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The Egyptian Mathematical Leather Roll. 2016-08-07 05:35:31 Post No. 1517634
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The Egyptian Mathematical Leather Roll. 2016-08-07 05:35:31 Post No. 1517634
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This thread is cross-posted from /sci/ (it's not doing well there, and I think there may be legitimate interest here), and it concerns a very small document in the history of mathematics (subject related): the "EMLR".
In fact, what the document actually consists of are a few dumb-simple arithmetic calculations, small enough that you could fit them onto one sheet of modern paper (the roll itself is a bit bigger than that, measuring about 10" x 17").
The story of the document goes like this. Once upon a time, there was a young, sickly Scottish lawyer by the name of Alexander Henry Rhind, born in 1833. Rhind also had an interest in antiquities, and so he took a trip to Egypt for some "fresh air", as people living in the 19th century liked to say. The trip was also an opportunity to make a purchase. Although the historical details on this point are unclear, the basic idea is this: circa 1858 A.D. , at a market in the modern city of Luxor (which in ancient Egypt was known as Thebes), Rhind purchased at least two items: a very long and much more involved papyrus which is a treatise of elementary mathematics, and therefore acquired the name of the "Rhind Papyrus", and a tiny, extremely brittle rolled-up piece of leather, which was impossible to unroll without destroying it (the EMLR). Both items are thought to have been "discovered" (i.e. illegally looted) from the nearby ruins of ancient Thebes, such that they were made availble for purchase to whitey.
Rhind goes back to Britain, bringing the documents with him, to live for a few more years. Being a sickly man, he promptly drops dead in 1863 (possibly pulmonary disease), aged 30. C. 1863/1864, Rhind's estate directs that both items are bequeathed to the British Museum, where they are kept today.
The topic of this thread is therefore closely related to the so-called Rhind papyrus, which I had previously discussed in an archived thread:
https://warosu.org/sci/thread/8027713
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Due to its central location in London, westerners from 1863/4 going forward thus had ample opportunity to study, translate, and bitch about the Rhind Papyrus, which they proceeded to do on-and-off for about 70 years. Meanwhile, the Leather Roll sat quietly curled up, its contents unknown.
Fast forward to 1927. Some scholars and a chemist start spot-testing a chemical treatment, in an effort to unroll this ancient piece of leather. It works, and they do indeed soften and unroll this ancient document. The writing turns out to be simple arithmetic, which is very much of a piece with that of the Rhind papyrus. Specifically, both documents are thought to date from c. 1650 B.C. (+- 100 years, the Rhind papyrus itself contains historical details which date it more precisely in terms of ancient Egypt). The EMLR performs very similar calculations to those found in the Rhind papyrus (or RMP, where the "M" stands for "mathematical"). And both documents are written in the ancient Egyptian "hieratic" script. Hieratic is basically a cursive form of ancient Egyptian hieroglyphics, which allowed scribes to write more rapidly.
Pic related is a historical blurb which comes from volume two of Chace's two-volume exposition on the Rhind Papyrus, published 1927/1929, when the EMLR had JUST been unrolled. the OP pic and this pic both derive from that volume, and they therefore represent among the first looks that anyone had had at the EMLR, in a publication.
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As for the hieratic writing system, and specifically hieratic numerals, the following page and image are extremely useful:
http://www-history.mcs.st-and.ac.uk/HistTopics/Egyptian_numerals.html
Using pic related, it is feasible for a determined reader to look closely at the OP (or a much cleaner drawing of same which is soon to follow), and translate the calculations. The short version is that the document consists of 26 simple equations among fractions, which are duplicated on the document. By using the existing wiki, you can quickly start to pick out which bits are which, and compare original with translation:
https://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll
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or if you don't feel like squinting, I found this drawing of the OP, somewhere (although its treatment of columns seems to be backwards/confused).
Small fragmentary bits of information are missing from this drawing, however.
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...aaand pic related gives you a very clear look at the duplicated information in the document. Between this picture and what was written in >>8252949 , it begins to become clear that the duplication of most of the information is a big-yet-reasonable /assumption/, since the bulk of "column 1" seems to have been broken off or lost. Yet the bits that we do have match 1-1 with those of column 3.
The document reads from right to left, by the way.
I'll probably continue this later, but there is one other article about the EMLR that I'll point out. It is, however, very poorly written and itself contains multiple numeric mistakes. The wiki is a better source of info, at present, although its treatment derives in part from this link:
https://www.academia.edu/517467/Egyptian_Mathematical_Leather_Roll
Although the author has the right idea of grouping the roll's equations into a few different groups (and using the RMP and other ancient documents to historically substantiate his attributions of how the scribe thought, which he refers to as "attestation"), I think I can improve upon his scheme. The tension with such a process is in whether one is doing history, or doing (babby) mathematics.
An exercise for the reader: examine column 4 (the far left-column in pic related), and using the information in this thread, verify that it matches column 4 as reported at wiki. Everything that you need to do this has already been posted in this thread, if you are so inclined.
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Bringing this little thread home, I went through Gardner's proposed proto-algebraic methods whereby the scribe who wrote the EMLR actually performed and understood his operations. Gardner comes up with five species, based on the historical evidence of work shown elsewhere (the RMP especially), which he characterizes.
I model Gardner's view versus my view in pic related. The colors in the right columns don't mean anything in particular, except that a block of problems have the same form, depending on who you ask.
I also went to the troubel of organizing the EMLR in a sensible fashion, at left. each number reprsents a natural number whose reciprocal is taken. The bits at far left (that is, their reciprocals) are then summed, and compared with the reciprocal on the RHS, in this case. A simple logic-test in Excel compares the two, and they are mostly true, with a single exception which involves the fraction 2/3 (also allowed in ancient Egyptian fractions, as an extra-special case that they just liked for some reason). The scribe also made two mistakes which are held apart for that reason, though intent would seem to be clear.
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and pic related is a parallel screen-grab from the /sci/ thread, which quickly and lazily advances this duplicated thread to about the same place, for any who may care to read it.
Inspired in part since /his/ can't render latex as /sci can.
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Interesting shit but Egyptology is out of my depth
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