
Re: RMP # 48
Posted:
Oct 8, 2007 9:27 AM


Franz,
Your points are listed first. My responses follow by a **comment**. This style was used in this thread concerning Lee's points, so this style should be acceptable.
Milo, the Rhind Mathematical Papyrus announces all secrets, and then offers nothing more than rather dry calculations.
** False. Ahmes announced that all secrets were solved, and then he solved his problems without detailing his methods (in modern or ancient notation). There is nothing dry concerning Ahmes solutions to his 2/p series, a method that was used over and over again in his algebra problems (ie RMP 33, as we are discussing).
F. Hultsch was first to fairly and accurately report Ahmes' garbled shorthand in mathematical terms in 1895. Transliteration linguists, that only read the surface level of the mathematical texts still do not consider Hultsch and Bruins' views to be correct in 2007. I do, since I also have independently confirmed the method, by working the 2/p problems upside down, as Bruins may have done in 1944! **
This can't be all there is about Egyptian mathematics.
**I agree.**
I claim that we see only the level for beginners, but the same problems can be solved on the level of advanced pupils.
**The EMLR text was intended for beginners. The RMP and its 2/n table was for advanced students. Are you ready to discuss the EMLR in the context of the RMP and the Liber Abaci? Wikipedia offers an introduction to the 26 line text by:
http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll **
RMP 33 on the advanced level can be formulated like this (my proposition): A cube measures 37 by 37 by 37 fingers. Calculate the diagonal of the volume in palms, and then the volume itself in cube cubits. First you have to draw up the number column for the square root of 3, or consult it as written on a wall of the scribe's office. Choose the line 56 97 168. Multiply 37 fingers by 168 and divide the result by 97 and you get the diagonal in fingers. Divide it again by 4 and you get the diagonal in palms. Abbreviate the calculations by dividing 37 fingers by a factor of 97/42 = 1 "3 '2 '7 and you get the diagonal in palms (numbers given in the RMP). Now calculate the volume in cube cubits. If you carry out the rather demanding calculation involving high numbers, and if you proceed correctly, you are rewarded with a fine surprise: the volume measures practically 1 "3 '2 '7 cube cubits.
**It that appears that you have proposed, without proof, to not read Ahmes' shorthand, and find alternative Ahmes' reasons and methods. Clearly you have substituted your own views, without discussing alternatives. What you have done is create your own shorthand methods that have little or nothing to do with Ahmes' actual methods and answers (though your methods are often interesting in several respects. When are you going to discus the HultshBruins method, and the 2/p table, and other pertinent issues relating to Peet and the 1920's additive authors  that tried to compromise with the linguists, thereby throwing the baby (Ahmes' math) out with the bath water (reading only hieratic words  skipping over hieratic finite math). Neugebauer is another case in point. In 1927 Otto incorrectly concluded in "Exact Sciences in Antiquity" that all of Middle Kingdom Egyptian fraction arithmetic had gone into intellectual decline, compared to the Old Kingdom  sad, sad, sad. Otto did not even try to directly read any of Ahmes work! Otto only declared that he knew the meaning of the 2/n table, and thereby dismissed its methods and meanings. You have 'de facto' done the same thing, though with an interesting twist.
For example, in RMP 33, Ahmes converted 2/97 and added 26/97 into one Egyptian fraction series without using a duplation (multiplication method, the historical reason that you show unit fraction multiplication in your work). Had Ahmes started with 2/97 and doubled to 4/97, 8/97, 16/97 (or 32/97), he would have stopped at 16/97, and added 8/97, and 4/97. Or, if he had stopped at 32/97 he would have subtracted 4/26. Ahmes did not do either. Clearly, he also did not use the 'false position' method to solve:
x = 14 + 28/97
Ahmes converted 2/97 + 26/97 to a finite Egyptian fraction series by using the HultschBruins method, as I have cited in detail yesterday.**
I claim that we see can only see a small part of Egyptian mathematics and have to restore the rest, namely wooden objects visualizing a problem (in the case of RMP 33 perhaps a chest whose outer measures are 41 by 41 by 41 fingers, and whose inner measures are 37 by 37 by 37 fingers) on whose surfaces the problems were written, and the apparatus of simple yet clever additive algorithms such as number columns and number sequences. We have to combine math history with math archaeology and experimental math history, as too many sources (papyri, leather rolls, and the teaching materials) are lost.
**yes, we can only see a small part of Ahmes math, as described in Ahmes' footnotes. Ahmes' footnotes/shorthand is very hard to read. Ignoring the shorthand and assuming, as you do, that Ahmes' arithmetic is lost, or degraded (ala Otto) in ways that may have made Ahmes' data unreadable has allowed many well meaning scholars (such as yourself) to substitute their own modern logic, or illogic, for Ahmes' historical methods.
False position is another example of a modern idea (the algorithm method of finding roots) has been inappropriately applied to attempt to find Ahmes' first degree equation roots (a method that works most of the time, but not always  ie RMP 33). False position is a well known medieval method that had been improved by Arabs, and listed sidebyside with Egyptian fraction methods in the 1202 AD Liber Abaci. Gee, since Fibonacci used Egyptian fractions and the False position method, sidebyside, why did not Ahmes do the same?
Well, Ahmes' shorthand and answers must determine the methods that Ahmes used. Ignoring Ahmes work is a common mistake. Assumptions, such as yours, are interesting to introduce a broader RMP discussion. However, not one Middle Kingdom text can be declared 'read' until it is validated against the other Middle Kingdom texts. All of my decoding and math translation work links the Middle Kingdom texts, from the oldest, EMLR, AWT, Reisner, et al, to the newest, the Greek, Coptic and yes, even the medieval texts that continued to use Egyptian fraction arithmetic. **
Best Regards,
Milo Gardner

