Hi Mark,

Thanks for your reply. I was really looking for the seked more in fractional form. I was really surprised and somewhat disappointed not a single individual on three different message boards could answer, what I thought to be a simple question.

Mark Heaton Wrote:
-------------------------------------------------------
> In my monograph on the Grand Gallery I have shown
> that that a slope built accurate to 1 part in a
> 1000 on both the rise and run must have an angle
> well within 3 arc minutes of the intended angle
> even if the rise was plus 1 part in 1000 and the
> run minus 1 part in 1000, or vice versa.
>
> This means that the slope of the Grand Gallery,
> which is close to 26 degrees 18 minutes, cannot
> have been intended to be a rise of 1 cubit for a
> run of 2 cubits which is nearly 26 degrees 34
> minutes unless the build error was so great as to
> make any theoretical interpretation a waste of
> time.

I agree, the descending passageway is not a 14 seked. I am working on it.

>
> We may view the angle as a cotangent to get closer
> to understanding the seked. The cotangent of a run
> of 2 cubits for a fixed rise of 1 cubit is 2,
> which converts to the seked of 14 palms (2
> cubits).
>
> The Grand Gallery has a rise of 39 cubits for a
> slope length of 88 cubits so it may be that the
> Grand Gallery was constructed using a template.
> This corresponds to a rise of 1 cubit for a run of
> 2.0227.. cubits or the seked of 14.159.. palms or
> a run of 56.636 digits.
>
> This decimal analysis needs several decimal places
> to show the difference from the symbolic slope:

I know it is a pain, but unit fractions will serve much better to understand the Ancient Egyptians use of seked.

>
> In round numbers the square of the cotangent to
> square the circle is 360/88, so the cotangent
> equates to sr360/sr28pi fot the pi approximation
> 22/7 which is the symbolic cotangent to square the
> circle exactly for the pi approximation 22/7 and
> equates to a cotangent of 2.0226 or a run of 2.026
> cubits or seked of 14.158 palms or a run of 56.632
> digits.

Sorry Mark not to skewer your work, but a seked of 14.158 palms seems innocuous enough in decimals, but in unit fractions it can become a nightmare for those Ancient Egyptians trying to understand it. A seked of 14.158 palms = 14 + 1/7 + 1/125 + 1/140 palms = 56 + 1/2 + 1/8 + 1/200 + 1/500 digits or 2 + 1/47 + 1/875 + 1/6580 cubits. To me this clearly demonstrates one of the pitfalls of using decimals in lieu of unit fractions when relating to Ancient Egyptian mathematics and what they intended.

In practical terms calculate how long a line would have to be to express your seked in terms of whole numbers. As demonstrated by problem #56 RMP by reversing the order of calculations the seked divided by 7 will always equal the run rise of the gradient in whole numbers. i.e.
5 1/25 / 7 equals a run of 18 to a rise of 25: G1 a seked 5 1/2 seked a run of 11 to a rise of 14: G2, a 5 1/4 / 7 = 3 run 4 rise: G3 a seked 5 1/2 + 1/10 = 5 3/5 / 7 = 4 run 5 rise. Once you have the run rise of a gradient in whole numbers it will work with any unit be it, cubit, palm or digit it is easier to communicate instructions to workers.

Every unit fraction, fraction or integer has a reciprocal. The reciprocal of 28 is 1/28.

>
> The architect did not need to calculate these
> numbers. The template had a run of 15 palms for a
> rise equal to sr55 in palms, with the latter
> constructed from a triangle with a hypotenuse of 8
> palms and a base of 3 palms.
>
> For the purpose of analysis this angle equates to
> 26 degrees 18 minutes 30 seconds.
>
> If you marked off twice the diameter of any circle
> on this slope then the vertical rise is equal to
> the side-length of an equal area square, exactly
> so for the pi approximation 22/7.
>
> Let's try a simple example to compare with the
> Rhind Mathematical Papyrus in which a circle with
> a diameter of 9 cubits is calculated as if it had
> an area equal to a square with a side length of 8
> cubits which is a decent approximation in
> calculating the volume of a cylindrical granary
> with a height of 10 cubits. Volume = 640 cubic
> cubits.
>
> Mark off twice diameter of 9 cubits on the slope,
> or 18 cubits, which is the exact profile length of
> the north end wall of the grand gallery, so you
> can imagine running a tape measure along the
> profile and marking off the total length on the
> floor of the gallery. The rise is 7.9776.. cubits
> for the template mentioned above, not that the
> architect need to know this number, as the method
> simply gave the side-length of the square by
> measurement which is just over 223 digits in
> theory, so AE took 224 digits (8 cubits) for a
> simple method of calculation.
>
> We can calculate as sr (pi x r x r) or sr (22/7 x
> 4.5 x 4.5) = 7.9776..
> Volume of granary = 636.43 cubic cubits (as
> 7.9776.. x 7.9776.. x 10)
>
> or using pi to 4 decimal places 7.9760.. x
> 7.9760.. x 10 = 636.17 cubic cubits
>
> I expect the architect could have estimated the
> volume of any granary at a glance.
>
>
> Mark

Time dedicated to this for me is limited, so I will study the balance of your post over the next few days.

Regards,
Jacob