effectively) the same problem thrice over. Notice that multiplier, multiplicand and product are all given as Egyptian fractions.

[math]

\displaystyle P.7: \bigg( \frac{1}{4} + \frac{1}{28} \bigg)S = \frac{1}{2} \\
\displaystyle P.7B: \bigg( \frac{1}{4} + \frac{1}{28} \bigg)S = \frac{1}{2} \\
\displaystyle P.8: \frac{1}{4} E = \frac{1}{2} \\
\displaystyle P.9: \bigg( \frac{1}{2} + \frac{1}{14} \bigg)S = 1 \\
\displaystyle P.10: \bigg( \frac{1}{4} + \frac{1}{28} \bigg)S = \frac{1}{2} \\
\displaystyle P.11: \frac{1}{7} S = \frac{1}{4} \\
\displaystyle P.12: \frac{1}{14} S = \frac{1}{8} \\
\displaystyle P.13: \bigg( \frac{1}{16} + \frac{1}{112} \bigg)S = \frac{1}{8} \\
\displaystyle P.14: \frac{1}{28} S = \frac{1}{16} \\
\displaystyle P.15: \bigg( \frac{1}{32} + \frac{1}{224} \bigg)S = \frac{1}{16} \\
\displaystyle P.16: \frac{1}{2} E = 1 \\
\displaystyle P.17: \frac{1}{3} E = \frac{2}{3} \\
\displaystyle P.18: \frac{1}{6} E = \frac{1}{3} \\
\displaystyle P.19: \frac{1}{12} E = \frac{1}{6} \\
\displaystyle P.20: \frac{1}{24} E = \frac{1}{12} \\

[/math]

The egyptians were black too werent they?

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>>8028115
No. Does their visual art make them look black?

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>>8028158
Yes.

>>8028225
They're just tanned (those in your pic who even have darker skin, which is only like half of them all). Their facial features don't look negroid at all.

>>8028158
Ancient egyptians were black as in skin color but not negroid.

>>8028249
There's no such thing as negroid. if they were black Africans they were black Africans.

>>8028264
Do you think only africans can have black skin color retard ?

>>8028271
More proof that defining people by their skin colour is dumb as shit.

>>8028264
>anthropologists had been so wrong for centuries, thankfully tumblerites finally arrived and set anthropology straight

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>>8028287
>19th century social theorists were never wrong
Funny how if /sci/ agrees with a theory it's the right one, if /sci/ doesn't agree with it (race, climate change, string theory) it's a big conspiracy by the joos to hide the real theory.

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>>8028158

Not him but traditional visual art of royalty is a shitty 1:1 representative of the composition of a given nation's population.

Using this logic one would surmise there were virtually no native americans and very few blacks in america using traditional monument/statues and paintings as evidence.

Obviously this would be wrong since they did exist in decent numbers but had little representation in the ruling government.

Same shit could have happen in egypt which would make sense since Nubia did trade with them and eventually got it's shot in ruling over the nation for a bit.

>>8028301
Come on, the whole "race is skin-deep" nonsense is just political agenda that was pushed way futher than political agendas should ever be allowed to be pushed if science was to remain integrity. Anyone who's not a halfwit and not commited to that political agenda knows it.

>>8028098

Things get slightly more interesting in passing from the first 20 problems to the next 20; we pass now from arithmetic into /algebra/, and although "a quantity" is mentioned more than once in a given problem, all we are really doing throughout is solving for x in what all(?) ultimately simplify to linear equations. Ahmose seems to understand "solve for x" and his own manipulation methods for actually doing so, but he doesn't seem at this point to have quite our same notion of a linear equation.

That said, I am simply summarizing the mathematics. What I'm doing is to re-state the problems in modern form, then wade through each problems' "work" to pick out what looks like the solution, and check whether I agree. So far I haven't noticed a straight-up goof (Ahmose is supposed to have made these as well); I'm through problem 30 and I haven't caught a bad "final answer" yet, though perhaps there are goofs in his work-steps, which I'm not reading.

The /language/ of these problems' statement is quite similar to our abstraction in that "a quantity" is usually spoken of; only a few are concrete story-problems (divide these loaves, use those hekat...) A "hekat", by the way, is a volume unit of about 5 liters that they used for grain.

https://en.wikipedia.org/wiki/Hekat_%28unit%29

I shall present modern summaries of problems 21-30 in the sequel, followed by 31-40. Then the Rhind Papyrus passes into geometry, which should be a bit more interesting.

Actually, even the ordering of the material also makes some modern sense: rote tables -> arithmetic -> algebra -geometry -> presumably some slightly more complex problems at the end.

Here is a modern summary of problems 21-30. One item of interest here is that the lcm of the denominators used in the long problem 23 is precisely 360, a number we can thank Babylon for IIRC.

[math]

\displaystyle P.21: \bigg( \frac{2}{3} + \frac{1}{15} \bigg) + x = 1 \;\;\; \rightarrow \;\;\; x = \frac{1}{5} + \frac{1}{15} \\
\displaystyle P.22: \bigg( \frac{2}{3} + \frac{1}{30} \bigg) + x = 1 \;\;\; \rightarrow \;\;\; x = \frac{1}{5} + \frac{1}{10} \\
\displaystyle P.23: \bigg( \frac{1}{4} + \frac{1}{8} + \frac{1}{10} + \frac{1}{30} + \frac{1}{45} \bigg) + x = \frac{2}{3} \;\;\; \rightarrow \;\;\; x = \frac{1}{9} + \frac{1}{40} \\
\displaystyle P.24: x + \frac{1}{7} x = 19 \;\;\; \rightarrow \;\;\; x = 16 + \frac{1}{2} + \frac{1}{8} \\
\displaystyle P.25: x + \frac{1}{2} x = 16 \;\;\; \rightarrow \;\;\; x = 10 + \frac{2}{3} \\
\displaystyle P.26: x + \frac{1}{4} x = 15 \;\;\; \rightarrow \;\;\; x = 12 \\
\displaystyle P.27: x + \frac{1}{5} x = 21 \;\;\; \rightarrow \;\;\; x = 17 + \frac{1}{2} \\
\displaystyle P.28: \bigg( x + \frac{2}{3} x \bigg) - \frac{1}{3} \bigg( x + \frac{2}{3} x \bigg) = 10 \;\;\; \rightarrow \;\;\; x = 9 \\
\displaystyle P.29: \frac{1}{3} \Bigg( \bigg( x + \frac{2}{3} x \bigg) + \frac{1}{3} \bigg( x + \frac{2}{3} x \bigg) \Bigg) = 10 \;\;\; \rightarrow \;\;\; x = 13 + \frac{1}{2} \\
\displaystyle P.30: x \bigg( \frac{2}{3} + \frac{1}{10} \bigg) = 10 \;\;\; \rightarrow \;\;\; x = 13 + \frac{1}{23} \\

[/math]

>>8028115
>>8028158
>>8028225
just as black as they are today

>>8027713
This is a really neat thread. That Wikipedia article (among other things ) encouraged me to start learning ancient Egyptian a while ago. I never really stuck with it though.

>>8028631

This is actually false. The Nubians would have been highly intermingled with Egypt at this time. There have been a few negroid groups that once lived in Egypt and have since been dispersed back into continental Africa.

>>8027716
>>8027746
>>8028648

Happy to hear that some anons appreciate the thread. I thought the thread might have a we wuz kingz spate (it did) but hopefully we're past that.

Grinding into this, my pace is slowing down, which is a nice sign. The remaining problems up through 40 are solutions of linear equations as expected, but a bit more "culture" is involved: a /ro/ is 1/320 of a hekat, as given above, and certain solutions are alternately expressed in terms of "Horus-eye" fractions. These are simply reciprocals of low powers of 2, which were culturally associated with parts of pic related. We are veering dangerously close to pseudoscience /x/ territory (in a modern context), but I make the mention since this is a historical document after all.

Problem 35 in particular involves finding the volume of an unknown unit measure, versus the established hekat measure, and has a somewhat confused original wording. I shall gloss over these to present 31-35, and eventually (once I've checked them), 36-40. Then the geometry, later.

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>>8028732

This is the picture meant to go with this post, whoops. no harm done.

>>8027759
>Unfortunately the Egyptians' attitude towards math was purely utilitarian.
Yes, engineers are not scientists; they are not engineers who love to think that it is worthy dwelling in your mental proliferation.

Problems 31-35 can be paraphrased as follows. 31 has a particularly onerous solution, but Ahmose seems to like using factors of 28 and certain small primes for his examples. Or at least, his predecessors did - the scroll is described as repeating methods previously handed down.

[math]

\displaystyle P.31: x + \frac{2}{3} x + \frac{1}{2} x + \frac{1}{7} x = 33 \;\;\; \rightarrow \;\;\; x = 14 + \frac{1}{4} + \frac{1}{56} + \frac{1}{97} + \frac{1}{194} + \frac{1}{388} + \frac{1}{679} + \frac{1}{776} \\

\displaystyle P.32: x + \frac{1}{3} x + \frac{1}{4} x = 2 \;\;\; \rightarrow \;\;\; x = 1 + \frac{1}{6} + \frac{1}{12} + \frac{1}{114} + \frac{1}{228} \\

\displaystyle P.33: x + \frac{2}{3} x + \frac{1}{2} x + \frac{1}{7} x = 37 \;\;\; \rightarrow \;\;\; x = 16 + \frac{1}{56} + \frac{1}{679} + \frac{1}{776} \\

\displaystyle P.34: x + \frac{1}{2} x + \frac{1}{4} x = 10 \;\;\; \rightarrow \;\;\; x = 5 + \frac{1}{2} + \frac{1}{7} + \frac{1}{14} \\

\displaystyle P.35: \bigg( 3 + \frac{1}{3} \bigg) x = 1 \;\;\; \rightarrow \;\;\; x = \frac{1}{5} + \frac{1}{10} \\

[/math]

btw I encourage checking of thes babby-problems.

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>>8028777

Let me next present 36-39. I want to hold 40 separate, as it appears to ACUTALLY BE SOMEWHAT INTERESTING!!!

[math]

\displaystyle P.36: \bigg( 3 + \frac{1}{3} + \frac{1}{5} \bigg) x = 1 \;\;\; \rightarrow \;\;\; x = \frac{1}{4} + \frac{1}{53} + \frac{1}{106} + \frac{1}{212} \\

\displaystyle P.37: \bigg( 3 + \frac{1}{3} + \frac{1}{3} \frac{1}{3} + \frac{1}{9} \bigg) x = 1 \;\;\; \rightarrow \;\;\; x = \frac{1}{4} + \frac{1}{32} \\

\displaystyle P.38: \bigg( 3 + \frac{1}{7} \bigg) x = 1 \;\;\; \rightarrow \;\;\; x = \frac{1}{6} + \frac{1}{11} + \frac{1}{22} + \frac{1}{66} \\

\displaystyle P.39: Given \;\; 6x = 4y = 50, find \;\; y-x. \;\;\ \rightarrow \;\;\; y-x = 4 + \frac{1}{6} \\

[/math]

>>8028844

Problem 40 concludes the "simple algebra" section, and has the added bonus of having enough stuff going on to be significantly more interesting than what has preceded it (in our accepted babby-terms).

After #40, the next several problems are the "geometry" section, and it is clear to me from skimming that this middle part (including this problem 40) is the "sexy" part of the text. Following the geometry section are some more prosaic "miscellaneous" bits.

Problem 40 has two phrasings because Chace provides both the original (much shorter) Egyptian transliteration, and a (relatively) modern English paraphrase of his own. But Chaces' paraphrase is complex/modern enough (sufficiently out of character with what has gone before) that I wanted to check the original, and there is a discrepancy in language, as I expected. Basically Ahmose was using a method and making assumptions that he doesn't "show work" on.

Problem 40 goes like this. Loaves are divided among men, and some bulleted conditions and questions are put on this.

1) 100 loaves are unevenly divided among 5 men according to the below,
2) the 5 mens' shares are in arithmetic progression, (according to Chace and Ahmoses' after-the-fact statement, this is "assumed", /not actually stated by Ahmose/, and fixed per 4), but we should consider a more general case for the fun of it)
3) 1/7 of the sum of the three largest shares is equal to the sum of the smallest two shares,
4) /The difference of the shares is "arbitrarily" fixed by Ahmose as 5 1/2, before he works the problem, almost after done stating it.

Simplifying this philology, the essential statement of the problem is as follows, where x_1 is the (presumably) largest share, x_5 the smallest:

[math]
\displaystyle x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 100 \\
\displaystyle \frac{1}{7} \big( x_{1} + x_{2} + x_{3} \big) = x_{4} + x_{5} \\
[/math]

The arithmetic progression and Ahmoses' "5 1/2" are overlaid on this, which I will poke around with next.

>>8028887
But there was a hidden problem!


Problem 41:
n>2

a^n+b^n=c^n -> a=..., b=... and c=... ? It says proof is left to the reader but I can't find any simple proof.

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>>8028734
There's more to that symbol

>>8028282
There's nothing dumb about using skin color as a statistical feature.

It has good predictive properties by itself and when combined with other features you have an even better prediction.

Leaving it alone for tonight, 40 has a few different angles to it and I'm drunk.

If you like this thread, be a lamb and bump it overnight.

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Good stuff OP!

>>8028916

Don't be a Fer-memer.

>>8027909
https://en.wikipedia.org/wiki/Egyptian_fraction#Early_history

This wiki page says some interesting things about their notation, including:
>The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size of numbers greater than 1/2 when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation.

So I guess if I had some fraction like 5/4 then I would write it as a sum of 3/4 + (stuff smaller than 1/2 in proper egyptian fractions).

The wiki also mentions some other special notation for numbers of the form 1/2^k that were used when dealing with real world stuff. Makes sense, sounds like how we use tenths, hundredths, thousandths, etc..

>>8028158
It would've helped your argument if you had actually posted a statue that didn't look like a black woman.

>>8028312
>thinks the argument is "race is skin-deep".
How embarrassing for you. Let me help you out anon. Either learn population genetics or gb2/pol/.

>>8028916
I lol'd.

>>8029120
You're wrong. If you disagree with the fact that "race is skin-deep" and look for genetic variances you should gb2/pol/. Because we're all identical and phenotypes aren't real.

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>>8029133
>This is what /pol/ actually thinks /sci/ actually thinks.

>>8029140
Racist science isn't science you bigot. Science must be politically correct without discrimination. And yes, this is what /sci/ thinks.

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>>8029143
>moooooom come look at my awesome false flag!

Keep going retard. This is the best laugh I've had all day. You /pol/esmokers sure are dumb.

>>8029150
Are you retarded or something ? /pol/ is for political incorrectness, /sci/ is for science. Science can't be politically incorrect.
> false flag
> reaction image replies
You sure sound like /pol/ yourself.

>>8029150
The International Academy of Science, The National Academy of Science and the Human Genome Project have all officially rejected "Scientific Racism" as being pseudoscience.
They have provided evidence and proofs.

Deal with it. Your parents and community lied to you, as did Stormfront. Back to pol please.

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>>8029153
>/pol/ is the only board with reaction images and image macros.
You should return to your containment board. It misses you.

>>8029154
Replied to the wrong anon but yes you are correct.

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>>8028225
Your evidence is shitty

>>8029157
Political correctness is what we follow to avoid racial discrimination, which applies to science as well. Discussing inexistent pseudoscience like genetic variances and population genetics is what we call scientific racism.
see : >>8029154

>>8027909
strange (becoz inconsistent) that they would write
[math] \displaystyle \frac{7}{10}= \frac{2}{3}+ \frac{1}{30}[/math]
... rather than
[math] \displaystyle \frac{7}{10}= \frac{1}{2}+ \frac{1}{5}[/math]
... ainnit?!

>>8029167
Population genetics has nothing to do with race. Race doesn't exist this isn't because of political correctness, but because genetics actually turned out to be much more complex.

"Racial science" and "scientific racism" are what tinfoil hat retards who can't into science (i.e. /pol/) use to justify their racism. Like you pointed out though, those things are now considered pseudoscience.

>>8028933
the dumb, it hertz

>>8029175
Population genetics is just a codeword for "racial science" or some other bullshit like that. Discussing genetic variances among ethnic groups is what causes discrimination and highlights differences between groups of people. Nobody cares if you call it something else than "race".
Why do you even care about it so much that you have to discuss it ?

>>8028964
OP can you Indian and Chinese math next?

>>8029170
I'm pretty sure OP went to sleep but I think the reason lies with this.
>>8029102

What surprises me more is that they didn't write 3/4 in 8/10 and 9/10. I wonder if that symbol just wasn't used as often.

That wiki page says some really interesting things under the equally distributing objects section. I can actually see why these Egyptian numbers are practical and not just a quirky limitation of their writing system.

https://en.wikipedia.org/wiki/Egyptian_fraction#Equally_distributing_objects

>>8029177
This.

I actually know two autistic /pol/ retards at my university. They constantly spout memes irl, talk about /pol/ and 4chan, and rant racist shit with each other and at anyone who will listen.

>>8029181
In population genetics you can group populations in any way. For instance you can look at two different villages belonging to the same tribe as different populations and study their genetic distance.

Traditional notions of race on the other hand instead rely on physical features (nowadays considered shitty due to findings related to convergent evolution) and then try to make inferences about the most retarded shit.

They are different things, anon. I don't think you could even abuse population genetics to study traditional notions of race if you wanted to.

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>>8028264
>>8028312
>>8028933
>nooo, IAS, NAS, and HGP are all wrong
>La La La La Laaa, I can't heeeear you!
fgt pls

>>8028887

#40, cont.

Before I go further, the OBJECT of the problem is to find the difference of shares (which are in arithmetic progression, so you want a single number).

This is also the first situation where I feel compelled to comment on Ahmose's work. Upon review, I mis"spoke" earlier, and he does both "assume" /and actually state/ an expected arithmetic progression throughout. To solve, he takes a nicer version of his original problem that instead adds up to 60, and then simply scales it. He knows that this (scaling) works, but it passes without comment. Let's put on our modern-cap for a bit.

Let x_5, the smallest term, be 1. then just keep adding 5.5 (11/2) for each successive term, thus taking care of the arithmetic progression. x_1, the largest term, is 23 just now. The sum of the lot is 60, and the 1/7 relationship checks out.

Now to scale the lot up to 100, multiply by 5/3 (or, 1 + 2/3). Everything checks out again, and then you can convert into relatively easy egyptian fractions to wind up with the same thing that Ahmose did. But he doesn't state the solution as-such!

The answer is simply (11/2)(5/3) = 55/6, or, as the Egyptian would say, 9 + 1/6. An ugly-looking set of conditions turns out to be consistent, and may be expressed thus:

[math]

\displaystyle P.40: x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 100 \\

\displaystyle \bigwedge x_{1} + x_{2} + x_{3} = 7( x_{4} + x_{5} ) \\

\displaystyle \bigwedge x_{1} > x_{2} > x_{3} > x_{4} > x_{5} \\

\displaystyle \bigwedge x_{1} - x_{2} = x_{2} - x_{3} = x_{3} - x_{4} = x_{4} - x_{5} = \Delta \\

\displaystyle \rightarrow \Delta = \frac{55}{6} = 9 + \frac{1}{6}

[/math]

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>>8029133
>correlation proves things
Found the undergrad. Wait at least until after stats to post.

>>8029572
Out of curiosity, are the arguments in the text written out in prose or do they use some sort of syntax or what?

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>>8029572

And yet, this single tersely-worked problem is unsatisfying to a modern for two reasons: it is the first problem in the set to suggest a generalization of any interest (which we present in pic related), and the author seems to have pulled his solution out of thin air (or perhaps chose his problem to coincide nicely wiht an already-worked solution). In other words, one ought to be able to derive a solution without having Ahmoses' specific-case method spoon-fed.

We can begin a modern solution by taking three of the above four conditions as a simpler starting point, and working towards the fourth. This is worked out to my satisfaction in pic related. The short version is that in Ahmoses' problem as stated his solution is not only correct, but more interestingly, unique.

THERE. Now I'm done with #40, and what feels like the first half of the Rhind Papyrus. 41-87 remain, being simple geometry and some miscellany (more algebra I guess).

>>8029700

The original Egyptian (just to skim it) and its modern English transliteration are not flowery. It's actually pretty to-the-point: set up problem, display calculations, declare answer. The hieroglyphs make the ideas being expressed look more complex than they really are. So far it's the usually the simple statement of the problem in sentence form (perhaps with some number-forms, just as we would do), followed by the calculations, just as a schoolboy would turn in his homework, but with more-adult motivational notes a la "we did this because this, okay now add that to this", etc.

For example, what follows is the literal modern-English transliteration of the problem #40 that I just spent some time with, and I agree that it captures the spirit of the original:

"PROBLEM 40: Divide 100 loaves among 5 men in such a way that the shares received shall be in arithmetical progression and that 1/7 of the sum of the largest three shares shall be equal to the sum of the smallest two. What is the difference of the shares?"

Even that treatment is too wordy. The Egyptian was more like "Loaves 100 for man 5, 1/7 of the 3 above to man 2 those below. What is the difference of share?"

So we're (originally) dealing in relatively tight language, there are some units of measure, and we even speak of "a (unknown) quantitiy" rather than using some concrete-noun every time. But we haven't quite (yet) advanced to the point of writing someting like " 3x = 2, solve for x".

Apart from the language itself, the most persistent mathematical syntax are the aforementioned units of measure, and the arbitrary conventions for how to express Egyptian fractions. The glyphs themselves and syntax end up expressing the same thing as "1/7", "1/8" or whatever unit fraction, and with about as much effort as it took me to write those.

I salute you for this quality thread, OP.

GEOMETRY: 41-60

Grain silo time. Problems 41-43 are calculations of volumes of (right circular) cylinders, which in itself is straightforward in modern terms.

These stick out for three reasons: first inexact and incorrect answers (approximating pi), first serious mathematical mistakes besides (#43), and tacking on more units after each "main" problem, so a little dimensional analysis. Thus we review both units and volume methods used. Yes, we're using literal cubits as units, now. :D

A CUBIT is a unit of LENGTH, from which a unit of VOLUME, a CUBIC CUBIT, may be derived in the usual way.

2/3 of ONE CUBIC CUBIT = 1 KHAR (another VOLUME unit)

A HEKAT (mentioned earlier, ctrl+f for the wiki, it's useful here) is a unit of VOLUME.

1 KHAR = 20 HEKATS, 2 HEKATS = 1 DOUBLE HEKAT, and 4 HEKATS = 1 QUADRUPLE HEKAT.

Note that wiki's treatment of khars doesn't apply here. It's one of those goofy things that changed as time passed.

The pertinent volumes are given below, and the expanded forms are meant to show the original methods. V41,42 is a nice one desu, it's within 1%. Since the problems work in terms of diamaters, we supply the exact version for comparison. the "Elsewhere" volume is something that Ahmose seemed to copying (from another text) for problem 43 to be cute and shortcut a certain unit conversion, but he fucked up by doing the 1/9 step, resulting in an 8/9^2 error term.

[math]

\displaystyle V = \pi r^{2} h = \frac{ \pi }{4} d^{2} h \\

\displaystyle V_{41,42} = \bigg( d - \frac{1}{9} d \bigg)^{2} h = \frac{64}{81} d^{2} h\\

\displaystyle V_{elsewhere} = \frac{2}{3} h \bigg( d + \frac{1}{3} d \bigg)^{2} = \frac{32}{27} d^{2} h = \frac{3}{2} V_{41,42} \\

\displaystyle V_{43FUBAR} = \frac{2}{3} h \bigg( \bigg( d - \frac{1}{9} d \bigg) + \frac{1}{3} \bigg( d - \frac{1}{9}

d \bigg) \bigg)^{2} = \frac{2048}{2187} d^{2} h = \bigg( \frac{8}{9} \bigg)^{2} V_{elsewhere} \\

[/math]

>>8030488

Modern treatment of problems 41-43. I only express the Egyptian fraction results given for the "units" of these, which are cubic cubits (the others generally seem to check out and I'm uninterested in them):

See if you can spot any goofs between this and the above setup post!

P. 41: "Given a cylinder with d = 9 cubits and h = 10 cubits, find V and V_{41,42}, and express the latter in terms of cubic cubits, khar and quadruple hekats."

[math]

\displaystyle V = \frac{405}{2} \pi \;\; cubit^{3} \;\;\; ; \;\;\; V_{41,42} = 640 \;\; cubit^{3} = 960 \;\; khar = 4800 \;\; quad.hekat

[/math]

P. 42: "Given a cylinder with d = 10 cubits and h = 10 cubits, find V and V_{41,42}, and express the latter in terms of cubic cubits, khar and quadruple hekats. Further express the latter cubic cubit result in Egyptian fractions."

[math]

\displaystyle V = 250 \pi \;\; cubit^{3} \;\;\; ; \;\;\; V_{41,42} = \frac{64000}{81} \;\; cubit^{3} = \frac{32000}{27} \;\; khar = \frac{160000}{27} \;\; quad.hekat \\

\displaystyle = 790 + \frac{1}{18} + \frac{1}{27} + \frac {1}{54} + \frac{1}{81} \;\; cubit^{3}

[/math]

>>8030594

Due to its extant mistakes and various forms of unit conversion, problem 43 has multiple angles which have been set up above and which I have paraphrased like so:

P.43: "Ahmose has made a mistake, and so let's check his existing work and fix it. Given a cylinder with d = 9 cubits

and h = 6 cubits, do the following:

a) Find V, [math] V = \frac{243}{2} \pi \;\; cubit^{3} [/math]

Find V_{43FUBAR} and express it (thereby "verifying" what Ahmose did) in

b) cubic cubits, [math] = \frac{8138}{27} \;\; cubit^{3} [/math]
c) khar, [math] = \frac{4096}{9} \;\; khar [/math]
d) and quadruple hekats... [math] = \frac{20480}{9} \;\; quad.hekat [/math]

...Then, fix Ahmoses' mistake by finding V_{elsewhere} and expressing it in

e) cubic cubits, [math] = 384 \;\; cubit^{3} [/math]
f) khar, [math] = 576 \;\; khar [/math]
g) and quadruple hekats. [math] 2880 \;\; quad.hekat [/math]

44 and 45 are mercifully simple, after this.

P.44: A cube measures 10 x 10 x 10 cubits. Find its volume in terms of quadruple hekats.

Ans: 7500, with straightforward conversions.

P.45: A cube's volume is 7500 quadruple hekats. Find its edge length.

Ans: 10 cubits. This is an exact reversal of 44.

P.46: A rectangular prism's volume is 2500 quadruple hekats. describe its three dimensions in terms of cubits.

Ans: This is of course amenable to an infinity of solutions, but " 10 x 10 x 10/3 cubits" is preferred, and given. I actually need to think about this one a bit more, so this post is slightly premature.

47-49

Problem 47 amounts to a small multiplication table, done in certain units (including the small /ro/, check the hekat wiki), all of which check out. A phrase suggests cylindrical vs. rectangular volume (whether in a rectangular or circular granary...), but this seems to be a way of indicating generalization on the "volume" concept of previous problems, and not a relevant restriction from a modern POV.

P.47: "Give the products when 100 is multiplied by 1/10, 1/20, 1/30 ... 1/100". Ans: (and, they are correctly given).

P.48: "Compare the area of a circle of diameter 9 and of its circumscribing square, which also has side length 9."

The comparison is given flatly as 64/81, as opposed to pi/4. This is the most explicit statement yet of (this) Egyptian method for finding the area of a circle; this slight overestimation is consistent with the method of problems 41-42.

Problem 49 is a rectangle computation, and introduces another unit of measurement, a khet, which is either a unit of area or length, and bears a happy "metric" relationship to either one. A khet is 100 square(?) cubits, or perhaps 100 cubits, and while we're at it, here's a helpful page on Egyptian units:

https://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurement

The problem's phrasing is slightly goofy by our usage, and between this and wiki, . P.49: "A rectangle ("10 khets by 1 khet") is like so. Express this area in terms of square cubits". Ans: 1000 square cubits. The setup of course suggests a khet as a unit of length, but this problem is dumb-simple enough that I'm not worried about conversions or minutae, except to point out that "khet" may be interpreted as length or area depending on what's going on (wiki seems to want to say it's a unit of area).

Problem 50 strengthens Rhinds' internal logic that a khet is a unit of length. We have a round field of diameter 9 khet (length!) and we are to find its area. It is 64 setat (square khet). ~37 to go and I mean to finish it.

overnight bump. same encouragement.

>>8031144
Bump for OP

>>8029801
I'm loving these posts, OP

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>>8031144

I wanted to hand-wave about 49, but I was unsatisfied with my treatment of it both because I didn't understand the units being used at-a-glance, and also because my answer of "1,000 square cubits" is clearly wrong.

Ignore wiki for the purposes of 49, and refer to pic related. For right now...

A KHET is a unit of LENGTH, equal to 100 CUBITS.

ONE SQUARE KHET, therefore, is equal to 10,000 SQUARE CUBITS, both obviously units of AREA.

There is another name for ONE SQUARE KHET, itself manifestly a unit of AREA. ONE SETAT is equal TO ONE SQUARE KHET, or 10,000 SQUARE CUBITS, as just stated.

Here is the confusion about 49, which Chace clears up, and which I misrepresented earlier. What's being counted are strips of area measuring 1 cubit by 100 cubits. For clarity, Chace dubs these "CUBIT-STRIPS", to differentiate them from the above. So let's try 49 again.

P.49 (take 2): A strip measures 10 khet by 1 khet. Express its area in terms of CUBIT-STRIPS. Ans: 1,000 CUBIT-STRIPS.

Pic related clears this up, it's basically alloting acreage or plots on a given block, or parcel, as we'd say today My confusion also seems indicitive of Egyptian confusion on concepts of length vs. area and their dimensional representations.

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>>8032115

Now that we've got our units straight, 50 offers no confusion. It is just a repetition of the "64/81" rule for squares and circles that we've seen multiple times already, with a "unit conversion".

P.50: "A circle has a diameter of 9 khet. Express its area in setat." Ans: 64 setat, a nice 1/1 "unit conversion" as 1 setat = 1 khet^2.

Now, for problems 51-53, you can compare the original, especially the diagrams, by taking a GOOD CLOSE LOOK at the OP >>8027713 picture, which is decent quality albeit with some annoying watermarks. The areas in the middle of the picture contain 51 (triangle), 52 (trapezoid) and 53 (triangle with tiers on it), going down.

There seems to be a little confusion in 51, over whether the author had a grasp on the notion of a right triangle versus an isosceles, or scalene triangle. That drawing looks pretty close to a right-triangle, but not quite, and you can judge for yourself. :^)

I believe just from skimming that Ahmose uses a straightforward A = (1/2 b"h") argument in 51, taking one of the sides as the "height", (and the diagram would seem to bear this out) but I'll have to check.

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Short version of 51:

P.51: A triangle has a base of 4 and an altitude of 10. Find its area. Ans: 20.

Long version of 51: pic related.

The motivation for this autism is that the figure almost, but not quite! suggests a right triangle. We then wonder just how well Ahmose understood finding areas of triangles, in general, and it prompts a review of certain things.

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If 51 suggests some problems, 52 is a literal "Thanks Common Core!" trapezoid problem. Short version first, making certain assumptions and giving Ahmose the benefit of the doubt:

P.52: A trapezoid has two bases 6 (khet) and 4 (khet) , and an /altitude/ of 20 (khet). Find its area. Ans: 100 (setat, or square khets, via the usual method).

The issue is much the same as 51, pic related. I'm not going to beat this one to death just now like I did the other.

53 is of a piece with 51 and 52, has its own challenges, and I'm still thinking about it. In the interest of productivity, let me paraphrase the much simpler 54 and 55 (dumb-simple division problems masquerading as geometry), and get them out of the way.

P. 54: There are 10 different fields, and you have to carve equal areas out of each one, all 10 of which sum to 7 setat. What is the area of one (each) of them? Ans. 0.7 setat (expressed as 1/2 + 1/5).

P. 55: There are 5 different fields, and you have to carve equal areas out of each one, all 5 of which sum to 3 setat. What is the area of one (each) of them? Ans: 0.6 setat (expressed as 1/2 + 1/10).

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53 now. It has a diagram, but is plagued by the same issues as 51 and 52.

"This simple (charitable!) paraphrase: "Given an isosceles triangle of altitude 14 and base 9/2, divided into sectors per the figure, find the two bases and the three areas of the sectors."

Ans: pic related.

My big personal takeaway from this recreation is that the Egyptians were just fine about manipulating fractions, but really sucked hard at geometry.

Problems 56-60 deal with, of all things... pyramids! And they have some fancy diagrams to yield up. However given my pace with the above and a real urge to finish this thread in one go for better or worse, I've made a decision in the interest of productivity for now to skip 55-60 for the time being, and skip ahead to 61-87, the miscellany.

61 is a very simple table of multiplications of arbitary (Egyptian) fractions, all of which is correct and checks out. As one representative example, " 2/3 x 1/3 = 1/6 + 1/18 ".

61B comes up, and is a rule for taking 2/3 of the reciprocal of an odd number, using 1/5 as a brief example (add 1/10 and 1/30, is what it amounts to), "and so for any other odd number". It's right in this case and demonstrates a correct method, and although odd, I can let it pass for now. He wants to express it in egyptian fractions is the issue, but the sum works.

62 is some linear combination involving a money-unit and a weight-unit of three precious things, in a heap. I'm probably not turned in yet but I'm quite serious about finishing this so any further overnight bumps are appreciated (thx guy >>8031490 !) I hope to dispense with the miscellany and be into the pyramids to wrap it up between tonight and tomorrow, but personal life intrudes, to your surprise.

overnight bump.

Page 4? Bumpitty bump bump then. This is the only /sci/ worthy thread I've seen in weeks.

Now I've got my head all the way around 61B (Ahmose barely writes anything, we have to discern context!)

P.61B: Give a general procedure for converting the product of 2/3 and the reciprocal of any odd number 2n+1 into an egyptian fraction of two terms, e.g.

[math]

\displaystyle \frac{2}{3} \cdot \frac{1}{2n+1} = \frac{1}{p} + \frac{1}{q}

[/math]

With natural p and q.

In Ahmose's words, the procedure is: multiply 2 by your odd number (odd denominator) (which is p), then take the reciprocal of that. Also multiply 6 by your same odd number (q), and take the reciprocal of /that/. The sum of those two is your Egyptian fraction in this case. (And we can clearly see this in modern terms, with a few examples of Ahmoses' discovery, viz.

[math]

\displaystyle \frac{2}{3} \cdot \frac{1}{2n+1} = \frac{2}{3(2n+1)} = \frac{1}{2(2n+1)} + \frac{1}{6(2n+1)} \\

\displaystyle \frac{2}{9} = \frac{1}{6} + \frac{1}{18} \\
\displaystyle \frac{2}{15} = \frac{1}{10} + \frac{1}{30} \\
\displaystyle \frac{2}{21} = \frac{1}{14} + \frac{1}{42} \\
\displaystyle \frac{2}{27} = \frac{1}{18} + \frac{1}{54} \\

[/math]

P.62:

A bag of three precious metals, gold, silver and lead, has been purchased for 84 sh'tay, which is a monetary unit. All three substances weigh the same, and a deben is a unit of weight (or mass if you like, but either way it's immaterial to this simple problem :^).

1 deben of gold costs 12 sh'tay, 1 deben of silver costs 6 sh'tay, and 1 deben of lead costs 3 sh'tay. Find the common weight W of any of the three metals.

Ans: divide the bag's purchase price by the sum of the unit price-per-weight for the three metals to find 4 deben, e.g.

[math]

\displaystyle \frac{84 \;\; sh'tay}{ 21 \frac{sh'tay}{deben} } = W = 4 \;\; deben

[/math]

>>8034331
Ahmose is getting cleverer here

>>8034421

Yes, I thought so too once I took another look and understood what he did. He uses 5 in one example, and basically says " 2x5 with 6x5, and so on".

63 is another linear equation in disguise.

P.63: 700 loaves are to be divided unevenly among 4 men, in 4 unequal, weighted shares. The shares stand to one another in this relationship (I then give the solution in this same chunk of TeX:

[math]

\displaystyle \frac{2}{3} : \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \\

\displaystyle Soln \;\; (shares): \frac{800}{3} , 200, \frac{400}{3} , 100

[/math]

Since the fractions themselves do not sum to 1 but 7/4 (so we don't start out by just carving up 700), and since some x would act as a scalar on the fractions, preserving the relationships, Ahmose recognizes the need to solve for x, which is 400 (convenient next to the 700 and 7/4 terms). The solutions are then given as above.

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I had hoped to have some content ready by now, but I've been busy with RL stuff today which went well - and which now frees me to get srs about this. I sat with the old Chace in my lap in an auto repair shop earlier today.

64 is a redux of 40 with an even number of x's, but I've stupidly been hung up on some details. 65 is another linear combination which I thought to skip ahead and check, but I haven't got it worked out just yet.

I appreciate all the positive comments on this treatise of dumb-simple elementary mathematical history, but I really must be producing more and completing this project. Coming back shortly.

Despite the "we wuz kingz" suggestion of the link in the pic, the diagram may prove helpful for the pyramid-stuff that I skipped.

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P.64: 10 numbers in arithmetic progression sum to ten. The difference between consecutive terms is 1/8. Find this solution.

Ans: {7/16,9/16,11/16,...,25/16}.

The problem is a redux of 40, this time with an odd number of terms. The mathematics is mostly the same, and a generalization is (again) pic related.

P. 65. 100 cash-money-bread are divided unequally among 10 employees. 7 of the slaves (er, employees) each gets a single share, while the other 3 happier employees receive double shares. Compute each share.

This becomes an elementary linear algebra problem entailing 2 equations with 2 unknowns, which doesn't even need gaussian elimination, simple substitution suffices. Ahmoses' method is not to do the substition that we would gravitate towards, and instead to consider "13 (primitive) shares", a natural step, and then he gets it right. The peons get 100/13, and the bigwigs get 200/13.

P.65: Recall that the hekat is a unit of volume. There is also a little-bitty tiny volume of unit called a "ro", and their relationship is 1 hekat = 320 ro. Furthermore, the Egyptian is aware that a year is (commonly) 365 days so just assume 1 year = 365 days for this one.

The pharoah deigns to give out 10 hekat of fat over the course of a year to so-and-so, in an equal amount every single day. Express the fat given out in (arbitrary!) terms of hekat and ro as you see fit, just make sure your (Ahmoses') version is right.

(Ahmoses') Ans: Take into account the unit conversion of hekat and ro above, and also take into account the 1 yr = 365 day above. Do your unit conversions and so-and-so gets 2/73 hekats per day. Or (what checks out)...

[math]

\displaystyle \frac{2}{73} \;\; hekat = \frac{1}{64} \;\; hekat + \bigg( 3 + \frac{2}{3} + \frac{1}{10} + \frac{1}{2190} \bigg) \;\; ro

[/math]

>>8035751

THE PREVIOUS PROBLEM WAS CLEARLY PROBLEM 66. A GOOF ON MY PART.

P.67. Shepherd-dude has to give the lord-conqueror-dude (2/3)(1/3) of his flock, which is 70, to the lord-dude as tribute. So figure out the size of shepherd-dude's original flock before paying the tribute.

Ans: solve the linear equation and the original flock was 315 mang.

P.68: 4 overseers have gangs of men, and the men in their gangs work at identical linear rates so go ahead and assume their labor is fungible, etc.

Anyway, working on the same interval of time these 4 overseers and their gangs have collectively produced 100 units (quad.hekat-of-such-and-such, doesn't matter).

One overseer has a gang of 12,
the next has a gang of 8,
the next has a gang of 6,
the next has a gang of 4.

How much did each gang produce during the interval (that the overseer can take credit for)?

Ans. Straightforward linear decomposition of the orignal 100, since "everyone's the same". 100/30 as a man-labor unit...

and so the gang of 12 produced 40 units,
the gang of 8 produced 80/3 units,
the gang of 6 produced 20 units,
and the gang of 4 produced 40/3 units,

which of course sum to 100 units.

The next few problems entail a new unit measure called a "pefsu", used in the context of food prep. These appear to entail mostly volumes and amounts, but I've been stuck on things before so let me not be premature.

Chace says that "pefsu" is supposed to denote "units of food-stuff that you get from such-and-such material", but we'll see.

>>8035817
if an 8ball of crack leaves houston on a bentley travelling hood miles per hour howdunna fast delivery to chicago and how much street value in that 8ball in chicago man dindu nuffin

>>8035544
Wait so they described angles in terms of length? Trigonometric ratios of lengths more accurately

>>8035865

Go to Anchorage instead for premium rates? I stayed in a motel room in Chicago recently with a bullet hole in the window.

P.69: 3 + 1/2 hekat of meal make 80 loaves. Give two things: meal per loaf, and the "pefsu".

Ans: the "pefsu" is given as the 80/3.5. This suggests a reciprocal of what I just said.

The meal per loaf is the flip of that, and he converts it into units easy enough that I bother to repeat them: 1/32 hekat + 4 ro.

>>8035888

I haven't drilled into that just yet (I skipped the pyramids problems for last), but Chace suggested something along those lines, thinking of angles or whatever in terms of units, being "palms", and maybe not making the abstract leap.

The starting point of math is to describe everything you can in terms of number, of quantify. So since length is in its own way coterminous with number, this should come as no surpise to you.

What else is 2π, but a length? What are 30, 45, 60, 90, but lengths in relation to each other and such-and-such?

P.70: 7 + 1/2 + 1/4 + 1/8 hekat of meal make 100 loaves of bread, give meal/loaf and the "pefsu", again.

Ans: The pefsu is given as loaves over hekat, or 800/63, circa 12.7-something. Meal/loaf: 63/800, given in the units, checks out (ro...)

>>8035909
Still loving it. Bump before bed.

>>8035949

The Egyptians, Ra bless them, were avid beer drinkers, and this one uses beer. A few more problems circa #80 will use some of these units again. A little setup though.

It takes 1/2 hekat of "besha" to make 1 "des"-measure of beer. Again, these units are purely cosmetic to the problem, I just illustrate them for completeness. I can't even figure out what "besha" actually is, but it doesn't matter, it could be any of the ingredients needed, so it's a "generic" ingredient as far as I'm concerned. If in doubt I'll just call it hops. A des-measure is just a standard measure, so you could just as well call it a pint-glass, a 40, a barrel, whatever works. ANYWAY.

P.71: It takes 1/2 hekat of besha to produce 1 full des-measure of beer. Such a produced unit of beer has a 1/4 glass poured out, and then diluted back to capacity with water. What is the "pefsu", that is, quantity of such unit-diluted beer glasses that can be produced, using exactly /one/ hekat of besha? Assume the 1/4-glass of full-strength that was poured out is captured and re-used later - that is, that it does /not/ take exactly 1/2 hekat of besha to make one diluted glass.

Ans. So to make that 3/4 glass of full strength that results, just before it is tragically diluted, that clearly takes (1/2)(3/4) = (3/8) hekat of besha to do that. Then you just add water so no more besha, and that's your unit-diluted glass. Which can be expressed as (3/8) hekat/dilute.glass.

But this is not what we've been asked to get, which is the pefsu. (food-units per ONE hekat of stuff) And now that I'm on my third one of these, that latter context is clear to me now, he always wants to end up there. So just take the reciprocal, 8/3 dilute.glass/hekat = 8/3 pefsu and that's the answer.

>>8036788
bumppppppppp

>>8036788
It's too bad that they only have story problems and little abstraction.

>>8036917

Agreed; the most abstract things thus far have been the "discovery" of 61B (trivial once you have elementary algebra) and the interval problems, but even those are really specific cases. I have "filled in" the abstraction on multiple problems to keep myself amused. The pre-occupation with Egyptian fractions (uncommon denominators) appears to have slowed down their ability to compute; I'm beginning to appreciate why according to mathworld's article on the topic, Andre Weil described these fractional representations as "a wrong turn". Ahmose is competent with manipulating fractions and phrasing just-so problems which are tailored to the type of manipulation he wants to make (or, perhaps, the writers he is copying were), but there seems to be almost no concern for establishing general results. Also their geometry was very confused, as I've indicated.

Now, keep using 1 pefsu = "1 generic food-unit per 1 hekat of raw material" as a derived unit...

P.72: 100 loaves "of 10 pefsu" are to be evenly exchanged for an equal x loaves "of 45 pefsu". Find x. Ans: 1 Pefsu in this context is 1 loaf per hekat of whatever raw material-grain, flour etc, doesn't matter, like before.

So 100 loaves per 10 pefsu means that heap A of loaves was gotten from 10 hekats (of stuff). We can then infer that for heap B, an equal amount of stuff, that is 10 hekats, gives 45 loaves per hekat, or x = 450 loaves.

P.73: 100 loaves of 10 pefsu are to be evenly exchanged for an equal x loaves "of 15 pefsu". Find x. Ans: x = 150 loaves. (identical thought process)

Getting "dangerous" now...

P.74: 1000 loaves of 5 pefsu are divided evenly into two heaps of 500 loaves each. Each heap is to be evenly exchanged for two other heaps, of x loaves of 10 pefsu, and y loaves of 20 pefsu. Find x and y.

Ans: x = 1,000 loaves, y = 2,000 loaves, same thought process.

P.75: 155 loaves of 20 pefsu are exchanged for x loaves of 30 pefsu. Find x. Ans: 465/2 loaves.

P. 76: 1000 loaves of 10 pefsu, one heap, will be exchanged for two other heaps. The other two heaps each has an equal number of x loaves, one being of 20 pefsu, while the other is of 30 pefsu. Find x. Ans: x = 1200 loaves.

A hint of 'abstraction'! Pefsu, previously used for beer and bread, /can be (whoa) used interchangeably, regardless of food-item under discussion/. Because the hekat is the basis of these equations, in both the noun-and-verb sense of that word. And so, we trade beer for bread and vice verse.

P.77: 10 des-measures of beer, of pefsu 2, are to be evenly exchanged for x loaves of bread, of pefsu 5. Find x. Ans: x = 25 loaves.

P.78: 100 loaves of pefsu 10 = x des-measures of beer of pefsu 2. Find x. Ans: x = 20 des-measures.

We're now at the extreme back-end of the Papyrus, and things get a bit more scattershot. A few more food problems to feed animals, and some not-very-mathematical "problem" segments which I will probably simply transliterate. Then I swing back round to the pyramid problems that I skipped and that's all. 79 is so "cute" that it deserves its own post, and a bit of TeX.

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P.79: (Modern treatment, together with Ahmose's version and a bit of culture, all in a few posts):

Consider the finite geometric series [math] \displaystyle a_{1} + ... + a_{n} [/math] with the first term being equal to the common ratio of terms, or [math] \displaystyle \frac{a_{k}}{a_{k-1}} = a_{1} = 7 \;\; ; \;\; 2 \leq k \leq n [/math]

Observe that, as Chace points out, since the first term is equal to the common ratio, the following holds:

[math]

\displaystyle \sum\limits_{k=1}^{n} 7^{k} = 7 \bigg( \sum\limits_{k=1}^{n-1} 7^{k} + 1 \bigg)

[/math]

Let n = 5 and do the following:

a) Find the sum by computing the LHS by directly summing its terms, and
b) Find the same sum by computing the form inside parentheses on the RHS (state this result), and finally by multiplying by 7.

Ans, a&b): 7+49+343+2401+16807 = 7(2801) = 19607. Ahmose gives the sum when n = 5 by employing both methods, although he simply states "multiply 2801 by 7" in as many words without mentioning the derivation of 2801, which suggests some cleverness, per the above.

Now, the fun bit. :D Ahmose presents a) as a table, together with some familiar words:

[math]

\displaystyle \begin{bmatrix}

houses & 7 \\
cats & 49 \\
mice & 343 \\
spelt & 2401 \\
hekat & 16807 \\
Total & 19607 \\

\end{bmatrix}

[/math]

Even Chace cannot help making the comparison right in the midst of his text: "As I was going to Saint Ives, I met a man with seven wives. Every wife had seven sacks..." A spelt, btw, is just a thing of wheat. This will need another post to do it justice.

>>8037119
It's just so frustrating that they didn't generalize their solutions. He has plenty of examples of a general process in these problems. I guess it's too large of a conceptual leap?

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>>8037321

Wiki says it all, complete with a section on Rhind which agrees with what I'm seeing in Chace. Incidentally, the wiki on "spelt" suggests that attribution of its cultivation to Ancient Egypt is erroneous (diff. species of wheat or somesuch), but even if that's an old philological goof, it's immaterial to the amusement of the problem.

https://en.wikipedia.org/wiki/As_I_was_going_to_St_Ives

But it gets even better. Immediately following his own St. Ives comment, Chace pulls Fibonacci into the conversation, and suggests the thing that we would /like/ to believe, historically true or not, and I quote:

"Rodet (1882, page 111) found in the Liber Abaci of Leonardo of Pisa (see Bibliography, 1857) a problem of geometrical progression expressed in much the same way, and having the ratio 7, and he suggests that Problem 79, absurd as is its heterogeneous addition, has perpetuated itself through all the centuries from the times of the ancient Egyptians."

Problem 79 is a meme. Praise Kek.

Actually, this purported "transmission" of the meme requires a historical analysis. That "St. Ives" was published long /before/ the Rhind Papyrus got back into the world spotlight suggests a spoiler of this theory. For those who may not be aware, the villian in one of the Die Hard movies uses "St Ives" as a riddle to taunt pic related, hence my picture choice here.

>>8037358

Last thing on 79 for now: both Chace and wiki, indeed moderns in general, have suggested an /interpretation/ of 79 as a savings-problem over a large estate. How many hekats do you save over such-and-such time by having those cats to eat those mice, etc.

However, this is merely an interpretation, possibly reading in too much to the problem, which is actually little more than the above table and "2801x7".

>>8037325

The other possibility that one must keep in mind is that Ahmose himself is a scribe first and a mathematician second. The title page indicates transmission of the material from earlier sources, and although Ahmose seems capable of registering certaing goofs and factual errors, he makes more of his own.

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>>8028098
>Kingdom of Kush

>>8029113
It doesn't though. Small nose, retracted jaw, small lips and non-swept brow. These are not negroid features.

P.80: One hekat is also equal to 10 hinu, yet another unit of volume. Suppose that you have a "Horus' eye fraction's" worth of hekats, where a Horus eye fraction may be any of the six fractions in >>8028734 . Express these quantities in terms of hinu (in Egyptian fractions, of course).

Ans: Of course, you're just multiplying by 10 (hinu/hekat). But this is simple enough that I will format the table. It was probably of interest to Ahmose that several "Horus eye fractions" repeat, following the conversions:

[math]

\displaystyle \begin{bmatrix}

1 \;\; hekat & = & 10 \;\; hinu \\
1/2 \;\; hekat & = & 5 \;\; hinu \\
1/4 \;\; hekat & = & 2 \frac{1}{2} \;\; hinu \\
1/8 \;\; hekat & = & 1 \frac{1}{4} \;\; hinu \\
1/16 \;\; hekat & = & \frac{1}{2} \frac{1}{8} \;\; hinu \\
1/32 \;\; hekat & = & \frac{1}{4} \frac{1}{16} \;\; hinu \\
1/64 \;\; hekat & = & \frac{1}{8} \frac{1}{32} \;\; hinu \\

\end{bmatrix}

[/math]

Note that two fractions being written next to each other denoted addition and not multiplication, much like our mixed-fraction conventions.

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Problem 81 is a large-ish table, and partial solution, which extends the basic idea of the table in 80. In fact, the first part of the problem is a simple verbatim repetition of that table. Chace's version of the statement is needed now.

"P.81: Another reckoning. Express fractions of a hekat as 'Horus eye' fractions and in terms of the hinu."

Basically, Ahmose lists not quite 30 combinations (additions) of the six Horus eye fractions

[math]

\displaystyle \frac{1}{2} \;\;\;\;\; \frac{1}{4} \;\;\;\;\; \frac{1}{8} \;\;\;\;\; \frac{1}{16} \;\;\;\;\; \frac{1}{32} \;\;\;\;\; \frac{1}{64}

[/math]

, and also gives their equivalents in hinu (and sometimes also in ro), a tedium.

What this problem suggests to me by way of failed generalization, is that either Ahmose or his source were trying to exhaustively list all combinations of the above six fractions, falling short by over 50%, not making the leap, to another anon's point about the frustration over failure to generalize. Consider the expression I just wrote, above. Ahmose partially considers situations where each term is either /present/, or /absent/. this of course gives 2^6 = 64 possibilities, in general, but he doesn't seem to register that possibilities are missing. In fact, any and all combinations of the above are strictly less than 1, staying in the neighborhood of investigation. One cannot help but think of Zeno's paradox, and the associated convergent series.

I'm going to drill into the table now and see if I can find a pattern in the information, specifically the information which is missing.

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>>8037659

Depressing! I laid out a binary combination of all the 64th's from 0/64 - 63/64, and went hunting through the table. This is maybe about 90% accurate but even I gave up after a time, too tedious, no discernable pattern. MANY entries are double-counted, and then sometimes variously expressed in different Egyptian fractions. The only relevant columns here are the "Present?" columns with some entry from P.81 in dark green.

I had hoped to establish that the table at least lists all combinations with "3" (6 choose 3) terms, "2" (6 choose 2) terms etc, but those aren't even true. Just a smattering of information.

Just to give this its due diligence, I'm just going to pick two that look juicy and spot-check, then move on.

1/2 + 1/4 + 1/8 hekat is 8 + 1/2 + 1/4 hinu (multiply by 10), yep, and

( (1/2 + 1/8 + 1/32) hekat plus (3 + 1/3) ro ) = (6 + 2/3) hinu (Yep, do conversions per above). The former quantity was of interest because it's exactly 2/3 hekat.

82-84, which include the final "B" problem ,82B, apparently involve giving animals certain amounts of feed, and are messes in their own right.

82 tosses around a few "wheat-like" objects, and sets a problem. I will merely sketch the problem in simple terms, following Chace.

P.82: "Estimate in wedyet-flour, made into bread, the daily portion of feed for geese."

"10 fatted geese eat daily 2.5 hekat,
"Taking for 10 days 25 hekat,
"Taking for 40 days 100 hekat.

There then follow a series of somewhat related terms which I simplify.

Amount of spelt that must be ground: 500/3 hekat
Amount of wheat: 200/3 hekat
SUBTRACT 1/10 OF THE ABOVE FOR SOME REASON BECAUSE WE'RE EGYPTIANS WE WUZ BAKERS N SHEET HOW DO U NOT KNOW DIS: 20/3 hekat

Ans: "Grain required: 280/3 hekat, or 140/3 double hekat."

The whole problem is obscure because Chace is somewhat at a loss himself, but the speculation is that multiple raw materials are used and some well-established Egyptian cooking rules are assumed and presented. The relationships of the raw materials are confusing, however the "tenth to be subtracted" was taken from the established baseline of 100 hekats.

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82B is short and immediately related, so let me just type out what's in front of me.

P.82B: "Estimate the amount of feed for other geese."

"If to fatten 10 geese it takes daily 1 + 1/4 hekat,
it will take for 10 days 12 + 1/2 hekat
and for 40 days 50 hekat.

"The amount of grain to be ground in double hekat is '23 + 1/4 + 1/16 + 1/64 hekat 1 + 2/3 ro' ."

The first few lines are a straightforward progression a la 82's opening lines, halved this time. The trick is to understand how the latter line is derived, given what has just been established in 82. It works out to 140/3 hekats (remember to multiply by 2 for the double hekat bit to get back into logical territory).

As the rate of feeding less-important geese (to feed slaves? cucks?) has halved, so-to has the amount of "necessary stuff" halved. 280/3 becomes 140/3, without further comment. Same deal.

P.83: "Estimate the feed necessary for various kinds of birds."

Some comments about various birds and feed-rates come first, followed by a table of daily allowances for : "a goose, a terp-goose, a crane, a set-duck, a ser-goose, a dove, a quail", their daily allowances generally decreasing. It is so totally dull that I cannot exposit it any further, and I've been pretty good.

84 is wholly unintelligible. Per Chace, "One can only agree with Peet that "with this problem the papyrus reaches its limit of unintelligibility and inaccuracy." We have in the main followed Peet's readings and explanations, but have made certain guesses which differ from his."

P.84: "Estimate the feed of a stall of oxen."

All in hekats...

"4 fine Upper Egyptian bulls eat 24 hekat...2 hekat...
"2 fine Upper Egyptian bulls eat 22 hekat...6 hekat...
"3 common...cattle eat 20 hekat... 2 hekat...
"1...ox 20 hekat...
"Total of this feed: 86 hekat, 10 hekat...
"It makes in spelt 9 hekat, 7 1/2 hekat...
"It makes for 10 days 1/2 1/4 of 100 hekat... 1/2 1/4 of 100 hekat...
...15 hekat...
It makes for a month 200 hekat... 1/2 1/4 of 100 hekat
...15 hekat...
It makes in double hekat 1/2 of 100 hekat... 1/4 of 100 hekat...
...11 1/2 1/8 hekat... 5 hekat
3 ro...

...

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85-87 are errata and mystery, which conclude the document. There is a rather sad feeling as the document grinds to a halt in such a pathetic way, but do not fret! I left the pyramid-problems for last, so we'll skip back that way afterwards. Also, these mysterious bits at the end, following somewhat logical arguments and pronouncements in a short document, remind one of how Wittgenstein's Tracatatus ends.

It will be simplest if I just let Chace drive, now.

"Eisenlohr gave the number 85, 86, 87 to certain fragments that are not a part of the mathematical work of the papyrus, but are of interest, although incomplete and more or less unintelligible. As they cannot in any sense be called problems I will use the word "Number" when referring to them, but will give them these numbers, as does Eisenlohr, in continuation of the numbering of the problems.^1 1: Strictly Number 87 should be numbered before Number 86, which is at the very end of the papyrus. See Diagram in the Second Volume (never fear, your humble OP shall reproduce this diagram directly - pic related)."

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"Number 85"

"This is a group of cursive hieroglyphic signs, written upside down on the back of the papyrus. It has been suggested that the scribe was merely trying his pen (see Peet, page 128). Eisenlohr attempted to give a meaning to the group, calling it a "Motto," and translating it somewhat as follows:

'Kill vermin, mice, fresh weeds, numerous spiders. Pray the god Re for warmth, wind and high water.'

Gunn, however, claims (Page 136) taht we have here an early example of the so-called enigmatic writing, and gives as a a tentative translation, [this is the one Chace repeats in his volume on the original Egyptian]:

...

...

'Interpret this strange matter, which the scribe wrote ... according to what he knew.' "

A deeply poignant coda to a historically important yet deeply confused document that I've spent a few days with. If we accept this latter version, it also suggests a barest hint of the ABSTRACTION of mathematics, and if it is Ahmoses' hand, it even suggests a humility of sensing lack of understanding, an original interpretation on my part, now that I understand the Rhind Papyrus a little bit.

Once again, it's best I let Chace drive, to complete the back of the book. However, "86" was something that was literally pasted onto the backside to hold the document together.

"Number 86"

"This seems to be from some account or memorandum. There are three pieces which appear separated in the British Museum Facsimile, but since that was made they have been placed together in their proper relative positions as may be seen in Photograph 31, Volume 2. There are eighteen lines, but parts are missing from both ends of the lines. The following is a translation of the words that remain in the eighteen lines:

1...living forever. List of the food in Hebenti...
2...his brother the steward Ka-mose...
3...of his year, silver, 50 pieces twice in the year...
4...cattle 2, in silver 3 pieces in the year...
5...one twice, that is, 1/6 and 1/6. Now as for one...
6...12 hinu, that is, silver, 1/4 piece, one...
7...(gold or silver) 5 pieces, their price therefor, fish, 120, twice...
8...year, barley, in quadruple hekat, 1/2 1/4 of 100 hekat 15 hekat, spelt, 100 hekat...hekat...
9...barley, in quadruple hekat, 1/2 1/4 of 100 hekat 15 hekat; spelt, 1 1/2 1/4 times 100 hekat 17 hekat...
10...146 1/2; barley, 1 1/2 1/4 times 100 hekat 10 hekat, spelt, 300 hekat... hekat...
11...1/2, there was brought wine 1 ass(-load?)
12...silver 1/2 piece; 4; that is, in silver...
13...1 1/4; fat, 36 hinu, that is, in silver...
14...1 1/2 1/4 times 100 hekat 21 hekat; spelt, in quadruple hekat, 400 hekat 10 hekat...
15-18: [these lines are repetitions of line 14]

"Number 87"

"This seems to be a memorandum of some incidents, not very coherent, but apparently complete.

"Year 11, second month of the harvest season. Heliopolis was entered. The first month of the inundation season. 23rd day, the commander (?) of the army (?) attacked (?) Zaru.

"25th day, it was entered.

"Year 11, first month of the inundation season, third day, Birth of Set; the majesty of this god caused his voice to be heard.
"Birth of Isis, the heavens rained."

Somewhere elsewhere the text, I believe it was suggested that this episode is something contemporaneous with Ahmoses' time, which was jotted on the document. There is a speculation that the hand of certain of 85-87 are different, which is reasonable given their wild departure.

This ends the Rhind Papyrus. But remember as I said, we still have a bit of math left to plumb. The pyramid problems >>8033615 await, but I am out of energy tonight.

Since this thread is >90% complete, I repeat my appreciation for the kind attention of those of you who have bumped. I'm going to sleep soon, and I encourage any anons to keep the thread bumped while North America sleeps. I won't promise, but the very strong urge is to finish tomorrow.

I was very pleased to learn about problem 79 today desu, interesting.

>>8037944
>tfw he ahmose discovered binary representations of numbers

>>8038300

If your "ahmose" is to be interpreted as a pun on "almost", then I might agree. :^) But no, that sketch was a reading-in on my part. They were obsessed with horus-eye fractions and reciprocals of powers of 2, but no binary numbers as-such were articulated, to my knowledge.

But while I am unironically on the subject of "ethnomathematics", let me instead suggest that a Polynesian people more legitimately beat Leibniz to the punch [of discovering binary representations], although it was an elementary blended system and desu I haven't read the article.

http://www.pnas.org/content/111/4/1322.full

I should also start setting the stage for 56-60, also for my own benefit because I really haven't been through them yet. A "Seked" appears to be a unit of angular measurement, which is quantified in different ways. Now I'm going to bed.

https://en.wikipedia.org/wiki/Seked

>>8038340
Yup, I was making a joke and meant almost! He's at the point where he says "yes, this power of two, not this one, ..." But just needs to write 1 for yes and 0 for no, as far as I can tell. Or rather, he doesnt seem to far off.

Praise anon. Praise kek. Bump.

Bump after eight hours.
Great thread.

OP here, gonna get lunch and then I'll get serious about closing on this. Some more prelims before I actually look at the problems 56-60.

It's worth reviewing what an angle actually is. An angle is a (quantified) amount of rotation between two line segments (or rays, or...) which already meet at a point (and about that point) that would be necessary to cause them to coincide again. Any number system, or period, that encapsulates 1 full turn and its multiples (720 deg, 1080 deg) will do. In other words, the closed real interval of primitive (less than one) turns is mapped to whatever other closed (or perhaps half-open if you want to be picky) real interval you may care to use

In particular, we are familiar with radians and degrees. However wiki also mentions turns, and /gon/, which is just some fancy new(?) word for gradians. And even "minutes" and "seconds" as other units of angular measurement can be used independently of degrees if we wish:

https://en.wikipedia.org/wiki/Angle#Units

a rough schema looks like

[math]

\displaystyle [0,1] \;\;\; turns = [0,2 \pi] \;\;\; (radians) = [0,400] \;\;\; gradians = [0,360] \;\;\; degrees = [0,21600] \;\;\; minutes \;\;\; = [0,1296000] \;\;\; seconds

[/math]

by which I simply mean to indicate that each real interval, equipped with its unit, is independently capable of representing /any real-valued angle/ on the interval and mapping all the same angles, or turns, although when using units, we are of course accustomed to speaking in terms of integer multiples, fractions, decimal approximations.

So, my expectation for the last few problems is that I will be able to subsume the seked, and maybe some smaller units of angular measurement (their own min/sec?), into the above schema. Given the other confusion in the geometry section, odds are high that what we will originally have will not be consistent, but that we will be able to derive a consistent generalization of the problems, as I've done throughout the thread.

>>8039081
Thanks OP. Great thread.

>>8039081
I'm excited to read about it

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Pic related is a pretty-good detail of 56 (top) through 60 (bottom). 59 (right) and 59B share the same row. You can blow it up bigger at page 14 of

http://www.f.waseda.jp/sidoli/MI314_02_Egypt_Babylon.pdf

Basically, these problems give us a hint of legit trig, and inverse trig at that!

A ROYAL CUBIT is a length unit, equal to 7 PALMS, another length unit. Fingers are involved later, but I'm going to look at the context of the other problems before I pronounce on the latter - wiki has been wrong with respect to units as they are used in Rhind more than once.

P.56: A (four-sided) pyramid is 250 (royal) cubits high, and the side of its base is 360 (royal) cubits long. What is its /seked/?

Ans: 360 is halved, to give 180. We then consider the right triangle in the interior of the pyramid, with (adjacent) leg 180 (royal cubits), running from the middle of a base to the bottom center of the pyramid, and (opposite) leg 250 (royal cubits), running from that point to its peak. The hypotenuse, immaterial to the problem, bisects a face of the Pyramid, but the angle theta that it forms with the adjacent leg, which is just the inverse cotangent of 18/25, is what we want.

There is a little dimensional analysis to express the seked first as "so many (royal) cubits", and later as "so many" palms, which I need to think about a bit more.

I thought the Egyptians used hieroglyphics? Why do some of those letters look like Arabic script?

>>8039460

The script is known as "Hieratic", and it's basically cursive Egyptian, which the elites (if you're literate, you're elite) used to write faster.

https://en.wikipedia.org/wiki/Hieratic

A great side-by-side comparison/specimen of Ahmoses'/Rhind's Hieratic versus the standard-hieroglyphs that Chace and Co. decomposed the text into, to make sense of the thing, is at >>8027824 .

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>>8039363

A bit embarassing that it took this long but now I have a clear definition of a seked, and why it is expressed in length units. Anyway here's a clear (if busy) working of 56.

Now. One PALM = 4 FINGERS. Now I agree with wiki on that one.

P.57: The seked of a pyramid is 5 palms (plus) 1 finger (per unit royal cubit), and the side of its base is 140 cubits. Find its altitude a.

Ans: now that I've got my head around the terms I did this one before checking and wound up with the same. Ahmose tries a different roundabout calculation but it works since he concludes that

a = 93 + 1/3 (royal) cubits, or 280/3 (royal) cubits.

Method: The seked amounts to 3/4 cubit, so imagine a right triangle with short leg 3/4 and "vertical" leg 1. This is similar to what we want, and you take half of 140, 70, then scale. a = 280 / 3. He didn't even bother expressing it in "palms, fingers" etc (a gruesome thought!), so that was nice.

P.58: A pyramid is 93 + 1/3 cubits high and the side of its base is 140 cubits long. Find its seked.

Ans: Exact straightforward reversal of 57. The seked is finally expressed as 5 palms 1 finger, again.

P.59: A pyramid's altitude is 8 cubits, and a side of its base is 12 cubits. Find its seked.

Ans: the resulting right triangle has vertical leg 8 and horizontal leg 6, so this is the identical " 3/4 : 1 " triangle we've just seen twice now. Scale 8 down to 1, 6 down to 3/4. The seked is again finally expressed as 5 palms 1 finger, or 3/4 cubit.

P. 59B: A pyramid's seked is 5 palms 1 finger (per cubit), and a side of its base is 12 cubits long. Find its altitude a.

Ans: Getting boring! We've fundamentally done the same thing four times over, but it's an excuse for Ahmose to try different routes to get to the same correct answer(s). The altitude is 8 cubits.

At least this has been a suggestion of trig and notions of slope, dihedral angle etc, to close out the geometry section nicely. And this part isn't nearly as confused as 51-53. Finally, 60.

P.60:

"If a pillar (?) is 30 cubits high and the side (diameter?) of its base 15 cubits, what is its seked?"

A little translation confusion here. So, take a good look at the pyramid-figures in >>8039363 . The one at the bottom left does not have a double-struck, "rectangular" bottom. And our notion of seked is inherently run/rise/right-triangle, being applied uniformly to a geometric object from any side...

Chace and Peet believe that the figure in 60, the "pillar", is a /cone/. And this makes sense so we roll with it.

A right circular cone with altitude 30, diameter 15, and consequently radius 7.5. The associated right triangle then has vertical leg:horizontal leg 30:7.5, or 1:1/4. The seked is 1/4 royal cubits, or however else you might wish to express the same length. Ahmose lets it sit as 1/4 royal cubits, since it's a nice Horus-eye fraction, perhaps.

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This completes my exposition of the Rhind papyrus. Thank you for your kind attention, ladies and gentlemen. I had a few loose ends I wasn't happy with in my own derivations so I may go back over the thread later and add a bit, but the thread's purpose has been effectively completed.

What did we learn?

-A single document, being a sample size of one, is not a good starting point for a statistical analysis. We have very few mathematical documents from ancient Egypt. However, the sheer scale of Rhind by itself is on a par with a modern textbook, and contains enough information that, now that we've been through it, we can draw some conclusions, or at least form informed opinions of the period.

-Ancient Egyptians (the literate elite, anyway) were indeed preoccupied with practical everyday-tier problems, and didn't do much abstraction. The most abstract results in the text are possibly 61B >>8034331 and 79 >>8037321 , although the latter has some novelty value. Still, I agree just a little with Chace that Egyptians "did math for its own sake" insofar as they related it to their religion and culture.The Horus-Eye fractions clearly fascinated Ahmose.

-Literate Egyptians were competent at manipulating fractions and properly performing unit conversions, which is better than can be said for many literate moderns.

-Ahmose and his sources had a nice approximation procedure for pi, for situations involving circles. The error of <1% did not seem to be of interest, but they could have pushed the argument since they were clearly capable of correctly adding tiny fractions ad nauseum viz. 31 >>8028777

-The whole text also gives me a flavor for primordial beginnings of linear algebra, but over half such cases ultimately reduce to linear equations.

-Ahmose sucked at geometry, or at least his source did, and yet interestingly, the pyramid-seked problems immediately suggest primordial beginnings of trigonometry. Not Euclid, but important, primordial beginnings of our activity.

>>8030823

The "answer" for b) here should be 8192/27 cubic cubits.

>>8039788
Thanks for posting. This was a really interesting thread and I learned about something that I never knew existed.

>>8039979

You're very welcome.

Personal notes/errata:

-already picked out the above goof on 43.
-may briefly reconsider the 2/N table since I have a hard copy at hand and all. 50 instances of division by (2n+1), where n: 1-50, giving 3-101. Why pick those (fractional representations)? Wiki suggests more.
-my initial idea that a seked is an angular quantity, which I wrote a little about, is WRONG WRONG WRONG. A seked is a physical length, which can be given in any "physical" length-units. The diagram's suggestion that a seked is an "angle" in >>8035544 is a step backwards, and confused me. Even the context of 5.25 palms next to that given altitude is all wrong. The image sucks, is critically misleading, and I regret posting it.
-old researchers from Chaces' time had another theory of what the "seked" problems were all about, which Chace mercifully rejects. Review this alternate theory.
-I was slightly unsatisfied behind-the-scenes in my treatments of 64 and the big detour I took about 51. I forgot how I managed to do something in 40 so I thought about 64 longer than I should have. I was unable to manually compute a elementary derivative relating to 51, and it annoyed me. Also didn't derive "g" for myself, had to look up something that worked.
-maybe take another look at 81 and 83, but those would be the most tedious.

Polite sage.

note: manually checked Chaces' 2/N table against the wiki, spot-checked original. I agree that wiki's table accurately reflects what was originally done in the 2/N table (another thing I wanted to check before turning the library books back in, so I can play with it after if I want).

The (old) alternate theory about the seked problems goes like this.

"Eisenlohr in his translation takes the "ukha-thebet" as the diagonal of the base and the "per-em-us" as the lateral edge, while he takes "sentet" and "kay-en-heru" as the side of the base and the altitude. Borchardt (1893) contends that the ukha-thebet and sentet both mean the side of the base, and that per-em-us and kay-en-heru both mean altitude.

"In the first interpretation the ratio involved in the seked in problems 56-59B is the cosine of the angle which the lateral edge makes with the diagonal of the base, and in problem 60 the tangent of the angle which the lateral face makes with the base. In the second interpretation the seked means the cotangent of the latter angle in all of the problems.

"Borchardt argues with considerable force from practical considerations and most Egyptologists have now accepted his interpretation, although it necessitates the assumption that the Egyptian called the same line by different names in successive problems. In my Free Translation it will be convenient to translate these terms in accordance with Borchardt's theory. The whole matter is not very important from the point of view of Egyptian mathematics. The important point is that at the beginning of the 18th century BC and probably a thousand years earlier, when the greater pyramids were built, the Egyptian mathematician had some notion of referring a right triangle to a similar triangle, one of whose sides was a unit of measure, as a standard.... (Chace, writing about 90 years ago.)

Thank you OP.

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I took the books back to the library. Playing with the 2N table, now.

Doing my part to get Necrotizing fasciitis off the front page.

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>>8041410

...And now that I've actually read wiki again and gone through the data a bit more, Rhind's beginning, which I had initially skipped, /suggests/ one more tantalizing rudiment: the significance of primes in Egyptian computation.

Recall that 61B >>8034331 give a general procedure for "multiplying 2/3 by the reciprocal of any odd number", resulting in an Egyptian fraction of exactly two terms. This "procedure" in its generality, is most closely related to (but distinct from) the 2N table, as most of the 2N table involves an Egyptian fraction expression involving exactly two terms.

But a minority of terms, in Ahmose's expression, involve an exceptional three or four terms? And a suspiciously high number of times, MOST but not ALL the time, when he chooses this different expression, what is 2 being divided by?

A prime. Pic related (in pinks. The first few aren't done this way, and 95's 3-term representation is an outlier).

Now I'm interested again. By way of review, what I've got going on here (eventually) is that 2 is divided by all odd numbers from 3-101. This is found equal to some Egyptian fraction, where the numerators are /always/ one, there's no 2/3 chicanery in the table, or bits I missed. For example, the first row is "2/3 = 1/2 + 1/6".

The idea is to analyze the data and suggest a (historical narrative, as opposed to scientific) motivation for why Ahmose (or his predecessor) chose the Egyptian fractional representations that he did, and since this has been more rigorously attempted multiple times before, wiki already has some theories which I'm simply following an appreciating now. The big thing right now is that there seems to me to be just a bit of primordial number theory buried in Rhind as well, insofar as these-3-4 term representations are not-randomly accompanied by primes.

This one is becoming more interesting than I thought, and is a much more gratifying table to explore than the errata at Rhind's back. And the reason why, is because some logic seems to be emerging. The more I play with the data, the more certain "rationales" for decomposing a given number suggest themselves to me. I no longer have Chace in front of me but some rules are popping out, with outliers.

No actual results yet, but I have more than enough patterns in front of me which make sense from a modern perspective, to be above pseudoscientific numerology and explain some sort of motivation. Must keep a cool head. :^)

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>>8042158

Meant to attach this state. At right are the terms dictated by 61B, which have a constant ratio of 3:1 or 1/3, depending on how you want to look at it.

Thank you based OP

Okay, let's back up a moment, and get a little speculative, and theoretical. I reserve the right to mix metaphors and time periods; I also want to lay a foundation, here.

Suppose that you are an ancient Egyptian scribe and mathematician. You know the rule 61B >>8034331 , which I will reiterate in modern terms, because it bears repeating here.

Rule 61B: The quantity 2/3, multiplied by the reciprocal of an odd number 2n+1 with natural n, has the following two-term representation as an Egyptian fraction:

[math] \displaystyle \frac{2}{3} \cdot \frac{1}{2n+1} = \frac{1}{2(2n+1)} + \frac{1}{6(2n+1)} [/math]

Now, the pharoah has set you a task. You will compute for the pharoah a table. The table will consist of Egyptian fractional representations of all quotients 2/1, 2/2, 2/3, ... , 2/101. He has given you some latitude in how you want to proceed, and he's a bright guy, so he probably won't cut your balls off if you screw up. Still, doing anything for the pharoah is serious business.

How to begin? Well, first of all, consider the original LHS numerators themselves, you think. They're always two. Now you're no god, but you've been crunching numbers long enough to know a pattern, which you can readily explain to the pharoah for half of these numbers. Whenever a denominator is a /double/ of some other whole number, that denominator can be re-written as something like... 2k. And then that 2 on top, and the one on the bottom, [math] \frac{2}{2k} [/math] It's almost is if they... yes, they /cancel each other out/. And what is left? [math] \frac{1}{k} [/math] ! Half the table is done, and you know THE FIRST RULE: "If the number to be divided by is itself a double of some other number, then two divided by the first number is equal to one divided by the second number." e.g. 2/24 = 1/12, 2/60 = 1/30, etc. /Yes, that is obvious. And I can explain it to pharoah in any number of ways./

Or,

[math] \displaystyle Rule \;\; 1: \;\;\; \frac{2}{2k} = \frac{1}{k} [/math]

>>8043693

That first part was so easy, that you barely wrote anything down. You know pharoah well enough to know that you will be able to explain those quickly, and he'll get it. But the rest of the table has been taunting you for a day.

You had just thought of the doubles of certain numbers. That rule 61B looks like it could be useful, somehow. But how? What is next? How could you knock out another large chunk?

There are no "doubles" left among the denominators, but there are several "triples": 3, 9, 15, 21, 27... You look back at 61B and you realize something:

[math]

\displaystyle \frac{2}{3} \cdot \frac{1}{2n+1} = \frac{1}{3} \cdot \frac{2}{2n+1}

[/math]

Where the factor at the far right is veeery similar to your general problem. In fact, if you were to /triple/ the RHS, and assuming 2n + 1 is a number that is a triple of some other whole number, it would be /precisely/ what is needed. But will it yield an Egyptian fraction again? You check, and it WILL. You even verify that since 2n+1 is always divisible by 3 in this case, nothing will be left over.

Therefore you USE Rule 61B, to get at this next case, and create THE SECOND RULE: "If the number to be divided by is itself a triple of some other number, then /triple the corresponding calculation for 61B/. Or, /after exhausting Rule 1, when 3 divides 2n+1/,

[math] \displaystyle Rule \;\; 2: \;\;\; 3 \cdot \frac{2}{3} \cdot \frac{1}{2n+1} = 3 \bigg( \frac{1}{2(2n+1)} + \frac{1}{6(2n+1)} \bigg) = \frac{1}{ \frac{2}{3}(2n+1)} + \frac{1}{2(2n+1)} [/math]

You /especially/ notice that the denominators at the end are precisely /one-third/ of the corresponding denominators when 61B is applied by itself, giving a quantity /three times as large/. So now you know how to generate your Egyptian fractions in this case (3) based on 61B. And it happens that (the appropriate fractions, at least) /are exactly what is given in this case, without exception (see my "green" cells in the above sketches).

>>8043741

Most of your table is done, at this rate. You reflect that quadruples, sextuples, octuples etc have already been taken care of, as they are all themselves doubles. You further reflect that nonuples have been taken care of, by dint of having dispensed with triples.

You pass from the case of 2, to the case of 3, now skipping 4 (and even 6, 8 and 9, say), to consider the cases of 5 and 7. /You are discovering prime numbers/. It even occurs to you that many large numbers on this table have no other whole-number factors apart from themselves and 1. You think that this will be useful to know, so you verify this for yourself in another list, and lay it off to one side. These will be the laborious calculations, it seems.

There are just 7 multiples of 5 left on the list, which are not themselves multiples of 3 or of 2. They are: 5, 25, 35, 55, 65, 85, 95. (Perhaps) you even bite off a few single instances by a stroke of luck, which you then leave as discovered***, but you know that this brute-forcing is not the way to get things done. You need another rule, like the one above...

***("Rule 2" works perfectly for the 3-case of the 2/N table, and the assumption of a "Rule 1" even explains the absence of even denominators altogether, as something that even the Egyptian perhaps deemed trivial. But understanding, or even venturing an educated guess as to how and why Ahmose populated the rest of the table as he did becomes more complicated and speculative from this point. In particular, a rule that I shall derive and speculate, was /not/ applied uniformly to the above remaining numbers in the way that a modern might expect.)

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>>8043773

Continuing this Rule 2 ad nauseum, particularly fitting formulae to data, give the following rules 3, 4, 5

[math] \displaystyle Rule \;\; 3: \;\;\; \frac{2}{2n+1} = \frac{1}{ \frac{3}{5}(2n+1)} + \frac{1}{3(2n+1)} [/math]

[math] \displaystyle Rule \;\; 4: \;\;\; \frac{2}{2n+1} = \frac{1}{ \frac{4}{7}(2n+1)} + \frac{1}{4(2n+1)} [/math]

[math] \displaystyle Rule \;\; 5: \;\;\; \frac{2}{2n+1} = \frac{1}{ \frac{6}{11}(2n+1)} + \frac{1}{6(2n+1)} [/math]

etc, all of which taken together with obvious patterns coming out above, suggests an /algorithm/ that Ahmose could have used to populated his entire table with fractions of exacty /two/ terms. Whereas what /actually/ happened, in my view, is that Ahmose followed this algorithim /about halfway through/ the problem, then started making changes which I have yet to investigate.

This is all a bit premature, there's something here for me to show by induction, and /I haven't actually double-checked the above yet/, so check it/take it with a grain of salt.

Two possible activities at this point are to build a "revisionist" table, different from what Ahmose actually wrote, where every case is decomposed into exactly two terms, and giving a general statement about the above, closely modeled on "61B". I seem to be retreading the ground mentioned on the wiki page.

Pic related amounts to the algorithm I'm describing, mirroring Eratosthenes' sieve, albeit in the service of a specific problem.

My "induction" thought was me just over-thinking things. The general statement of the simple result is this. One could cutely call it, "Ahmoses' 2/N lemma":

For all natural n, the expression [math] \frac{2}{2n+1} [/math] has a two−term Egyptian fractional representation of the form

[math] \displaystyle \frac{2}{2n+1} = \frac{1}{n+1} + \frac{1}{(n+1)(2n+1)} [/math]

Pf. (babby-re-arrangement, nothing else).

It is /this/ which Ahmose iteratively applies (in one sense or another) in much of his table, before eventually abandoning what would have worked for the whole thing, it in favor of longer expansions. And this, (probably) /because Ahmose lacked algebra as we understand it/, which is another insight on the Egyptian limitations. One could even call this special case, this algorithm (where you skip already-covered composite odds such as 9, 15...) "Ahmoses' little sieve", or similar.

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>>8044394

The result of applying the above /iterative/ algorithm (not to be confused with the expression in the above post, I've realized) is pic related. This is a "revisionist" version of Ahmoses' 2/N table.

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What remains then, of the original table, are the situations where Ahmoses' sieve was /not/ employed (together with outliers), the former being the larger primes.

Two meaningful things pop out when we check, in between this noise. Ahmose for whatever reason strongly preferred to use "very composite", if not highly composite numbers for his first denominator D1 in these cases. He then resorted to using integer multiples of p for the denominators of (the rest of) his Egyptian fractions.

Notes on all outlier cases are given at bottom.

>>8045669

And, finally, at long last, having made a record of isolating the primes, and not finding much rhyme or reason of interest to Ahmoses' methods, I am content that I have exhausted this topic to my personal satisfaction, short of the items I mentioned in >>8040019, most of which I've already done anyway and the others being things I made up that I can check offline. Short of another bump from an interested anon, I stick a fork in this thread (a crook, a flail?) and declare it done. Deleting my spreadsheet right now, polite sage.

I remember James Burke told of (and I'm sure backed up by scholars in related fields in his research), in his series Connections, ancient Egyptians needed math because the Nile would flood its banks every year, washing away any markings for fields etc. Math would make it easier for farmers to recreate the fields they had the from the previous harvest. These prime denominators would make it possible for more precise calculations/measurements to take place, thus making his job of governing the lands he presided over easier. This would also explain the lack of development of prime numbers, as there may be a mandate for how precise it should be and not more, or something to that effect. innovation ended when practical use did, as you pointed out.

Maybe theory of how math was formed at this time has changed, but it would be consistent with your findings.

note to self: cleared up my treatment of functions f and g re: #51, correctly derived g from scratch, found f' & g', answered general question about where both rise and fall most rapidly. Everything I said was right all along.

last conceivably interesting things to squeeze out are revisiting 64 vs. 40 (derivation, I skipped something) and alternate seked theory (should lead to confusion &/or contradiction, doesn't sound like this other theorist was a big math guy). sega

>>8040053

the suggestion of the alternate theory as stated here is /mathematically/ consistent in 55-59, but then it hits a snag on 60 (cotangent versus tangent), and the interpretation requires two cases anyway (and fell out of favor of course); the "simpler" explanation, which had already been presented in 55-60, is both mathematically consistent and the philologists like it as the best explanation too (makes the most sense out of the different words), which is why the wiki was written as it is to begin with.