Subject: Re: Linguistic stabs-in-the-dark???
From: Saida
Date: Thu, 26 Sep 1996 12:58:25 -0500
Piotr wrote:
> I, for one, am not convinced that we all came to any agreement over supposed
> loans from Egyptian in Anglo-Saxon, but I would like to add a bit to the
> discussion on the possibity that Ape and the like come from Egyptian. I have
> no idea what the origins of the word might be, but I think that it might be
> difficult to argue that it has to come from Egyptian. The Egyptian word in
> question, however one transliterates it, is a "culture word" in the ancient
> Near East . It is fairly certain that it originated in some other language,
> and then was loaned into a variety of languages. Hebrew kof, Akkadian pagu,
> and Sumerian ugubi are all reflections of this (Hebrew f is historically p,
> and note that in the Akkadian the consonants are inverted). This is not an
> Afrosiatic root, but a foreign loan in all these languages. If Ape would
> really be a loan, one would have to prove that it came from Egyptian and not
> some other language.
Or it may have originated in Egypt and borrowed by the others. Perhaps
the Egyptians got it from another African people who brought them apes
and monkeys, because I am not clear if such creatures are indigenous to
Egypt. Zoology is not a strong area with me, but I think someone will
know the answer to this. Still, who would have gotten the simians
first--the Egyptians or those peoples you mention to the east?
I said I would do some research on the word "monkey" . I had an idea
that I would find this came from Egyptian and, sure enough, I came
across the term "ma'akhau", which Budge gives as "a kind of animal". It
is spelled "owl, extended arm, plant, chick" with the determinative
being the hindquarters of an animal with a long tail. Since this
determinative is present in nearly all Egyptian words denoting monkeys
or apes (when the picture of the total monkey isn't there) I think this
is the word I am looking for. I am fairly certain the "ayin" sound
represented by the "extended arm" was heard and or transliterated by the
Greeks (and others) as a nasal. This is documented variously.
Therefore "ma'a" would become "mon".
Loren wondered how easily the "g" in "gaf, "gafi" or "gafu" would be
lost to become "Affe", and "ape". Actually the Egyptian glyph that
begins this word (the one I call the little hearth) is not a very strong
consonant. It is thought to be pronounced between a "g" and a "k"
(although Egyptian has a hard "k", too). The next sign in the sequence,
the "alif vulture" is, I think, another story. Nowadays it is thought
to represent the "r" or "ar" but, knowing the Egyptians, I think this
was just pronounced "ah" or "aw". In Late Egyptian, anyway, the "r" was
seldom pronounced anymore. IMHO the "alif vulture" was said in the
exaggerated way that a very posh Briton would say "Cahn't do it, old
chap" and that was what was heard more the the first consonant.
From what I have been able to gather about Egyptian pronunciation, I
have the weird suspicion that it WAS, in many ways, very much like posh
English intonations, except that it probably included sounds not used in
English like the "ayin", although we don't know for sure what sort of
"ayin" came out. One notices that, in British English, the "o's", for
example, are pronounced differently than our American one or from the
"o" in other European languages (not including Scandinavian!) Now
ancient Egyptian is not even supposed to have an "o" but, in Ptolemaic
times, a glyph supposedly pronounced close to an "o" was used for
writing this vowel. Maybe this was the equivalent of a BBC English "o".
I could say more but, let me just state, risking your derision, that it
wouldn't surprise me somehow if the Britons of long ago, whoever they
were, knew some Egyptians. Not only met them but admired their way of
talking (as we Americans admire the melifluous British accent and
intonation that can make the most prosaic statements sound as beautiful
as poetry), adopting it as a high-class accent of their own.
And wouldn't it be a hoot if one of those shadowy "Sea Peoples" were
from the British Isles? In fact, when I hear this term, I don't think
of Phoenicians or Greeks, I think of the English (Britannia rules the
waves) with their naval prowess. In modern history, the British sailed
into the Mediterranean in ships barely thought seaworthy by present
standards, fought and won the "Battle of the Nile" and kicked Napoleon
out of Egypt. And Napoleon, surely, had an immeasurably better navy than
any sea-hating pharaoh could have assembled.
Subject: Re: Sweet Potatos and Silver Bullets
From: yuku@io.org (Yuri Kuchinsky)
Date: 26 Sep 1996 17:14:21 GMT
Peter van Rossum (PMV100@psuvm.psu.edu) wrote to Yuri:
: Or how about
: you give any kind of recent research that refutes the possibility of
: a natural dispersal?
Here we go, Peter -- bits and pieces from my webpage, and some other
stuff:
Needham clearly believes that the human-assisted transmission for sweet
potato is the mainstream view. I trust Needham.
From Needham, TRANS-PACIFIC ECHOES:
"...in the case of the sweet potato (_Ipomoea batatas_) it is ... fairly
sure that there was a connection between S. America and Polynesia, though
whether the Peruvians took it, or the Polynesians came and fetched it,
remains quite unknown. But the transfer is accepted on all hands." (p. 61)
A wealth of material on this and other issues can be found in Riley, C.
L., et al, eds, MAN ACROSS THE SEA; PROBLEMS OF PRE-COLUMBIAN CONTACTS,
Univ. of Texas Press, Austin, 1971, p. 343. (Cited by Needham.)
I just got hold of this book, and it contains a whole lot about the
potato.
**********
The following comes from SEED TO CIVILIZATION, by Charles Heiser (New
edition, 1990, Harvard U. P.). He is quite a famous scholar of the
origins of agriculture.
It should be noted that Heiser is no diffusionist. In fact he's often a
significant _opponent_ of diffusionists. Yet he accepts human-assisted
diffusion for the sweet potato! Significant. For other plants in question
he's careful to hedge his bets and to qualify his language.
At the time of the European discovery of the New World,
the sweet potato (Ipomoea batatas) was widely cultivated
in tropical America and was also being grown on some of
the Pacific islands. ... The presence of the plant on
either side of the Pacific at such an early date poses
several interesting questions -- among them, how and when
did it get across the ocean? ... While either the
introduction of seeds by some natural means or an
independent domestication remain a possibility, it
seems _far more likely_ that people were responsible for
the introduction of the sweet potato from the Americas
to the Pacific region. There are two ways in which this
might have occurred. ... (p. 139)
I hope this answers your objections satisfactorily. He has a lot more on
this, so refer to the book if you're interested.
*********
More info about sweet potato can be found in the following work by
Lathrap.
Lathrap is an anthropologist and archaeologist with a strong
interest in paleo-ethno-botany -- the study of the origins of
domestication of earliest agricultural plants. In 1977, he published
a seminal work, OUR FATHER THE CAYMAN, OUR MOTHER THE GOURD. It is
included in the important volume, ORIGINS OF AGRICULTURE, C. A.
Reed, ed., Mouton (Proceedings of the IX International Congress of
Anthropological and Ethnological Sciences). This volume includes
many more important works about diffusion, including the article by
George C. Carter, A HYPOTHESIS SUGGESTING A SINGLE ORIGIN FOR
AGRICULTURE.
I hope these references will satisfy Peter, the stern taskmaster.
Yuri.
--
#% Yuri Kuchinsky in Toronto %#
-- a webpage like any other... http://www.io.org/~yuku --
Students achieving Oneness will move on to Twoness === W. Allen
Subject: Re: 200 ton Blocks
From: james denning
Date: Thu, 26 Sep 1996 14:49:12 -0700
Kevin D. Quitt wrote:
>
> On Mon, 23 Sep 1996 21:21:42 -0700, Jiri Mruzek
> wrote:
> >A slew of amazing problems had materialized.
> >If you actually tried to move the Hadjar el
> >Gouble, I am sure that a slew of problems would ensue, just as well.
>
> So am I, but we're not talking about one thing, moved once. We're talking
> about an industry of transporting stones. The workers would quickly learn
> the problems and find the solutions. Ask the people who moved the machine
> how hard it would have been to move it the second time, and the tenth.
>
> Again, I'm not trying to say it's trivial or easy; it's not. It's a hell of
> a lot of work, but it can be done in a straightforward manner by not an
> unresaonable number of people.
>
> >I have never disputed that. Of course there are limits to what you say.
> >For instance: To mount wheels, you would have several choices like
> >hoisting the block up, or dig holes, and then roll the wheel out,
> >or build three sides - roll the block, add the fourth side.
> >The problem is in getting enough people connected to the relatively
> >compact block to carry out all the chores.
>
> That's true. But again, it's not just once, for one block. It's a process
> repeated many times.
>
> >Spooling towing lines on the smaller diameter block produces
> >a mechanical dis-advantage. The length of rope needed to turn
> >the block once, will be shorter than the distance traveled by
> >the wheel, which will also spin around once.
>
> That's true, but it probably doesn't really matter. It might mean you need
> 11 people instead of ten (whatever), but isn't insurmountable. On a level
> surface I don't think it would make a lot of difference. Uphill it might,
> but then again you can add more people.
>
> >You would really need to spool your ropes somewhere near the outside
> >of the taller wooden wheel. This spells troubles for the project..
>
> It would give a better advantage, but I don't think it's a kill. I suppose
> one could add rounded blocks of wood that would turn the block shape
> cylindrical where the ropes are. Without going out and doing with those
> tremendous blocks, it's impossible to say what all the problems are. I am
> convinced, however, that this technique can be used to move, by human labor
> alone, much heavier loads than many people thought.
>
> >> Once again, I didn't mean for this to be *the* explanation. There have been
> >> other reasonable ideas posted.
> >
> >Such as?
>
> For moving the blocks up the ramps, sledges or mechanical advantage from A
> frames could be used. Lots of things could be; I haven't really looked at
> moving the directly up as opposed to rolling them up a ramp. I still feel
> that the ramp is the most likely.
>
> >Well, only on the side ramps skimpy on material usage.. One could have a
> >wide road atop a large-volume self-supporting ramp.
>
> I was thinking not of a ramp that spirals up the pyramid as much as a
> straigh ramp at 90 degrees to one of the sides. And of course, once the
> block is rolled roughly into place, there's still the problem of accurate
> placement; perhaps A frames were used here.
>
> >With a large ditch along the planned route, you could slip a large
> >spooler onto the wheel, and thus regain the mechanical advantage.
>
> That's what I was talking about, above, with the rounded blocks. As I say,
> I'm not sure they're necessary
>
> >Must I do problem-solving for the skeptical party? :)
>
> Hey! *You're* the skeptic in this discussion!
>
> >I just meant that there is no limit to skeptics simply scaling Lo-Tech
> >up to any desired size.
>
> I apologize for personalizing it. And you're right, you can't scale
> technology very far. That's what killed the Titanic.
>
> >Wanna launch satellites into orbit?
> >Build a sloping ramp high enough..
>
> Nah, just a tower, straight up. But it seems to me that was tried, once...
>
> >With no signs of such a ramp to the higher reaches of the Pyramid,
> >this subject becomes purely academical, and generally oriented.
>
> As far as I know, the builders never described how they did their work. Too
> mundane for the nobility to worry about and record. My goal was to show
> that the materials can be transported via low-tech means, without magic or
> alien intervention. I believe we've agreed on that much.
>
> >We still can't duplicate the Pyramid with Lo-Tech methods and
> >materials.
>
> Do you have any idea how much that would cost, to the labor unions alone?!
>
> You have to be careful when applying math to artifacts so that it doesn't
> become numerology instead. Someone once posted a wonder article on how his
> bicycle held all the mathematical, physical, and astronomical constants and
> ratios, and he was working on QM at the same time. A real pity I've
> misplaced it.
> --
> #include
> _
> Kevin D Quitt USA 91351-4454 96.37% of all statistics are made up
> Per the FCA, this email address may not be added to any commercial mail list
but when you concider the lentgh of time cheops live and the amount of
stone it would be one stone every ten minutes that was quite a feat and
to think cheops didnt like it he was buried in a valley furthur down the
road
jim denning
Subject: Re: Edgar Casey--The theory of civilization not yet known to man--undiscovered
From: millerwd@ix.netcom.com(wd&aeMiller;)
Date: 26 Sep 1996 22:51:35 GMT
In <3wOwiPAYGcSyEwlx@skcldv.demon.co.uk> Jon
writes:
>
>In article <527alg$9c4@dfw-ixnews5.ix.netcom.com>, wd&aeMiller;
> writes
>>In Jon
>>writes:
>>>
>>>In article <51qlag$3sj@dfw-ixnews5.ix.netcom.com>, wd&aeMiller;
>>> writes
>>>>In Jon
>>>>writes:
>>>>>
>>>>>In article <51n7v8$deu@dfw-ixnews8.ix.netcom.com>,
>>>>>millerwd@ix.netcom.com writes
>>>>>>
>>>>>>>>Fly on a plane that follows little red lines, of course. Then
to
>>>>>>make
>>>>>>>>it interesting...the plane won't land...we'll just parachute
out
>>>>the
>>>>>>>>back and happen to land about two trees away from the main
>>entrance
>>>>>>of
>>>>>>>>the city. Of course, we'll have to shoot a couple of nazi's on
>>the
>>>>>>way
>>>>>>>> before we can get to the door where we shout the ancient
>>password
>>>>of
>>>>>>>>entry :"Mellon!"
>>>>>>>>
>>>>>>>>Hey, this could become a great screenplay. hehe
>>>>>>>>
>>>>>>>>Amanda :)
>>>>>>>I am afraid it won't work. You see Atlantis is underwater. By
>>the
>>>>>>time
>>>>>>>we got two tree away from the entrance by parachute, we would be
>>>>very
>>>>>>>wet, and, more upsettingly, dead. Moreover, the only way that
we
>>>>>>could
>>>>>>>shoot Nazis on the way down is if they were in a submarine!
>>Tricky
>>>>>>this
>>>>>>>one. I suggest that the way forward is to get the Nazis drunk
in
>>a
>>>>>>bar
>>>>>>>in Cairo, then enslave them, and force them underground to dig a
>>>>>>>Transatlantic tunnel. If we happened to come across any
>>fossilised
>>>>>>>Egyptian sailors on the way, whose remains were loaded to the
>>gills
>>>>>>with
>>>>>>>cocaine, this would be a bonus. But I'm not going until you
agree
>>>>to
>>>>>>>the thigh length rubber boots!
>>>>>>>--
>>>>>>>Jon
>>>>>>
>>>>>>Well, well. Ok. As long as the thigh-high leather boots can be
>>>>purple
>>>>>>and green tye dye. :) As for the tunnel...good idea! Perhaps
we
>>>>can
>>>>>>use our enslaved nazi's for even longer working hours if we let
>>them
>>>>>>chop up and snort any mummies they find.
>>>>>>
>>>>>>Amanda
>>>>>>:)
>>>>>>
>>>>>>P.S. For all you people out there, who haven't followed this
>>thread
>>>>>>from the beginning....It's a JOKE!!!! DOH!!!! Laugh! Have
>>fun!!!
>>>>
>>>>>>Get bent!
>>>>>You mean - gasp, you're not serious. How can I find Atlantis
without
>>>>>you - who will wear the boots. No calm down Jon, surely she jests
in
>>>>>case any Nazis are looking in. No, the mummies have to be
preserved
>>>>>to confound the Egyptologists. Now any really expert
archaeologist
>>>>>should regularly confound Egyptologists - it's modern form of pig
>>>>>sticking!
>>>>>
>>>>>--
>>>>>Jon
>>>>
>>>>Sorry, but I had to be careful there for a day or two. I heard the
>>>>Nazi regime was reading our posts. Can't let them in on the
secret,
>>>>now can we? Darn, I really thought the mummy idea was good. :) I
>>>>guess we'll just have to settle for Nazi slave labor. Of course, I
>>>>could always drive the heel of my boot into the back of the slow
>>>>workers.....(grin)
>>>>
>>>>Amanda
>>>It's OK, I have the whip for that - you only have to kick them for
>>>fun, but watch out for the ones who enjoy it!
>>>--
>>>Jon
>>
>>We'll just have to make them scrub the toilets. :)
>>
>>Amanda
>TOILETS! Sub Atlantic cesspits. You have to remember to leave loads
>of clues, but no substantial information. This is the essence of all
>ologys. I eons to come, when our transatlantic tunnel to Atlantis
>lives on a mountain peak somewhere a geologist will find some
>fossilised human shit, a jackboot, and build an entire theory about
>the 20th century from it, then an archaeologist will find several
>human fossils bearing whip and boot marks, dating from the same era
>and come up with a totally different hypothisis. Then the telly and
>newspapers will get involved and there will be heated debates in
learned
>societies - we are not only in the business of discovery, but creating
>employment for thousands of otherwise unemployable people in millenia
>yet to come. It is an onerous burden Amanda!
>--
>Jon
My only comment today...hehehehehehehe :)
That and fossils of human skulls with piercings.
Amanda
Subject: Re: Egyptian standards of measure: Was Re: Egyptian Tree Words
From: whittet@shore.net (Steve Whittet)
Date: 27 Sep 1996 02:56:08 GMT
In article , petrich@netcom.com says...
>
>
> [A lot of stuff on unit-fraction decompositions...]
>
> A rather interesting curiosity, but hardly Euclid's _Elements_. I
>think it interesting that Mr. Whittet has yet to claim any Egyptian
>originals for the _Elements_, any Egyptian text that looks as if the
>_Elements_ had been copied from it.
>--
>Loren Petrich Happiness is a fast Macintosh
>petrich@netcom.com And a fast train
>My home page: http://www.webcom.com/petrich/home.html
>Mirrored at: ftp://ftp.netcom.com/pub/pe/petrich/home.html
>
>
Date: Mon, 23 Sep 1996 06:04:21 -0700 (PDT)
From: Milo Gardner
To: Loren Petrich
Cc: Steve Whittet
Subject: Euclid's Book VII and Unit Fractions
This paper presents an informal p/q generalized fraction point of view,
long proposed by Boyer and many other math historians as not having been
known in Babylon, Egypt or Greece prior to 600 BC. The informal evidence
that is being presented was taken from the 500 AD era, the Akhmim
Papyyrus, a time period when p/q was a known aspect of Classical
mathematics. Two tables of unit fractions, n/17 and n/19, will be "read"
in a manner that closely resembles Middle Kingdom exact innovations, as
discussed in the First and Second Installments as found on
1. http://www.teleport.com/~ddonahue/phresour.html and
2. http://www.seanet.com/~ksbrown/iegypt.htm
Please note the manner than 4/17 and 8/17 are computed, possibly by the
Euclidean algorithm. Comments would be appreciated by number theory or
hisotrian types that may be able to read Appendix I data in another
manner.
What I see, considering p/q as an unsolved Middle Kingdom problem, as an
aspect discussed by Richard Guy in his popular treatise on the subject,
4/y will not be directly proven to have been calculated by a 3-term unit
fraction. Nor will a 5/y or x/y unit fraction series be proven by
induction, or any other modern number theory method.
To justify one basis for presenting this informal proof, I would like to
cite, Makers of Mathematics, by Stuart Hollingsdale. Hollingsdale asks:
why did 500 AD Egyptians choose the awkward 7/29 = 6' 24' 58' 87' 232'
rather than the much simplier modern form of 7/29 = 5' 29' 145'?
Simply stated, Egyptians appear to have known and fully utilized:
5/29 = 1/6 + (6*5 - 29)/(6*29), as the Akhmim P. taught
= 6' 174'
and adding, 2/29 = 1/24 + (2*24 - 29)/(24*29), as the RMP taught
= 1/24 + (12 + 4 + 3)/(24*29)
24' 58' 174' 232'
Thus 7/29 = 6' 24' 58' (174' + 174') 232'
= 6' 24' 58' 87' 232', as the Akhmim P. taught
Given that this explanation may one day be confirmed, over 2,500 years of
unit fractions history, a set of n/p tables, can now be outlined by two
tables, n/17 and n/19, per Attachment I.
Attachment I
This officially unsolved number theory problem originates from ancient
Egypt, covers a period from 2,000 BC to 500 AD a period of 2,500
years. The source document is the Akhmim Papyrus, a Hellene 500 AD
to 800 AD papyrus, found along the Nile, as cited by Wilbur Knorr,
Stanford History of Science Department, Historia Mathematica, HM 9,
"Fractions in Ancient Egypt and Greece". Knorr's excellent paper
includes several x/y tables, as Guy described them, such as n/17 and
n/19. Both tables will be attempted to be "read" using rules that
that may have primarily originated in the Egyptian Middle Kingdom era.
n/17 Akhmim P. Value Ideas from 1650 BC Egyptian Fractions
---- ---------------- -------------------------------------
2/17 12' 51' 68' 2 a - p = 7 [a = 12, divisors 4, 3]
3/17 12' 17' 51' 68' 2/17 + 1/17
4/17 12' 15'17' 68' 85' (1/3 + 1/17)(1/4 + 1/5) or
(1/a + 1/p)(1/u + 1/v)*
or (1/3 + 1/17)(1/4 + 1/5) or
3/17 + 1/17 (1/a + 1/p)(1/u + 1/v) + 1/17
5/17 4' 34' 68' 5a - p [a = 10, divisors 2, 1]
6/17 3' 51' 6a - p = 1 [ new general form]
7/19 3' 17' 51' 6/17 + 1/17
8/17 3' 15' 17' 85' (1/3 + 1/17)(1/1 + 1/5) from 4/17
9/17 1/2 34' 5/17 + 5/17 - 1/17
10/17 1/2 17' 34' 5/17 + 5/17
11/17 1/2 12' 34' 51' 68' 10/17 + 2/17 (12' 51' 68')
12/17 1/2 12' 17' 34' 51' 68' 11/17 + 1/17
13/17 1/2 4' 68' 9/17 + 5/17 - 1/17
14/17 1/2 4' 17' 68' 13/17 + 1/17
15/17 1/2 3' 34' 51' 9/17 + 6/17
16/17 1/2 3' 17' 34' 51' 15/17 + 1/17
*u, v were NOT generally compute by the Euclidean algorithm
That is to say, Euclid's Algorithm was not copied from
Egyptian fractions, as a general rule.
On set of "decoding" of Hellene
n/19 Akhmim P. value unit fractions using 1650 BC rules
----- --------------------- ------------------------------------
2/19 10' 190' 2a- p = 1 is one of three methods
3/19 15' 20' 57' 76' 95' (1/3 + 1/4 + 1/5)(1/1 + 1/19) - 2/3**
or 2/19 + 2/19 - 1/19 [a = 30, divisors 6, 5]
4/19 5' 95' 2/19 + 2/19
5/19 4' 76' 2a - p [ one of three methods]
6/19 4' 19' 76' 5/19 + 1/19
7/19 3' 38' 114' 7a - p = 2 [ a = 3, divisors 2, 1]
or (1/2 + 1/6)(1/1 + 1/19) - 2/3**
8/19 3' 30' 38' 57' 95' (1/2 + 1/3 + 1/5)(1/1 + 1/19) - 2/3**
9/19 3' 12' 38' 57' 76' (1/2 + 1/3 + 1/4)(1/1 + 1/19) - 2/3**
10/19 1/2 38' 5/19 + 5/17
11/19 1/2 19' 38' 10/19 + 1/19
12/19 1/2 12' 38' 76' 114' 11/19 + 1/19 with new 5/19 + 5/19
[5/19 = 12' 76' 114', a = 12, divisors 3, 2]
13/19 3" 57' 3" = 2/3 and 7/19 + 7/19 - 1/19
14/19 3" 19' 57' 13/19 + 1/19
15/19 1/2 4' 38' 76' 10/19 + 5/19
16/19 1/2 4' 19' 38' 76' 15/19 + 1/19
17/19 1/2 3' 30' 57' 95' 18/19 - 1/19
18/19 1/2 3' 12' 57' 76' 1/2 + 9/19 - 1/38
** the 2/3 fudge factor as noted for 3/19, 7/19 and 9/19 may be
the most direct method to read these unit fractions.
Readers may wish to consider these additional facts:
1. Prior to 2,000 BC Egyptian fractions followed a binary structure,
with the notation being called Horus-Eye, as noted by
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2n + ...
2. Babylonian base 60 followed a very similar decimal fraction structure
such that zero was not required to be used. Only the fractions needed
were listed. Thus no zero place holders were required, as our base 10
decimal system was set down by Stevins in 1585 AD.
3. By 2,000 BC Babylonian algebra has been reported by the majority of
mathematical historians, such as Boyer in his popular text, that
this rhetorical algebra is equivalent to our modern algebra I.
4. By 1850 BC, as recorded in 1650 BC by Ahmes in the Rhind Mathematical
Papyrus an exact hieratic form of fractions alters greatly from the
earlier inexact Horus -Eye hieroglyphic fractions.
5. History of Science authors like Neugebauer, Gillings and Knorr have
cited a consistent composite number pattern, as I prefer to write as:
2/pq = (1/q + 1/pq)2/(p + 1) where p and q are prime with p > q.
6. Neugebauer notes the general algorithmic aspect to the composite
form, as does Gillings for the multiple of 3 case, and as does
Knorr and many others. Few if any used algebraic notation to
describe these easy to read unit fractions.
7. Disagreement between scholars is noted on two RMP levels for the
exceptions 2/35, 2/91 and 2/95.
a. First is the three composite case exceptions 2/35 and 2/91.
Knorr discusses the 2/35 and 2/91 in a friendly manner, while
Neugebauer and Gillings appear to be critical to this view.
Concerning 2/35 and 2/91, a well known pattern does emerge,
refuting the conclusions set down by scholars, a form that is
clearly an inverted Greek Golden Proportion, the product of
the arithmetic mean and the harmonic mean.
Note that the arithmetic mean A = (p + q)/2 and the
Harmonic mean H = 2pq/(p + q) can be seen as
2/AH = 2/pq = (1/p + 1/q)2/(p + q).
Fill in the values for p = 5, q = 7 for 2/35 and
p = 7 and q = 13 for 2/91 and see what I mean.
As a 500 AD to 800 AD Akhmim Papyrus point, the Egyptian
inverted Golden Proportion seems to be improved upon,
as Howard Eves noted in his AN INTRODUCTION TO THE
HISTORY OF MATHEMATICS textbook, by:
z/pq = 1/pr + 1/qr where r = (p + q)/2
b. Second is the exception 2/95, which is really:
2/19 stated as a prime unit fractions time 1/5.
Here the prime unit fraction algorithm is revealed by:
2/p - 1/a = (2a - p)/ap where
a is a highly divisible number, about 2/3rds the value of p,
with 2a -p being additively composed of the divisors of a.
Using 2/19 as an example a = 12 was chosen by Ahmes such that
of the two sets of divisors of a that add to 5 < 2a - p,
(2(12) - 19) = 5> 4,1 and 3,2 to be specific the largest
smallest term seemed to appeal to Ahmes.
Writing out 2/19 = 1/12 + (3 + 2)/(12*19)
= 1/12 + 1/76 + 1/114 or
or, as Greeks wrote,
2/19 = 12' 76' 114'
At this point is should be made clear that little time has been
spent on Classical or Hellene Greece. What is clear is that many
minor changes were made by Greeks, one I suspect is a general
form for primes and composite unit fractions, near this statement:
n/pq - 1/a = (na -pq)/apq,
where q can be 1, reducing the algorithm to the earlier Egyptian
form. Seen in this way the messy rules for composites could be
reduced by now using all the divisors of a, p and q to addively
compute the 2nd-4th partition value na -pq.
It should be noted, in conclusion, in terms of RMP prime number
patterns that two aspects may be significant. First, and most
importantly, all prime numbers in the RMP 2/nth table follow ONE
rule, 2/p - 1/a = (2a -p)/ap. Since there are no exceptions to
this rule is is very unlikely it was not know by Greek and
Egyptian mathematicians.
Second, the essentials of the prime aliquot part algorithm, divisors
of the first partition, was noted by B.L. van der Waerden in
SCIENCE AWAKENING, and by E.M Bruins and by Hultsch. Hultsch's 1895
number theory work by on Sylvester's 1880's work may now be
re-visited, right?
Milo Gardner
Sacramento, CA
Subject: Re: Egyptian standards of measure: Was Re: Egyptian Tree Words
From: whittet@shore.net (Steve Whittet)
Date: 27 Sep 1996 03:02:00 GMT
In article , petrich@netcom.com says...
>
>
> [A lot of stuff on unit-fraction decompositions...]
>
> A rather interesting curiosity, but hardly Euclid's _Elements_. I
>think it interesting that Mr. Whittet has yet to claim any Egyptian
>originals for the _Elements_, any Egyptian text that looks as if the
>_Elements_ had been copied from it.
>--
>Loren Petrich Happiness is a fast Macintosh
>petrich@netcom.com And a fast train
>My home page: http://www.webcom.com/petrich/home.html
>Mirrored at: ftp://ftp.netcom.com/pub/pe/petrich/home.html
>
>
Date: Tue, 24 Sep 1996 09:12:51 -0700 (PDT)
From: Milo Gardner
To: Steve Whittet
Cc: Loren Petrich
Subject: Re: Euclid's Book VII and Unit Fractions
Hi Steve (and Loren):
Smallness of the last term tended to dominate shortness of
series, at least that is how I read the RMP, Reisner, Hibeh and
other ancient unit fraction documents.
That is to say,
On Tue, 24 Sep 1996, Steve Whittet wrote:
> Milo Gardner wrote:
> >
> > This paper presents an informal p/q generalized fraction point of view,
> > long proposed by Boyer and many other math historians as not having been
> > known in Babylon, Egypt or Greece prior to 600 BC. The informal evidence
> > that is being presented was taken from the 500 AD era, the Akhmim
> > Papyyrus, a time period when p/q was a known aspect of Classical
> > mathematics. Two tables of unit fractions, n/17 and n/19, will be "read"
> > in a manner that closely resembles Middle Kingdom exact innovations, as
> > discussed in the First and Second Installments as found on
> >
> > 1. http://www.teleport.com/~ddonahue/phresour.html and
> >
> > 2. http://www.seanet.com/~ksbrown/iegypt.htm
> >
> > Please note the manner than 4/17 and 8/17 are computed, possibly by the
> > Euclidean algorithm. Comments would be appreciated by number theory or
> > hisotrian types that may be able to read Appendix I data in another
> > manner.
> >
> > What I see, considering p/q as an unsolved Middle Kingdom problem, as an
> > aspect discussed by Richard Guy in his popular treatise on the subject,
> > 4/y will not be directly proven to have been calculated by a 3-term unit
> > fraction. Nor will a 5/y or x/y unit fraction series be proven by
> > induction, or any other modern number theory method.
> >
> > To justify one basis for presenting this informal proof, I would like to
> > cite, Makers of Mathematics, by Stuart Hollingsdale. Hollingsdale asks:
> > why did 500 AD Egyptians choose the awkward 7/29 = 6' 24' 58' 87' 232'
> > rather than the much simplier modern form of 7/29 = 5' 29' 145'?
> >
> > Simply stated, Egyptians appear to have known and fully utilized:
> >
> > 5/29 = 1/6 + (6*5 - 29)/(6*29), as the Akhmim P. taught
> > = 6' 174'
> >
> > and adding, 2/29 = 1/24 + (2*24 - 29)/(24*29), as the RMP taught
> > = 1/24 + (12 + 4 + 3)/(24*29)
> > 24' 58' 174' 232'
> >
> > Thus 7/29 = 6' 24' 58' (174' + 174') 232'
> > = 6' 24' 58' 87' 232', as the Akhmim P. taught
> >
> > Given that this explanation may one day be confirmed, over 2,500 years of
> > unit fractions history, a set of n/p tables, can now be outlined by two
> > tables, n/17 and n/19, per Attachment I.
> >
> >
> > Attachment I
> >
> > This officially unsolved number theory problem originates from ancient
> > Egypt, covers a period from 2,000 BC to 500 AD a period of 2,500
> > years. The source document is the Akhmim Papyrus, a Hellene 500 AD
> > to 800 AD papyrus, found along the Nile, as cited by Wilbur Knorr,
> > Stanford History of Science Department, Historia Mathematica, HM 9,
> > "Fractions in Ancient Egypt and Greece". Knorr's excellent paper
> > includes several x/y tables, as Guy described them, such as n/17 and
> > n/19. Both tables will be attempted to be "read" using rules that
> > that may have primarily originated in the Egyptian Middle Kingdom era.
> >
> > n/17 Akhmim P. Value Ideas from 1650 BC Egyptian Fractions
> > ---- ---------------- -------------------------------------
> > 2/17 12' 51' 68' 2 a - p = 7 [a = 12, divisors 4, 3]
>
> 17/2 = 8 1/17, 9>
>
> 2/17 - 1/9
> 9*17 = 153
> 18/153 - 17/153 = 1/153
> 2/17 = 9' 153'
>
is shorter; however, the first partition denominator tended to be
selected from a small set of EVEN numbers --- based on the tradition
of the 2/nth tables from the RMP.
Makers of Mathematics author Stuart Hollinsdale from the Open University
asked the same question, as I may have stated in this post where 1/5 was
shown to be smaller than the selected series cited for 7/29 in the Akhmim.
> > 3/17 12' 17' 51' 68' 2/17 + 1/17
>
> 3/17 = 6' 102'
>
for all p > 13 the first partition was selected from highly divisible
numbers == abundant numbers. Note that 6 is perfect, and ONLY used in the RMP
for p < 13. I suspect the abundant number rule continued to be used
by Greeks, Romans and Copts.
> > 4/17 12' 15'17' 68' 85' (1/3 + 1/17)(1/4 + 1/5) or
> > (1/a + 1/p)(1/u + 1/v)*
> > or (1/3 + 1/17)(1/4 + 1/5) or
> > 3/17 + 1/17 (1/a + 1/p)(1/u + 1/v) + 1/17
>
> 4/17 = 6' 17' 102'
>
again, p > 13 and perfect numbers seem not to be allowed.
> > 5/17 4' 34' 68' 5a - p [a = 10, divisors 2, 1]
>
> 5/17 = 17/5 = 3 2/17, 4>
> 5/17 - 1/4
> 4 * 17 = 68
> 20/68 - 17/68 = 3/68
> 3/68 -1/68 = 2/68 = 1/34
> 5/17 = 4' 34' 68'
>
> > 6/17 3' 51' 6a - p = 1 [ new general form]
>
> 3/17 = 6' 102'
> 6/17 = 3' 51' doubling
yes, given a starting point, from an even number for the first partition,
doubling and halving for n/p tables then were easily available.
>
> > 7/17 3' 17' 51' 6/17 + 1/17
> > 8/17 3' 15' 17' 85' (1/3 + 1/17)(1/1 + 1/5) from 4/17
>
> 17/8 = 2 1/8, 3>
> 8/17 - 1/3
> 3 * 17 = 51
> 24/51 - 17/51 = 7/51
> 51/7 = 7 3/7, 8>
> 7/51 - 1/8
> 8 * 51 = 408
> 56/408 - 51/408 = 5/408 = 1/408 + 4/408, 1/102
> 8/17 = 3' 8' 102' 408'
>
the first key to understanding Egyptian fractions is the
choice of the first parition. The second rule appeared to
focus on the last term. A third rule, sometime equal in
importance to the second rule, was the length of the series --
always less than six, the length of the Horus-Eye fractions --
the notation that hieratic fractions superceded.
> (same number of terms, second term smaller)
>
> > 9/17 1/2 34' 5/17 + 5/17 - 1/17
>
> 10/17 = 2' 17' 34'; 9/17 = 2' 34'
>
> > 10/17 1/2 17' 34' 5/17 + 5/17
>
> 5/17 = 4' 34' 68'; 10/17 = 2' 17' 34'
>
> > 11/17 1/2 12' 34' 51' 68' 10/17 + 2/17 (12' 51' 68')
> [this expansion does equal 11/17 even though 10/17 + 2/17 = 12/17]
>
> 6/17 3' 51'
> 5/17 = 4' 34' 68'
>
> 11/17 = 3' 4' 34' 51' 68'
>
see, you have gotten the hang of Egyptian fraction tables. They
are very easy to assemble, given an allowed starting point.
> > 12/17 1/2 12' 17' 34' 51' 68' 11/17 + 1/17
>
> It appears that your calculation of 12/17 is affected by your
> method of calculating 11/17 and so should really equal 13/17
>
> 12/17 = 3' 4' 17' 34' 51' 68'
>
> but actually your expansion does equal 12/17 because 3' 4' = 2' 12'
>
> > 13/17 1/2 4' 68' 9/17 + 5/17 - 1/17
>
> 5/17 = 4' 34' 68'
> 9/17 1/2 34'
> 2/34 = 1/17
> 9/17 = 2' 4' 68'
>
> Yes,...
>
again, all of my series are historical -- from the 500 AD Akhmim P.
> > 14/17 1/2 4' 17' 68' 13/17 + 1/17
> > 15/17 1/2 3' 34' 51' 9/17 + 6/17
> > 16/17 1/2 3' 17' 34' 51' 15/17 + 1/17
>
> ... I think I am following you here
yes, the logic is straight forward. You have gotten it.
> >
> > *u, v were NOT generally compute by the Euclidean algorithm
> >
> > That is to say, Euclid's Algorithm was not copied from
> > Egyptian fractions, as a general rule.
> >
> > On set of "decoding" of Hellene
> > n/19 Akhmim P. value unit fractions using 1650 BC rules
> > ----- --------------------- ------------------------------------
> > 2/19 10' 190' 2a- p = 1 is one of three methods
> > 3/19 15' 20' 57' 76' 95' (1/3 + 1/4 + 1/5)(1/1 + 1/19) - 2/3**
> > or 2/19 + 2/19 - 1/19 [a = 30, divisors 6,
5]
> > 4/19 5' 95' 2/19 + 2/19
> > 5/19 4' 76' 2a - p [ one of three methods]
> > 6/19 4' 19' 76' 5/19 + 1/19
> > 7/19 3' 38' 114' 7a - p = 2 [ a = 3, divisors 2, 1]
> > or (1/2 + 1/6)(1/1 + 1/19) - 2/3**
> > 8/19 3' 30' 38' 57' 95' (1/2 + 1/3 + 1/5)(1/1 + 1/19) - 2/3**
> > 9/19 3' 12' 38' 57' 76' (1/2 + 1/3 + 1/4)(1/1 + 1/19) - 2/3**
> > 10/19 1/2 38' 5/19 + 5/17
> > 11/19 1/2 19' 38' 10/19 + 1/19
> > 12/19 1/2 12' 38' 76' 114' 11/19 + 1/19 with new 5/19 + 5/19
> > [5/19 = 12' 76' 114', a = 12, divisors
3, 2]
> > 13/19 3" 57' 3" = 2/3 and 7/19 + 7/19 - 1/19
> > 14/19 3" 19' 57' 13/19 + 1/19
> > 15/19 1/2 4' 38' 76' 10/19 + 5/19
> > 16/19 1/2 4' 19' 38' 76' 15/19 + 1/19
> > 17/19 1/2 3' 30' 57' 95' 18/19 - 1/19
> > 18/19 1/2 3' 12' 57' 76' 1/2 + 9/19 - 1/38
> >
> > ** the 2/3 fudge factor as noted for 3/19, 7/19 and 9/19 may be
> > the most direct method to read these unit fractions.
>
> The Egyptians used both 2/3 and 3/4, right?
> >
> > Readers may wish to consider these additional facts:
> >
> > 1. Prior to 2,000 BC Egyptian fractions followed a binary structure,
> > with the notation being called Horus-Eye, as noted by
> >
> > 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2n + ...
> >
> > 2. Babylonian base 60 followed a very similar decimal fraction
structure
> > such that zero was not required to be used. Only the fractions
needed
> > were listed. Thus no zero place holders were required, as our base
10
> > decimal system was set down by Stevins in 1585 AD.
>
> Also no place holders were needed because the signs for each decimal
> multiple
> are different "|" for units, "n" for tens, and "@" for hundreds; etc;
yes, Babylonian and Egyptian fractions, both Horus-Eye and hieratic,
required no place holders. Hieratic was particular clear -- having
no passing interest in zero -- the primary historical reason why
zero and base 10 decimals developed very late in teh Western Traditon,
1585 AD, by Stevins.
> >
> > 3. By 2,000 BC Babylonian algebra has been reported by the majority of
> > mathematical historians, such as Boyer in his popular text, that
> > this rhetorical algebra is equivalent to our modern algebra I.
> >
> > 4. By 1850 BC, as recorded in 1650 BC by Ahmes in the Rhind
Mathematical
> > Papyrus an exact hieratic form of fractions alters greatly from the
> > earlier inexact Horus -Eye hieroglyphic fractions.
> >
> > 5. History of Science authors like Neugebauer, Gillings and Knorr have
> > cited a consistent composite number pattern, as I prefer to write
as:
> >
> > 2/pq = (1/q + 1/pq)2/(p + 1) where p and q are prime with p > q.
> >
> > 6. Neugebauer notes the general algorithmic aspect to the composite
> > form, as does Gillings for the multiple of 3 case, and as does
> > Knorr and many others. Few if any used algebraic notation to
> > describe these easy to read unit fractions.
> >
> > 7. Disagreement between scholars is noted on two RMP levels for the
> > exceptions 2/35, 2/91 and 2/95.
> >
> > a. First is the three composite case exceptions 2/35 and 2/91.
> >
> > Knorr discusses the 2/35 and 2/91 in a friendly manner, while
> > Neugebauer and Gillings appear to be critical to this view.
> >
> > Concerning 2/35 and 2/91, a well known pattern does emerge,
> > refuting the conclusions set down by scholars, a form that is
> > clearly an inverted Greek Golden Proportion, the product of
> > the arithmetic mean and the harmonic mean.
> >
> > Note that the arithmetic mean A = (p + q)/2 and the
> > Harmonic mean H = 2pq/(p + q) can be seen as
> >
> > 2/AH = 2/pq = (1/p + 1/q)2/(p + q).
> >
> > Fill in the values for p = 5, q = 7 for 2/35 and
> > p = 7 and q = 13 for 2/91 and see what I mean.
>
> So were the arithmetic and harmonic means derived from unit fraction
> algorithms
> in use since 2,000 BC, (c 1850 BC min) or did they derive slightly later
> during
> the Hyksos period (c 1650 BC when Ahmes writes them down)?
It appears that the paritioning of p/q by proportions did spawn
arithmetic and harmonic series -- as well as Euclid's Book VII.
> >
> > As a 500 AD to 800 AD Akhmim Papyrus point, the Egyptian
> > inverted Golden Proportion seems to be improved upon,
> > as Howard Eves noted in his AN INTRODUCTION TO THE
> > HISTORY OF MATHEMATICS textbook, by:
> >
> > z/pq = 1/pr + 1/qr where r = (p + q)/2
> >
> > b. Second is the exception 2/95, which is really:
> >
> > 2/19 stated as a prime unit fractions time 1/5.
> >
> > Here the prime unit fraction algorithm is revealed by:
> >
> > 2/p - 1/a = (2a - p)/ap where
> >
> > a is a highly divisible number, about 2/3rds the value of p,
> > with 2a -p being additively composed of the divisors of a.
> >
> > Using 2/19 as an example a = 12 was chosen by Ahmes such that
> > of the two sets of divisors of a that add to 5 < 2a - p,
> > (2(12) - 19) = 5> 4,1 and 3,2 to be specific the largest
> > smallest term seemed to appeal to Ahmes.
> >
> > Writing out 2/19 = 1/12 + (3 + 2)/(12*19)
> > = 1/12 + 1/76 + 1/114 or
> >
> > or, as Greeks wrote,
> >
> > 2/19 = 12' 76' 114'
> >
> > At this point is should be made clear that little time has been
> > spent on Classical or Hellene Greece. What is clear is that many
> > minor changes were made by Greeks, one I suspect is a general
> > form for primes and composite unit fractions, near this
statement:
> >
> > n/pq - 1/a = (na -pq)/apq,
> >
> > where q can be 1, reducing the algorithm to the earlier Egyptian
> > form. Seen in this way the messy rules for composites could be
> > reduced by now using all the divisors of a, p and q to addively
> > compute the 2nd-4th partition value na -pq.
> >
> > It should be noted, in conclusion, in terms of RMP prime number
> > patterns that two aspects may be significant. First, and most
> > importantly, all prime numbers in the RMP 2/nth table follow ONE
> > rule, 2/p - 1/a = (2a -p)/ap. Since there are no exceptions to
> > this rule is is very unlikely it was not know by Greek and
> > Egyptian mathematicians.
>
> How is this different from what Sylvester came up with?
Sylvester and Hultsch were very close to this algorithm - -but they
only stated a seven step recurive form -- not the simple algorithm.
Reference: Robins and Shute, The Rhind Mathemtical Papyrus.
> >
> > Second, the essentials of the prime aliquot part algorithm,
divisors
> > of the first partition, was noted by B.L. van der Waerden in
> > SCIENCE AWAKENING, and by E.M Bruins and by Hultsch. Hultsch's
1895
> > number theory work by on Sylvester's 1880's work may now be
> > re-visited, right?
> >
> > Milo Gardner
> > Sacramento, CA
>
>
> really cool informational helpful post,
>
> steve
>
And thank you for reading the post so closely. I wonder if Loren has taken
any time to try to digest the math and references.
Loren, it is now your turn.
Milo Gardner
Sacramento, CA
steve
Subject: Re: Egyptian standards of measure: Was Re: Egyptian Tree Words
From: whittet@shore.net (Steve Whittet)
Date: 27 Sep 1996 02:58:44 GMT
In article , petrich@netcom.com says...
>
>
> [A lot of stuff on unit-fraction decompositions...]
>
> A rather interesting curiosity, but hardly Euclid's _Elements_. I
>think it interesting that Mr. Whittet has yet to claim any Egyptian
>originals for the _Elements_, any Egyptian text that looks as if the
>_Elements_ had been copied from it.
>--
>Loren Petrich Happiness is a fast Macintosh
>petrich@netcom.com And a fast train
>My home page: http://www.webcom.com/petrich/home.html
>Mirrored at: ftp://ftp.netcom.com/pub/pe/petrich/home.html
>
>
Date: Mon, 23 Sep 1996 13:46:17 -0700 (PDT)
From: Milo Gardner
To: Steve Whittet
Cc: Loren Petrich
Subject: Re: Euclid's Book VII and Unit Fractions
Hi Steve (and Loren):
Thanks Steve, for your email. Syvester first tried Fibonacci, and later
improved it by showing that the Euclidean Algorithm always computes
a unit fraction series. As a few example should clearly show that the
Eucidean Algorithm almost never computes a series as small as
Egyptians, Greeks and Romans wrote in their p/q tables.
Milo
On Mon, 23 Sep 1996, Steve Whittet wrote:
> Milo Gardner wrote:
>
>
> Hi Milo,
>
> Great post! Really excellent
....snip...
I need to more work on Book VII and
show its connection to Egyptian number theory. The no part is, I
have modified an old post --- one that I had almost forgotten about.
> As far as Sylvesters work, as I recall his theory was that an Egyptian
> could convert a fraction to a unit fraction by dividing the numerator
> into the denominator and rounding up to the next whole integer, then
> subtracting a unit fraction with that denominator from the original
> fraction and repeating the process until there was no remainder left.
>
> Is that essentially correct?
Sylvester tried several techniques, such as Fibonacci and
Euclid's Algorithm.
>
> 4/17
> 17/4= 4 1/17
> 4 1/17 > 5
>
> 4/17 - 1/5
>
> 5 * 17 = 85
> 20/85 - 17/85 = 3/85
> 85/3 = 28 1/85
> 28 1/85 > 29
>
> 3/85 - 1/29
>
> 85*29 = 2465
> 87/2465- 85/2465 = 2/2465
> 6485/2 = 3242 1/6485
> 3242 1/6485 > 3243
>
> 2/2465 - 1/3243
>
> 2465*3243 = 7993995
> 6486/7993995-2465/7993995 =4021/7993995
> 7993995/4021 = 1988.061428
> 1988.061428 > 1989
>
> 4021/7993995 - 1/1989
>
> 7993995*1989 = 15900056000
> 7997769/15900056000-7993995/15900056000= 3774/15900056000
> 15900056000/3774 =4213052.458
> 4213052.458> 4213053
>
> 3774/15900056000- 1/4213053
>
> 15900056000*4213053=
> 66987779000000000
> 15900062/66987779000000-15900056/66987779000000=6/66987779000000
>
> =1/11164630000000
>
> at which point I am exceeding my pocket calculators accuracy.
>
yes, starting at the 'wrong p/q value quickly runs into bid denominastors.
That is why a close review of p/q + or - 1/q, 2/q and so forth is required
to find the smallest table of n/q unit fraction series.
> ...
> 4/17 = 1/5 + 1/29 + 1/1989 + 1/3243 + 1/4213053 = .235294118
>
> 4/17 = 1/5 +1/29 +1/1989 + 1/3243 ... = .23529388
>
> Your algorithm gives
>
> 4/17 12' 15'17' 68' 85' (1/3 + 1/17)(1/4 + 1/5) or
> (1/a + 1/p)(1/u + 1/v)*
> or (1/3 + 1/17)(1/4 + 1/5) or
> 3/17 + 1/17 (1/a + 1/p)(1/u + 1/v) + 1/17
>
> (I think it is sharp to subtract 1/17 off the top)
>
Kevin Brown introduced this special case algorithm. As you know
it appears to be Euclid's algorithm,
>
> 4/17 - 1/17 = 3/17
> 17/3 = 5 2/3
> 5 2/3 > 6
> 3/17 - 1/6
> 6*17= 102
> 18/102 - 17/102 = 1/102
>
> 1/6 + 1/17 + 1/102 = 4/17
>
>
> >
> > Milo Gardner
> > Sacramento, CA
>
>
> steve
> >
> > This paper presents an informal p/q generalized fraction point of view,
> > long proposed by Boyer and many other math historians as not having been
> > known in Babylon, Egypt or Greece prior to 600 BC. The informal evidence
> > that is being presented was taken from the 500 AD era, the Akhmim
> > Papyyrus, a time period when p/q was a known aspect of Classical
> > mathematics. Two tables of unit fractions, n/17 and n/19, will be "read"
> > in a manner that closely resembles Middle Kingdom exact innovations, as
> > discussed in the First and Second Installments as found on
> >
> > 1. http://www.teleport.com/~ddonahue/phresour.html and
> >
> > 2. http://www.seanet.com/~ksbrown/iegypt.htm
> >
> > Please note the manner than 4/17 and 8/17 are computed, possibly by the
> > Euclidean algorithm. Comments would be appreciated by number theory or
> > hisotrian types that may be able to read Appendix I data in another
> > manner.
> >
> > What I see, considering p/q as an unsolved Middle Kingdom problem, as an
> > aspect discussed by Richard Guy in his popular treatise on the subject,
> > 4/y will not be directly proven to have been calculated by a 3-term unit
> > fraction. Nor will a 5/y or x/y unit fraction series be proven by
> > induction, or any other modern number theory method.
> >
> > To justify one basis for presenting this informal proof, I would like to
> > cite, Makers of Mathematics, by Stuart Hollingsdale. Hollingsdale asks:
> > why did 500 AD Egyptians choose the awkward 7/29 = 6' 24' 58' 87' 232'
> > rather than the much simplier modern form of 7/29 = 5' 29' 145'?
> >
> > Simply stated, Egyptians appear to have known and fully utilized:
> >
> > 5/29 = 1/6 + (6*5 - 29)/(6*29), as the Akhmim P. taught
> > = 6' 174'
> >
> > and adding, 2/29 = 1/24 + (2*24 - 29)/(24*29), as the RMP taught
> > = 1/24 + (12 + 4 + 3)/(24*29)
> > 24' 58' 174' 232'
> >
> > Thus 7/29 = 6' 24' 58' (174' + 174') 232'
> > = 6' 24' 58' 87' 232', as the Akhmim P. taught
> >
> > Given that this explanation may one day be confirmed, over 2,500 years of
> > unit fractions history, a set of n/p tables, can now be outlined by two
> > tables, n/17 and n/19, per Attachment I.
> >
> >
> > Attachment I
> >
> > This officially unsolved number theory problem originates from ancient
> > Egypt, covers a period from 2,000 BC to 500 AD a period of 2,500
> > years. The source document is the Akhmim Papyrus, a Hellene 500 AD
> > to 800 AD papyrus, found along the Nile, as cited by Wilbur Knorr,
> > Stanford History of Science Department, Historia Mathematica, HM 9,
> > "Fractions in Ancient Egypt and Greece". Knorr's excellent paper
> > includes several x/y tables, as Guy described them, such as n/17 and
> > n/19. Both tables will be attempted to be "read" using rules that
> > that may have primarily originated in the Egyptian Middle Kingdom era.
> >
> > n/17 Akhmim P. Value Ideas from 1650 BC Egyptian Fractions
> > ---- ---------------- -------------------------------------
> > 2/17 12' 51' 68' 2 a - p = 7 [a = 12, divisors 4, 3]
> > 3/17 12' 17' 51' 68' 2/17 + 1/17
> > 4/17 12' 15'17' 68' 85' (1/3 + 1/17)(1/4 + 1/5) or
> > (1/a + 1/p)(1/u + 1/v)*
> > or (1/3 + 1/17)(1/4 + 1/5) or
> > 3/17 + 1/17 (1/a + 1/p)(1/u + 1/v) + 1/17
> > 5/17 4' 34' 68' 5a - p [a = 10, divisors 2, 1]
> > 6/17 3' 51' 6a - p = 1 [ new general form]
> > 7/19 3' 17' 51' 6/17 + 1/17
> > 8/17 3' 15' 17' 85' (1/3 + 1/17)(1/1 + 1/5) from 4/17
> > 9/17 1/2 34' 5/17 + 5/17 - 1/17
> > 10/17 1/2 17' 34' 5/17 + 5/17
> > 11/17 1/2 12' 34' 51' 68' 10/17 + 2/17 (12' 51' 68')
> > 12/17 1/2 12' 17' 34' 51' 68' 11/17 + 1/17
> > 13/17 1/2 4' 68' 9/17 + 5/17 - 1/17
> > 14/17 1/2 4' 17' 68' 13/17 + 1/17
> > 15/17 1/2 3' 34' 51' 9/17 + 6/17
> > 16/17 1/2 3' 17' 34' 51' 15/17 + 1/17
> >
> > *u, v were NOT generally compute by the Euclidean algorithm
> >
> > That is to say, Euclid's Algorithm was not copied from
> > Egyptian fractions, as a general rule.
> >
> > On set of "decoding" of Hellene
> > n/19 Akhmim P. value unit fractions using 1650 BC rules
> > ----- --------------------- ------------------------------------
> > 2/19 10' 190' 2a- p = 1 is one of three methods
> > 3/19 15' 20' 57' 76' 95' (1/3 + 1/4 + 1/5)(1/1 + 1/19) - 2/3**
> > or 2/19 + 2/19 - 1/19 [a = 30, divisors 6,
5]
> > 4/19 5' 95' 2/19 + 2/19
> > 5/19 4' 76' 2a - p [ one of three methods]
> > 6/19 4' 19' 76' 5/19 + 1/19
> > 7/19 3' 38' 114' 7a - p = 2 [ a = 3, divisors 2, 1]
> > or (1/2 + 1/6)(1/1 + 1/19) - 2/3**
> > 8/19 3' 30' 38' 57' 95' (1/2 + 1/3 + 1/5)(1/1 + 1/19) - 2/3**
> > 9/19 3' 12' 38' 57' 76' (1/2 + 1/3 + 1/4)(1/1 + 1/19) - 2/3**
> > 10/19 1/2 38' 5/19 + 5/17
> > 11/19 1/2 19' 38' 10/19 + 1/19
> > 12/19 1/2 12' 38' 76' 114' 11/19 + 1/19 with new 5/19 + 5/19
> > [5/19 = 12' 76' 114', a = 12, divisors
3, 2]
> > 13/19 3" 57' 3" = 2/3 and 7/19 + 7/19 - 1/19
> > 14/19 3" 19' 57' 13/19 + 1/19
> > 15/19 1/2 4' 38' 76' 10/19 + 5/19
> > 16/19 1/2 4' 19' 38' 76' 15/19 + 1/19
> > 17/19 1/2 3' 30' 57' 95' 18/19 - 1/19
> > 18/19 1/2 3' 12' 57' 76' 1/2 + 9/19 - 1/38
> >
> > ** the 2/3 fudge factor as noted for 3/19, 7/19 and 9/19 may be
> > the most direct method to read these unit fractions.
> >
> > Readers may wish to consider these additional facts:
> >
> > 1. Prior to 2,000 BC Egyptian fractions followed a binary structure,
> > with the notation being called Horus-Eye, as noted by
> >
> > 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2n + ...
> >
> > 2. Babylonian base 60 followed a very similar decimal fraction
structure
> > such that zero was not required to be used. Only the fractions
needed
> > were listed. Thus no zero place holders were required, as our base
10
> > decimal system was set down by Stevins in 1585 AD.
> >
> > 3. By 2,000 BC Babylonian algebra has been reported by the majority of
> > mathematical historians, such as Boyer in his popular text, that
> > this rhetorical algebra is equivalent to our modern algebra I.
> >
> > 4. By 1850 BC, as recorded in 1650 BC by Ahmes in the Rhind
Mathematical
> > Papyrus an exact hieratic form of fractions alters greatly from the
> > earlier inexact Horus -Eye hieroglyphic fractions.
> >
> > 5. History of Science authors like Neugebauer, Gillings and Knorr have
> > cited a consistent composite number pattern, as I prefer to write
as:
> >
> > 2/pq = (1/q + 1/pq)2/(p + 1) where p and q are prime with p > q.
> >
> > 6. Neugebauer notes the general algorithmic aspect to the composite
> > form, as does Gillings for the multiple of 3 case, and as does
> > Knorr and many others. Few if any used algebraic notation to
> > describe these easy to read unit fractions.
> >
> > 7. Disagreement between scholars is noted on two RMP levels for the
> > exceptions 2/35, 2/91 and 2/95.
> >
> > a. First is the three composite case exceptions 2/35 and 2/91.
> >
> > Knorr discusses the 2/35 and 2/91 in a friendly manner, while
> > Neugebauer and Gillings appear to be critical to this view.
> >
> > Concerning 2/35 and 2/91, a well known pattern does emerge,
> > refuting the conclusions set down by scholars, a form that is
> > clearly an inverted Greek Golden Proportion, the product of
> > the arithmetic mean and the harmonic mean.
> >
> > Note that the arithmetic mean A = (p + q)/2 and the
> > Harmonic mean H = 2pq/(p + q) can be seen as
> >
> > 2/AH = 2/pq = (1/p + 1/q)2/(p + q).
> >
> > Fill in the values for p = 5, q = 7 for 2/35 and
> > p = 7 and q = 13 for 2/91 and see what I mean.
> >
> > As a 500 AD to 800 AD Akhmim Papyrus point, the Egyptian
> > inverted Golden Proportion seems to be improved upon,
> > as Howard Eves noted in his AN INTRODUCTION TO THE
> > HISTORY OF MATHEMATICS textbook, by:
> >
> > z/pq = 1/pr + 1/qr where r = (p + q)/2
> >
> > b. Second is the exception 2/95, which is really:
> >
> > 2/19 stated as a prime unit fractions time 1/5.
> >
> > Here the prime unit fraction algorithm is revealed by:
> >
> > 2/p - 1/a = (2a - p)/ap where
> >
> > a is a highly divisible number, about 2/3rds the value of p,
> > with 2a -p being additively composed of the divisors of a.
> >
> > Using 2/19 as an example a = 12 was chosen by Ahmes such that
> > of the two sets of divisors of a that add to 5 < 2a - p,
> > (2(12) - 19) = 5> 4,1 and 3,2 to be specific the largest
> > smallest term seemed to appeal to Ahmes.
> >
> > Writing out 2/19 = 1/12 + (3 + 2)/(12*19)
> > = 1/12 + 1/76 + 1/114 or
> >
> > or, as Greeks wrote,
> >
> > 2/19 = 12' 76' 114'
> >
> > At this point is should be made clear that little time has been
> > spent on Classical or Hellene Greece. What is clear is that many
> > minor changes were made by Greeks, one I suspect is a general
> > form for primes and composite unit fractions, near this
statement:
> >
> > n/pq - 1/a = (na -pq)/apq,
> >
> > where q can be 1, reducing the algorithm to the earlier Egyptian
> > form. Seen in this way the messy rules for composites could be
> > reduced by now using all the divisors of a, p and q to addively
> > compute the 2nd-4th partition value na -pq.
> >
> > It should be noted, in conclusion, in terms of RMP prime number
> > patterns that two aspects may be significant. First, and most
> > importantly, all prime numbers in the RMP 2/nth table follow ONE
> > rule, 2/p - 1/a = (2a -p)/ap. Since there are no exceptions to
> > this rule is is very unlikely it was not know by Greek and
> > Egyptian mathematicians.
> >
> > Second, the essentials of the prime aliquot part algorithm,
divisors
> > of the first partition, was noted by B.L. van der Waerden in
> > SCIENCE AWAKENING, and by E.M Bruins and by Hultsch. Hultsch's
1895
> > number theory work by on Sylvester's 1880's work may now be
> > re-visited, right?
>
steve