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.
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:
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.
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
Edited 1 time(s). Last edit at 06/08/2018 05:51PM by Mark Heaton.