L Cooper Wrote:
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> MJ wrote:
>
> "However, I am labouring under the impression that
> yourself and others are arguing that not only did
> the 4th Dyn AEs not know the Pythagoras Theorem,
> but also that even if they did know it they would
> not have incorporated it in the design of a king’s
> burial chamber."
>
> It does not necessarily follow that a knowledge of
> the 3-4-5 Pythagorean Triple, or any other Triple
> for that matter, means that the Egyptians knew of
> the Pythagorean Theorem, or that they knew that
> such a theorem might even exist.
>
> The 3-4-5 Triple, and many other Pythagorean
> Triples and "near" Triples, can be easily
> discovered empirically. All that is needed is the
> ability to create an accurate right-angle, and the
> ability to accurately subdivide a linear
> measurement tool. The Old Kingdom Egyptians had
> both of these capabilities. They were adept at
> field surveying and in the accurate laying out,
> and construction of, sophisticated structures.
>
> By entertaining the possibility that they were
> able to empirically discover many "true" Triples,
> and many more "near" or "apparent" Triples, it
> then becomes possible to arrive at explanations
> for many other design choices that one sees in
> Egyptian constructs. (I have been working on a
> draft of an initial paper on this subject dealing
> with OK pyramid design.)
>
> Two examples of apparent Triples possibly known
> about at the time are the 49-50-70 Triple, and the
> 70-70-99 Triple. If one accepts both Petrie's
> finding for the slope of the core of two sides of
> the Red Pyramid, and also the current estimate of
> 45° for the slope of the casing of this pyramid,
> then one can see that it is not outside the realm
> of possibility there was an intent to incorporate
> both of these Triple relationships in the Red
> Pyramid's exterior design.
>
> In this context, I also see it as being entirely
> possible for the remen and double-remen
> relationship to have been simply the practical
> application of an awareness of the above two
> apparent Triples. A square remen (i.e., 5 palms)
> was seen as having a diagonal of almost exactly 7
> palms (a royal cubit). A square royal cubit (7
> palms) was seen as having a diagonal of a
> double-remen (10 palms - with the 9.9 being fudged
> to 10. I suspect that in measuring large
> distances, or in situations where precision was
> desired, the more accurate 70-70-99 relationship
> would have been re-instated.)
There seems to be some misunderstanding whenever the remen is mentioned.
Using the knowledge available to Petrie a square of 1 royal cubit has divisions of .736 while the diagonal, if divided by 40, has divisions of .729. The contradiction becomes apparent when we realise that any cubit on the diagonal will always be about 5mm shorter than the cubit on the square.
I'm surprised that someone like Petrie even mentioned it.
Petrie's royal cubit of 20.620 ± .005 / 28 has divisions of .7362... - .7366...
The Diagonal of 29.1540... - 29.1681... / 40 has divisions of .7288... - .7292...
28 division royal cubit on the square 28 x .7362... - .7366...= 20.620 ± .005
28 division royal cubit of the diagonal 28 x .7288... - .7292...= 20.413 ± .005
No amount of juggling will bring the two together.
Put another way the diagonal accounts for 1 cubit + roughly a little more than 8.5 inches (11.5 digits), iow always half a digit short. of 40.
This would have been so basic to the egyptian that the diagonal would never have been divisioned into anything. The diagonal however would have been a great tool for halving and doubling land areas, as Gillings noted.
>
> All of this is, of course, theoretical. It may
> have been the way that they thought about things,
> and it may not. As there is much of value that can
> potentially be gained, I believe it certainly to
> be a line of inquiry worth pursuing.
>
> Lee Cooper