From: bosak@netcom.com (Jon Bosak)
Subject: Re: The Great Pyramid of Giza
Organization: Netcom, a Commercial Internet Service in San Jose
Date: Sat, 21 Nov 1992 09:12:02 GMT


Coming in late to this discussion -- don't want to get heavily
involved in the main thread, but would like to straighten out a couple
of the minor points --

Zerxes M. Bhagalia:

ZB> The quantity pi, by the way, had supposedly not been discovered yet

It's not clear who was doing the supposing here, but the proposition
that the Egyptians and Sumerians had working values for pi has been
generally accepted for a long time.

ZB> Sacred Cubit

As several people have pointed out, the "Sacred Cubit" of 25 inches is
a modern invention.  Belief in this nonsense was once quite
widespread, its supporters including at least one president of the
U.S. (Grover Cleveland, if I remember correctly).  For the record, the
mean value of the royal cubit actually used in the pyramids and
elsewhere was 20.62 inches (Petrie).

Chris Heiny:

CH> a Sacred Stadia

Stadium, please.  Stadia is the plural.

CH> Pi, by the way, had been discovered.  The Egyptians, being tidy
CH> and abhorring fractions, generally treated it as 3, although they
CH> did have means of approximating it when absolutely necessary.

Quite correct, but I would like to rephrase this a little more
strongly to dispel the popular impression (fostered by a passage in
the Bible that I won't get into here) that 3 was THE ancient value for
pi, which is an oversimplification.  In fact, the ancients were less
interested in pi than in pi/4, the constant used in calculating the
volume of a cylinder (which is about all that pi is really good for in
basic practical terms).  The actual value of pi/4 is 0.785398...,
whereas a value of 3 for pi yields 3/4 or 0.75 for pi/4.  We know from
written evidence that when they were being careful the Egyptians used
64/81 (0.790123...) for pi/4 and the Sumerians used 3 1/8 for pi
(giving 25/32 or 0.78125 for pi/4), both of which are quite a bit
closer to the true value.  I suspect that the Sumerians also used 5/6
(0.833333...) as a rough value for pi/4, but I don't have enough
evidence to prove this yet.

John Baskette (in an otherwise excellent summary of the facts
regarding pyramid measurements):

JB> The pi ratio in the pyramid is derived from the ratio of the 
JB> pyramid baseline divided by the height.  The average baseline 
JB> is 9,068.8.  Divide this by the height (5776 +- 7 inches) and you
JB> get 1.5701.  This value times two is 3.1402.  A better approximation
JB> of pi is obtained using the angle of the slope of the faces of the 
JB> pyramid.  The angle for the north slope according to Petrie is 
JB> 51 deg. 50 min. 40 sec. +- 1 min. 5 sec.  The same ratios in a 
JB> pyramid with this angle yield a value of 3.1427+-0.002.  
JB> 
JB> Petrie believed this feature was accidental.  The Egyptians, he
JB> said, chose to build the pyramid using a sloping angle that
JB> was 11 at the base and 7 at the height.  11/7 is 1.5714.  Times
JB> that by two and you get 3.1429 which is just about exactly
JB> the angle of the sloping stones.

I don't get the impression that Petrie believed this relationship
based on pi was accidental at all.  In _Pyramids and Temples of Gizeh_
he says:

    For the whole form the pi proportion (height is the radius of a
    circle = circumference of Pyramid) has been very generally
    accepted of late years, and is a relation strongly confirmed by
    the presence of the numbers 7 and 22 in the number of cubits in
    height and base respectively; 7:22 being one of the best known
    approximations to pi.  With these numbers (or some slight
    fractional correction on the 22) the designer adopted 7 of a
    length of 20 double cubits for the height; and 22 of this length
    for the half-circuit.  The profile used for the work being thus
    14 rise on 11 base.

If you know some other place where Petrie explicitly recanted this
theory, I would like to hear of it.  As far as I know, he found the
explanation based on pi to be among the most likely ones.  It is true
that the value of 22/7 for pi (or 11/14 for pi/4) is traditionally
attributed to Archimedes, but it seems highly unlikely that this value
had not been discovered previously by the Egyptians, from whom the
Greeks directly inherited several standards of weight and length.

Loren I. Petrich:

LP> And if there ever is a Great Pyramid FAQ, then it should discuss
LP> some of the fallacies of Pyramidology, like trying to obtain
LP> numbers of cosmic significance out of it. I'm sure that it has
LP> been satirized repeatedly by getting similar numbers out of other
LP> structures; any good ones?

Yes, Martin Gardner's _Fads and Fallacies_ has an amusing chapter on
the Pyramidologists that contains a devastating little satire based on
the measurements of the Washington Monument.  Gardner also recounts an
incident in which Flinders Petrie surprised a Pyramidologist in the
act of chipping away at a key measurement point inside the Great
Pyramid to make it better fit his Pyramid Inch theory.

+----------------------------------------------------------------------+
 Jon Bosak  Sunnyvale, California : bosak@netcom.com  jbosak@novell.com
+----------------------------------------------------------------------+