The Sphinx, from the collection of Lichtenstern and Harari in France
(http://silicon.montainge.u-bordeaux.fr:8001/IMAGES/EGYPT/CPA/LICHT/LHB205.jpeg)
Date: Tue, 14 May 1996 16:31:14 -0500 (EST)
Subject: Re: Egyptian geometry (reply to Paul Manansala)
To: athena-discuss@info.harpercollins.com
Message-Id: <01I4PASBLISI8X61V4@VAX1.JMU.EDU>
Paul Kekai Manansala wrote:
>From: IN%"polmansl@ix.netcom.com" 14-MAY-1996 15:34:58.91
>To: IN%"athena-discuss@info.harpercollins.com"
>Subj: Egyptian Geometry
[rest of header deleted; Paul's message to which this is
a reply is appended below]
Paul: Afraid not. I suggest you read in any good
history of mathematics book about the nature and
definition of pi, the development of algebraically
expressed formulas, etc. You might try the
second edition of the one by Carl Boyer, edited
by Uta Merzbach (note that my name is in the
index). However, there are better ones on
ancient mathematics.
Or better yet, take a look for yourself at
translations of the two papyri you mention. The
formula for the surface area of a sphere which
you claim is to be found in the Moscow papyrus
was not stated, or even statable in such a form
until sometime in the 17th century. There were
no "equations" as we know them now until then.
Techniques for *approximating* the surface area of
a *particular* sphere, with given numerical radius,
were known to Egyptians and other cultures of that and
other regions (e.g., probably in India) in
early times, but this is some distance from a
general formula or "equation", including the
very concept of a general formula or equation.
I haven't looked at the Moscow or Rhind papyri for
some years now, but I wouldn't doubt that there is
a verbally stated problem there (in hieroglyphics)
which one can interpret *now* as *approximating* the
area of a sphere with a radius given as a specific
number, not a variable or "unknown". (I wish my
wife were home. She knows the contents of the
Rhind papyrus very well, and uses material from
it in teaching her history of mathematics courses,
to racially mixed students --- she also uses in
a positive way some material from Bernal's vol I.)
No serious historian of mathematics that I know of
has ever doubted that such approximation techniques
were known long before the development of classical Greek
mathematics. But the axiomatic treatments by Euclid,
Apollonius, Archimedes and other lesser known Greek
mathematicians (to use an anachronistic name for them)
who preceded them, based on the work of many other
mathematicians --- Greek, Egyptian, Babylonian, probably
Sumerian, maybe by diffusion, Indian, or maybe they
arrived at their results independently --- is something else
again.
Note for example that the many, many theorems in
Euclid's *Elements* are *all* logically derived
from a small number of initially stated assumptions
(axioms and postulates). (This, in my opinion, is
what Aristotle based the development of his logical
theories on.) Some (actually, only a few) of the individual
results so derived are *general* statements (in words,
not relatively modern formulas) of procedures for finding
areas, volumes, etc., using proportions, so that a number
like pi was not explicitly involved, but can only be inferred
using mathematical techniques developed --- need I say it,
by many "races" --- since the 17th century or so.
There is no trace of such *general* techniques
(stated verbally) in any remains I ever heard of
from North Africa, Near Asia, India, or anyplace else,
before they appeared in classical Greece..
Nor is there any trace of anyone devising techniques
for handling incommensurables (now usually called
irrationals) before the Greeks of the classic era,
although the Babylonians, at least, had techniques
which can be interpreted (now!) as means for approximating
specific irrationals, such as the square root of 2.
(I'm not sure whether or not any Egyptian remains
show such a technique, but I wouldn't be surprised if
they did, or if the Egyptians had such a technique of
which we have no record, which they may have got from
the Babylonians (Chaldeans), or given to the Babylonians,
or developed by themselves, or some combination of these.)
Not even the Greeks had sines, cosines and tangents,
although in the *Almagest* of Ptolemy, c 150 AD, there
are techniques which are *equivalent* to using such
trigonometic functions. In fact, some writers a
few years ago showed that one can "translate"
various calculations made by Ptolemy into the
language of Fourier series, which are infinite series
made up using sines and/or cosines. But it would
be absurd to say that Ptolemy (or some of his
associates or predecessors) anticipated Euler and
the like in using such series (Fourier didn't
i
Which reminds me: there is no indication that I could
ever find in a number of years of looking into translations
of Egyptian, Babylonian, etc, "philosophical" and "religious"
and "scientific" (all words which promote anachronism)
documents which show the kind of concern and attitudes
toward *infinity* which shows up among the ancient Greeks.
But enough, enough. I guess everybody thinks they are
entitled to be experts in mathematics, just as in language,
race questions, etc.
Gordon Fisher fishergm@jmu.edu
**********************************************************
>Following are some examples of Egyptian knowledge of
>geometry and trigonometry about 2,000 years before the
>time of Archimedes according to the Rhind and Moscow
>Papyri. The purpose is to show that there is a basis
>for the claims of Greek historians that Greece borrowed
>its early geometry from Egypt, Chaldea, Babylon, etc.
>These are taken from Cheikh Anta Diop's _Civilization
>or Barbarism_.
>
>
>From _PAPYRUS OF MOSCOW_:
>
>Surface area of sphere:
>
> S = 4piR^2
>
>
>Volume of truncated pyramid:
>
> V = h/2(a^2 + ab + b^2)
>
>
>Calculation of slope of pyramid:
>
>This is calculated using a circle inscribed in the
>square composed from the base (b) of the pyramid:
>
> sine angle of inclination = height
> cosine " " " = base/2
> tangent " " " = height/cosine angle of inclination
>
> thus,
> cotangent angle of inclination = cosine angle of inclination/
> sine angle of inclination
>
>
>
>From _RHIND PAPYRUS_:
>
>
>Surface area of circle:
>
> S = piR^2
>
>
>Volume of cylinder:
>
> V = prR^2h
>
>
>Surface area of triangle:
>
> S = 1/2 hb
>
>
>Surface of rectangle (also in Papyrus of Moscow):
>
>
> S = Ll, where l = 1/2L + 1/4L
>
>
>
>
>
>Plutarch on the Pythagorean theorem:
>
>
>
> "The Egyptians conceive the world as a triangle,
> the most beautiful of triangles, just as Plato, in
> his _Republic_, uses it as a symbol of matrimonial
> union. The most beautiful of triangles has its vertical
> side of three, its base of four, and its hypothenuse
> as five..."
>
>
>
>Paul Kekai Manansala
-----------------
Subject: Re: Egyptian geometry (Paul & Gordon)
To: athena-discuss@info.harpercollins.com
Sender: owner-athena-discuss@info.harpercollins.com
Precedence: bulk
Gordon wrote:
Paul: I give up. If you think what the papyri, cuneiform tablets,
pyramids (as physical structures, e.g. tombs) show is a knowledge
of "equations" or "formulas", we can communicate no further on
this subject. Note, however, that this disagreement has nothing
to do with race, as far as I can see, but rather with what can
be legitimately inferred from remains of various sorts, without
being anachronistic.
Gordon
***************
I also do not want to get into a long-drawn out argument on this
issue. However, I'd like to give one example for those who may
not understand what I mean by "inference." Let's say a scrap
of paper was somehow preserved and found 500 years from now, when
for some reason all knowledge of our times was lost. This is
what was found on the paper:
----------------------------------
Problem: What is fifty times twenty?
Solution: 50
X20
00
1000
1000
-------------------------------------
>From this paper (if it could be translated), I could infer that a
place numeral system was used and also a system of cross
multiplication. I would not need a manual of mathematics to back this
up, or a rule stating "First arrange numerals according to ones, tens,
hundreds..."
The mathematical papyri of Egypt usually consist of problems placed
to a student, in which the solutions are also given. And these
solutions are usually *correct* (taking into account their value of
pi).
PKM
----------------
Date: Tue, 14 May 1996 20:52:08 -0500 (EST)
Subject: Re: Egyptian geometry (Paul & Gordon)
To: athena-discuss@info.harpercollins.com
Message-Id: <01I4PJVUA9IA8X5DOS@VAX1.JMU.EDU>
X-Vms-To: IN%"athena-discuss@info.harpercollins.com"
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Sender: owner-athena-discuss@info.harpercollins.com
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Paul wrote:
>From: IN%"polmansl@ix.netcom.com" 14-MAY-1996 19:13:01.85
>To: IN%"athena-discuss@info.harpercollins.com"
>Subj: RE: Egyptian geometry (Paul & Gordon)
>
>Gordon wrote:
>
>Paul: I give up. If you think what the papyri, cuneiform tablets,
>pyramids (as physical structures, e.g. tombs) show is a knowledge
>of "equations" or "formulas", we can communicate no further on
>this subject. Note, however, that this disagreement has nothing
>to do with race, as far as I can see, but rather with what can
>be legitimately inferred from remains of various sorts, without
>being anachronistic.
>
>Gordon
>
>***************
[Paul's reply to the above:}
>
>I also do not want to get into a long-drawn out argument on this
>issue. However, I'd like to give one example for those who may
>not understand what I mean by "inference." Let's say a scrap
>of paper was somehow preserved and found 500 years from now, when
>for some reason all knowledge of our times was lost. This is
>what was found on the paper:
>
>
>----------------------------------
>
>Problem: What is fifty times twenty?
>
>
>Solution: 50
> X20
> 00
> 100
> 100
>
>-------------------------------------
>
>
>From this paper (if it could be translated), I could infer that a
>place numeral system was used and also a system of cross
>multiplication. I would not need a manual of mathematics to back this
>up, or a rule stating "First arrange numerals according to ones, tens,
>hundreds..."
>
>The mathematical papyri of Egypt usually consist of problems placed
>to a student, in which the solutions are also given. And these
>solutions are usually *correct* (taking into account their value of
>pi).
>
>PKM
Yes, and all sorts of things about the Egyptian numeration system
and their algorithms for multiplication, division, etc., can be
inferred from the papyri and other sources. This again is taught
by my wife to her students, many of whom she convinces that the
Egyptian algorithm is simpler in many ways than the one in common
use today is most parts of the world. And I agree that the
Egyptians possessed algorithmic methods for these and other
numrical operations (e.g., with fractions) which they could apply
to any specific pair or other set of numbers. These algorithms
were general in that sense.
I also agree with your implied contention that the Egyptians
had algorithms for calculating areas, volumes, etc, which could
be applied to examples with specific numerical data given.
However, they seem never to have left statements or other
descriptions of these algorithms of the type "if r [or some
other symbol] is the radius of a circle, then the area of
the circle is the circumference of the circle as r is to 2",
much less one of the form (pi)r^2 is to 2(pi)r as r is to 2,
or (pi)r^2/2(pi)r = r/2. One might argue that what they
did was *equivalent* to this, but equivalence is not identity.
In my view, the transition from a step by step scheme illustrated
with numerical examples amounted to formulas and equations
of the sort you quoted from Diop amounts to an introduction
of algebra as applied to geometry, and this took some doing.
You may wish to say that in fact the Egyptians or others
knew about this sort of thing before, say, Vieta in the 15th
century, but kept it a secret. Of course, I can't deny that,
nor can you prove it. So I guess it comes down to a choice.
Evidently I choose one way, which involves a gradual (though
maybe punctuated, as some evolutionists say) cumulative
development of mathematics over time (with some notable
eclipses, recoveries, re-inventions, etc), whereas you,
if I understand you correctly, believe that certain
mathematics came quite fully formed to the ancient Greeks,
who added to it little or nothing of note, and eventually
it got passed on to us in much the same form.
In conclusion, let me say that inferences about numerical
algorithms, and algorithms for measuring things, don't
really touch the question of the development of axiomatic
systems. Also let me say that when I used the word "infinity"
in one of my messages, I forgot that this word means something
special to mathematicians, and we are open to all sorts of
misunderstanding when we use it. We use the word in quite narrow
ways, from some standpoints. For example, we make distinctions
between an infinite sequence (one enumerable with the so-called
natural numbers, or positive integers), an infinite sequence
of "partial sums" (the first term of the the first sequence, then
the sum of the first 2 terms, then the sum of the first 3 terms,
etc.), and the so-called "sum" of this sequence, which is *not*
obtained by adding together all the terms of the original infinite
sequence, but is shown to be a "limit" of the partial sums
in certain ways. It's concern with this sort of thing I was
thinking of when I said I hadn't found any traces of it in
remains before those we have from the classical Greeks. I
expect your "whoa, boy" in answer to my statement about
"infinity" referred to other uses of this term. We could
also talk about infinitesimals, which also concerned some
of the Greek mathematicians, but hasn't shown up earlier.
I apologize for stating all this stuff if it's known to you,
but some others on the list may be interested. Boy, did
this turn out to be longer than I intended!
Gordon
----------------
Date: Wed, 1 May 1996 18:27:57 -0500
To: athena-discuss@info.harpercollins.com
From: wagers@computek.net (Will Wagers)
Subject: Re: Egyptian Math (Long)
Sender: owner-athena-discuss@info.harpercollins.com
Precedence: bulk
This is a fairly narrow subject area, but no narrower than many now
on the list. Just forwarding a contribution from another list:
----------
Subject: Unsolved Problem, resolved = Greeks borrowed heavily
Hi Will:
I noticed your ANE post and thought you might like to log onto:
http://www.seanet.com/ksbrown/ and look under History of Math
and Why Egyptian unit fraction?
Kevin Brown shows that ancient Egyptian fractions used 'local'
yet provable algorithms. Even Greeks used this notation and
did not find 'global' solutions to this numeration issue,
as modern scholars search for a general factorization method.
As background the following paper may show a few details that
can be shared when Bernal-Lefkowitz get around the this
aspect of the debate. As I see it, the particulars of both
debaters are way off base. Bernal can win this point, but
he'll have to learn a little subtle number theory to do it.
Have a great day,
Milo Gardner
Sacramento, CA
A. This famous unsolved problem is a 'global' red herring:
If n is an integer larger than 1, must there be integers x, x, y,
such that 4/n = 1/x + 1/y + 1/z?
General considerations: A number of the form 1/x is an integer is
called an Egyptian fraction. Thus we want to know is 4/n is always
the sum of three Egyptian fractions, for n > 1. It should also be
recaled that the 2/p - 1/a = (2a -p)/ap 2-term case where 2a -p = 1 as
Hultsch showed in 1895 also applies for the 3-term and 4-term unit
fraction series as found in the RMP 2/nth table.
The problem is therefore considered resolved rather than solved since the
historical situation covers more than the RMP and over 2,500 years of
Egyptian fraction history. The ancient mathematicians easily solved
this problem by converting p/q into an appropriate Egyptian fraction
series by finding the smallest 2-term, 3-term or 4-term series
by an LCM rule set out by red auxiliary numbers as detailed in the
ATTACHMENT.
B. ATTACHMENT
As an outline this paper proposes to historically discuss the
EMLR and RMP 2/nth table as one document, a new idea, as mathematical
historians can easily determine. The draft paper's selected methodology
(Babylonian algebra) suggests that Old Kingdom Horus-Eye fractions and
Middle Kingdom hieratic fractions are algebraically closely related, but
unique in several important respects. That is to say, this one document
proposal may, one day, become an accepted classroom pedegogy and be
discussed as plausible worthy of humorous and serious study.
One assumption of this paper is that the EMLR and RMP can not be directly
refuted as belonging to the same Middle Kingdom tradition extending to
the Coptic era (detailed by the Akhmim Papyrus and other documents) by
modern mathematics. This conclusion, which any study group can easily test,
strongly suggests rational numbers were easily converted into exact unit
fraction series as early as 2,000 BCE, much as Horus-Eye fractions converted
p/q into its inexact decimal fraction series and the majority of RMP 2/nth
table members (though in a much more awkward manner). Given that the
Moscow Papyrus provides written evidence, such as its writing of 2/5 by
the same algorithm that is found in the improved EMLR and RMP, that pre-
hieratic fractions were poorly computed by the Horus-Eye duplation
multiplication, as now accepted by the Egyptology community (Shute). That
is a 'new' Middle Kingdom algebraic paradigm is outlined below:
1. EMLR, The Egyptian Mathematical Leather Roll, 26 equations (Gillings)
a. 1/p = 1/2p + 1/2p and 1/p = 1/3p + 1/3p + 1/3p
b. 1/p(1/2) = 1/p(1/3 1/6) and 1/pq(1/3 1/6)
c. 1/p(1) = 1/p(1/2 1/3 1/6) and 1/pq(1/2 1/3 1/6)
d. 1/p(2/q) where 2/q was taken from the RMP 2/nth table such as
(1) 1/p(2/7) = 1/p(1/4 + 1/28) and
(2) line 17's obvious error of 1/13 = 1/28 1/49 1/96, surely was not
(as Gillings suggested) an attempt to write -
(a) 1/13(1) = 1/13(1/2 1/3 1/6) = 1/26 1/39 1/78, using rule 1.c, but
(b) 1/13= 1/3(3/13) = 1/3(1/8 1/17 1/52 1/104) = 1/24 1/39 1/156 1/312
since 2/13 = 1/8 1/52 1/104 from the RMP, as also hinted by
e. Lines 1, 2 and 3 from the EMLR which shows
(1) 1/8=1/10 1/40 = 1/10(1/1 1/4) = 1/10(5/4)=1/pq= 1/p(1/(q+1)(1/1 1/q)
(2) 1/4 = 1/5 1/20 = 1/5(1/1 1/4) = 1/5(5/4) =1/pq =1/(pq+1)(1/1 + 1/pq)
(3) 1/3 = 1/4 1/12 = 1/4(1/1 1/3) = 1/4(4/3) = 1/p = 1/(p+1)(1/1 + 1/p)
2. RMP, 2/nth table (Shute, Gillings, Chace and others)
a. 2/p - 1/a = (2a -p)/ap where a is a highly divisible number,
usually about 2/3rd of p; with 2a -p additively composed of divisors of a.
Note: Every 2/nth table 2/p unit fraction series follows this one rule
much as Hultsch, Bruins and several others have proposed over the
last 100 years.
b. Otto Neugebauer, the dominate scholar on which the current Egyptology
view of Egyptian fractions rests, reported only a muddled version of
the easy to read composite case:
(1) 2/pq = (1/p + 1/pq)2/(p + 1),
than computes all composite 2/nth table member except 2/35, 2/91, 2/95.
It is important to note that the only positive Egyptian fractions view
that Neugebauer included in his Exact Sciences in Antiquity analysis
is the acceptance of algorithm(1) with the form:
"2/n=1/3(1/n) + 5/3(1/n)... (with the comment)...
in this way, more and more cases of the table can be reached
and it appears to me there is little doubt that we have found in essence
the procedure which has lead to these rules of replacement of 2/n by
the sum of unit fraction." As a counter example to Neugebauer note that
(2) 2/pq =(1/p + 1/q)2/(p + q)
is clearly present in the 2/35 and 2/91 cases, as not seen by Neugebauer,
even though algorithm 2.b.(2) is simply read as the product of the
arithmetic mean (A) and harmonic mean (H), seen in the form 2/AH,
a common Ancient Near East pattern.
(3) Wrapping up the final exception, 2/95, is achieved by the trivial
form 2/95 = 2/19(1/5) where 2/19 was taken from equation 2.b.(1).
c. In conclusion, the EMLR and RMP as proposed as one document presents an
interesting set of patterns. Two clues that tend to closely link the EMLR
and RMP, mathematically and historically, beyond Henry Rhind bringing
both back to England in 1855, can be summarized by:
(1) Three of the 26 EMLR lines contain RMP 2/nth table members. One error
contained on line 17 is interesting in that a student may have been
confused in the writing of 1/13 as 1/3(3/13) as also suggested by lines
1: 1/8 = 1/10 1/40 = 1/10(1/1 1/4) = 1/10(5/4)
2: 1/4 = 1/5 1/20 = 1/5(1/1 1/4) = 1/5(5/4)
3: 1/3 = 1/4 1/12 = 1/4(1/1 1/3) = 1/4(4/3)
1 p + 1 1 1 p + 1
which infers 1/p = ---- x ----- or --- = ---- x -----
p + 1 p pq p + 1 pq
was known. Secondarily, by line 17, for an unknown reason the EMLR
included 1/13 = 1/7(3/7) = 3/49 rather a value such as 1/13 = 1/7(7/13)
leading to a valued for 7/13 times 1/7. One 7/13 statement, that the
EMLR student seemed not to grasp is noted by the Coptic general rule:
n/pq - 1/a = (na -pq)/apq, or selecting a = 2 and q = 1 allows
7/13 - 1/2 = (7*2 - 13)/(2*13) or
7/13 = 1/2 + 1/26 which shows that 1/7(1/2 + 1/26) = 1/14 + 1/182
probably would have pleased the EMLR instructor.
(2) The final RMP 2/nth table line, 2/101, contains an EMLR type
algorithm, as easily identified by:
2/101 = 1/p(1/1 + 1/2 + 1/3 + 1/6).
These proposals should be considered tentative historical statements.
Additional research during Hellenic and Classical Greek periods will be
required to confirm and or reject particular statements. In the interim a
tentative pedagogy is offered to to mathematical education community, one
that appears to connect 2, 500 years of history, from 2,000 BC to 500 AD,
in a manner that students of Euclid or other Hellene applied mathemetics
may enjoy considering, for fun, or as a serious project.
References:
Boyer, C.B., 1968, History of Mathematics, John Wiley, 1985 re-print
Princeton University Press.
Bruckheimer, M, and Salomon Y., The RMP Unit Fraction System,
Historia Mathematica, Nov. 1977.
Chace, A. B., 1927, Rhind Mathematical Papyrus, National Council of
the Teachers of Mathematics, 1979 reprint.
Gillings, Richard J., 1972, Mathematics in the Time of the Pharaoh's,
Dover Publications, 1982 re-print.
Klee, Victor and Wagon, Stan, 1991, Old and new Unsolved Problems
in Plane Geometry and Number Theory, Mathematical Association of
America, Dolciani Mathematical Expositions-No. 11.
Knorr, Wilbur, Historia Mathematica, HM 9, "Fractions in Ancient Egypt
and Greece, 1982.
Neugebauer, Otto, 1962, Exact Sciences of Antiquity, Harper and Rowe.
Ore, Oystein, 1948, Number Theory and its History, McGraw Hill
(Dover reprint is available).
Robins, Gay and Shute, Charles, The Rhind Mathematical Papyrus, Dover
Publications (a reprint of a 1987 British Museum publication).
Author:
Milo Gardner
Cryptanalyst
7255 Sumter Drive
Fair Oaks, CA 95628
April 2, 1996
(916) 967-8977
(916) 388-1000 (fax)
Qualifications
==============
As background information concerning my cryptanalysis qualifications to
discuss the mathematical side of Egyptian fractions it may be important to
note that I worked for several years as a Cryptanalytical Specialist.
My daily and ongoing assignments were twofold:
1. To sort through encoded electronic messages and place practice
traffic in one pile and language text traffic in another.
2. To attempt to predict the next day's practice traffic, which
my partner(s) and I achieved from time to time, and secondarily
process ciphered messages using the latest cryptanalysis
techniques, frequently obtaining readable messages.
References:
Friedman, William F., Military Cryptanalyis, Parts I, II, III and IV
reprints available from Aegean Park Press, PO Box 2837, Laguna Hills, CA
92654, (800) 736-3587, USA and Canada, FAX (714) 586-8269
Friedman, William F. and Callimahos, Lambros, Military Cryptanalytics,
Part I, Vol. 1, 2 and Part II, Vol. 1, 2. Aegean Park Press.
Friedman, William F., Elementary Military Cryptography, Aegean Park Press.
Kahn, David, Codebreakers, a none technical book that went through
several editions, the last one in 1974. Kahn discusses Friedman
and modern cryptanalysis as well as Roman and Egyptian ciphers.
----------------
From: S. Thomas ()
Subject: Re: evading evidence
Sent On: 05/23 11:36 PM PM ET
Date: Thu, 23 May 1996 23:33:42 -0400
From: S. Thomas [sthomas@erols.com]
Sender: owner-athena-discuss@info.harpercollins.com
[owner-athena-discuss@info.harpercollins.com]
Subject: Re: evading evidence
paul manansala wrote:
(( cuts ))
>
> However, I'm firmly convinced that it is mostly intuitive and
> mathematical thinking that leads to discovery in these sciences,
> and that axioms and analysis usually come later. So, to me its not
> much of a big deal whether they used axioms or not.
I quite agree. As it happens, I have been to Egypt, also to
Greece (India and China also but that's beside the point right
now). I remember having the same reaction that Ben-Jochanan
has spoken of. "But these Egyptians were black people!" I
had absorbed a Western education which told me first came the
Egyptians, who btw were white, then came the Greeks. The Egyptians
developed empirical methods of geometric mensuration because
the Nile flooded every year, and that served as a spur for
the methods developed. But it was the Greeks who took it to
a higher level, conceptually and theoretically. Well, the
first lie that was exploded was that the Egyptians were white.
By the overwhelming evidence of statuary and paintings that
I could see with my own eyes, it was clear that they were black
people, certainly what would count as black in the United
States or the Caribbean. The second was the expectation
that the glory of Greece would somehow surpass Egypt. Not so
at all. Not even by a long shot. You have to stand within
the temple, say, at Karnak, then go to the Acropolis, to very
quickly realize that the latter is first of all a copy, and
*much* less impressive in scale. And if you walk around the base
of the Great Pyramid at Giza, and contemplate the sheer
vastness of that structure, you quickly realize that this, and the
other pyramids, were built by men who knew what they were doing.
It was a matter of plan and execution -- calculation -- rather than
of general idea followed by a lot of empirical muddling through.
To see it is to be convinced that these master builders knew their
geometry, trigonometry, and statics. Geometry is to the pyramids
as climbing Mt. Everest is to building a Hilton Hotel atop it.
The first, impressive though it is, is as nothing compared to
the latter; and the latter may be taken as proof that you had
truly mastered the former. Nothing I saw in Greece came even close to
matching the Egyptian accomplishment. Which is why I have very
little difficulty crediting the Egyptians by inference from
indirect evidence. And I too do not feel the need for the
"smoking gun" direct evidence that would remove all doubt.
I do suspect, however, that the disparagers of a black ancient
Egypt would do the dance of distortion and denial even if there were
direct evidence.
> Paul Kekai Manansala
Regards,
S. F. Thomas
----------------
Date: Wed, 05 Jun 1996 22:02:31 -0400
From: "S. Thomas"
X-Mailer: Mozilla 2.0 (X11; I; Linux 1.2.11 i486)
MIME-Version: 1.0
To: Athena Discuss
Subject: Re: questions of origin
Scott A. Simmons wrote:
> 2) I am also still bothered by the neglect which the appeal to determine
> a sense for 'influence' has received. I think Meadows is justified in asking
> along these lines. If, indeed, we are to take 'influence' as code for 'cause',
> then we are entering a notoriously difficult area in predicate logic which
> calls for a higher-order and modal approach. Perhaps S. F. Thomas could
> assist us in negotiating these waters?
For me, the question of "influence" and what it means in this context
resolves fairly easily, and in a very common-sense way. I don't think
there is any cause to agonize over it. I had the experience of visiting
Egypt and Greece both, on the same trip. Mere days after having stood
surrounded by the massive stone columns at Karnak temple, I stood
surrounded by those at the Acropolis in Greece. If you would do the same,
I think any question of "influence" would rather quickly resolve itself.
The Greeks copied from the Egyptians. Independent reinvention is simply
not credible as a hypothesis.
And, as in this one particular, so also more broadly
over a wide array of fronts. Diop cited a range of evidence to suggest
that Greek plagiarism was a widespread practice, and theorems that today
are attributed to Archimedes, Pythagoras, Thales, and other Greeks were
known by the Egyptians hundreds of years before. Moreover, there is
evidence that these individuals went to Egypt to study with the priests.
Is it mere coincidence? Hardly, I should think. So, I don't
think an exegesis on "causality" or some other notoriously
hard philosophical nut is necessary to come to grips with what is at issue in the
present debate. Any jury of "12 men good and true"--women also--would have no
difficulty understanding the indictment (plagiarism, ie. unacknowledged
intellectual borrowing), or the evidence adduced. If the level of
proof required is "beyond a reasonable doubt", it is possible that one
or other Greek may have to be acquitted. But if the level of proof
required is "preponderance of the evidence"--which of two competing
hypotheses appears more likely given the evidence--then the finding of
any jury good and true has to be with the complainant in the case. For
me, Karnak and the Acropolis say it eloquently: at least in that
particular, not only did the Greeks copy from the Egyptians, the copy
was not as good.
More broadly, I still reflect on the Egyptian seven-point
division of human abilities thus (from an earlier post):
1 - Ba, the ability to experience omnipresence
based on the existence of the universal spirit
2 - Khu, ability to intuit the truth of a
logical premise, the oracular faculty of
prophets
3 - Shekem, ability to affect nature through
the use of spiritual power
4 - Ab,
+ the ability to see the interdependence
between all things, to love
+ the ability to analyze, to see the
abstract difference between things
+ the ability to think circumspectly, ie.
to coordinate the activity of all the
faculties of the spirit, to reason.
5 - Sahu,
+ imagination and congregative thinking--
aesthetics
__________________________________________
| + syllogistic, logical and segregative |
| thinking |
------------------------------------------
+ memory and imitative faculty, learning
6 - Khaibit, the animal soul, emotions, sense
perception, the sensual, physical movement
7 - Khab, the physical body which gives us the
illusion that we are separate beings
which the Egyptians were able to intuit, and the complete inadequacy
of modern Western civilization as to the first three abilities, along
with a fetishistic--with nod here to Mr. Kaufman--obsession with the
the kind of syllogistic mode of thinking that the Egyptians put
relatively low on the scale of human abilities.
It suggests to me at least the possibility that the Greeks learnt well
their lessons up to and including that having to do with the
Ab. They copied, but perhaps as with Karnak/Acropolis, the
copy was not as good as the original, and we today are inheritors
of what the Greeks, alas, only imperfectly copied from the Egyptians.
> Scott A. Simmons
Regards,
S. F. Thomas
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